Recent Developments in Algebra and Related Areas ALM 8, pp. 155–200

c Higher Education Press ° and International Press Beijing-Boston

Γ-Leading Homogeneous Algebras and Gr¨ obner Bases ∗ Huishi Li †

Abstract

L Let Γ be an ordered semigroup by a total ordering, and R = γ∈Γ Rγ a Γ-graded algebra over a field K. By extending [24, Proposition 2.1] to a quotient algebra A = R/I of R, we present a study of A via its Γ-leading homogeneous algebra AΓ LH = R/hLH(I)i, that leads to effective applications of Gr¨ obner bases to the structure theory of A.

0

Introduction

This work is a continuation of [22]. We start the introduction by recalling E.S. Golod’s definition of a standard basis. Let Γ be a ≺ - ordered semigroup whose elements satisfy the descending chain condition with respect to the ordering ≺, K a commutative ring, and let R be a K-algebra equipped with a Γ-filtration {Fγ R}γ∈Γ consisting of K-submodules S Fγ R such that R = γ∈Γ Fγ R and Fγ R · Fγ 0 R ⊆ Fγγ 0 R for all γ, γ 0 ∈ Γ. Then L R has the associated Γ-graded K-algebra GΓ (R) = γ∈Γ GΓ (R)γ with GΓ (R)γ = S Fγ R/Fγ∗ R, where Fγ∗ R = γ 0 ≺γ Fγ 0 R. Since for each f ∈ R there exists a unique γ ∈ Γ such that f ∈ Fγ R − Fγ∗ R, the residue class of f in GΓ (R)γ , denoted by σ(f ), is called the leading term of f . If I is a two-sided ideal of R equipped with the Γ-filtration F I induced by F R, then the associated Γ-graded ideal GΓ (I) of I in GΓ (R) is generated by the set of leading terms σ(I) = {σ(f ) | f ∈ I}. Definition. ([17], 1986) Let S be a set of generators of I in R, if GΓ (I) can be generated by σ(S) = {σ(f ) | f ∈ I}, then S is called a Γ-standard basis of I. ∗

Project supported by the National Natural Science Foundation of China (10571038). Department of Applied Mathematics, College of Information Science & Technology, Hainan University, Haikou 570228, China. E-mail: [email protected] †

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(For the reason of considering different types of filtration in this paper, here we modified Golod’s original definition of a standard basis as a Γ-standard basis for emphasizing the Γ-filtration). In terms of the first homology module of the Shafarevich complex (E.S. Golod and I.R. Shafarevich, On the class-field tower, Izv. Akad. Nauk. SSSR, Ser. Mat. 28, 1964 (in Russian)), respectively the Koszul complex in the commutative case, E.S. Golod gave a homological characterization of standard bases. As applications of this characterization, three celebrated results concerning algorithms were reproved in [17], namely the confluence or diamond lemma by G.M. Bergman (The diamond lemma for ring theory, Adv. Math. 29, 1978) and A.I. Shirshov (Selected Works, Nauka, Novosibirsk, 1984 (in Russian)), the Gr¨obner basis algorithm in commutative polynomial algebras by B. Buchberger (Gr¨obner Bases: An Algorithmic Method in Polynomial Ideal Theory, CAMP-Publ., No. 83-29.0, Nov. 1983), and the algorithm for the construction of Gr¨obner bases of ideals in enveloping algebras of Lie algebras by V.N. Latyshev (On the equality algorithm in Lie-nilpotent associative algebras, Vyisn. Kiyiv. Univ., Mat. Meth. 27, 1985 (in Ukrainian)). Considering a quotient algebra A = KhXi/I of the free K-algebra KhXi = KhX1 , . . . , Xn i, in [24] it was proved that with respect to the natural N-gradation of KhXi, GN (A) ∼ = KhXi/hLH(I)i as N-graded K-algebras [24, Proposition 2.2.1], where LH(I) is the set of N-leading homogeneous elements of I (see the definition in Section 1 below); and it was observed that if S is an N-standard basis of I with respect to the natural N-filtration of KhXi, then hLH(I)i = hLH(S)i [24, Proposition 2.2.4]. In order to obtain an N-standard basis for I, it was observed in the same paper that under using a graded monomial ordering ≺gr on B with respect to the natural N-gradation of KhXi, where B is the standard K-basis of KhXi consisting of all monomials (words in the alphabet X1 , . . . , Xn , including the empty word), any Gr¨obner basis of I with respect to the data (B, ≺gr ) is an N-standard basis of I. Moreover, the following result was obtained: Theorem. [24, Theorem 2.3.2] G is a Gr¨obner basis of I with respect to the data (B, ≺gr ) if and only if the set of N-leading homogeneous elements LH(G) of G is a Gr¨obner basis for the ideal hLH(I)i with respect to the data (B, ≺gr ). Thus, it turns out that if G is a Gr¨obner basis of the ideal I with respect to the data (B, ≺gr ), then ∼ KhXi/hLH(G)i, (i) G yields an N-graded K-algebra isomorphism GN (A) = which, indeed, not only amounts to be a version of the classical PBW theorem (Poincar´e-Birkhoff-Witt theorem concerning enveloping algebras of Lie algebras) for A = KhXi/I, but also provides an algorithmic way for finding a PBW-deformation of a given N-graded K-algebra R = KhXi/hHi with H consisting of N-

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homogeneous elements with respect to the natural N-gradation of KhXi (see a remark given in the subsequent Section 3 for an interpretation); and ∼ KhXi[t]/hG ∗ i, where e= (ii) G yields an N-graded K-algebra isomorphism A L e = ep (with A ep = Fp A) is the Rees algebra of A with respect to the A A p∈N

natural N-filtration F A, and G ∗ is the homogenized Gr¨obner basis of G in the polynomial algebra KhXi[t] with respect to the commuting variable t, and, indeed, this provides an algorithmic way for finding a regular central extension of an Ngraded algebra R = KhXi/hHi with H consisting of N-homogeneous elements with respect to the natural N-gradation of KhXi (see the subsequent Section 8 for details). Motivated by the study of the N -type PBW property in connection with (homogeneous and non-homogeneous) Koszulity (see [10], [7] and [13]), the idea contained in the above results was made clearer in [22] to propose a more general PBW property for quotient algebras of a Z-graded algebra, and when quotient algebras of a path algebra (including free algebra) are considered with respect to the natural N-filtration defined by the length of paths, a solution to the general PBW problem was given by means of Gr¨obner bases in path algebras. L Let Γ be an ordered semigroup by a total ordering, and R = γ∈Γ Rγ a Γgraded algebra over a field K. By extending [24, Proposition 2.1] to a quotient algebra A = R/I of R, in this paper we present a study of A via its Γ-leading homogeneous algebra AΓLH = R/hLH(I)i, that covers the consideration of [22] and leads to effective applications of Gr¨obner bases to the structure theory of A. The contents of this paper are arranged as follows: 1. The Γ-graded isomorphism GΓ (A) ∼ = AΓLH ; 2. Γ-standard basis and Γ-PBW isomorphism; 3. Realizing Γ-PBW isomorphism by Gr¨obner basis; 4. Recognizing A via AΓLH ; Γ 5. A working chart from AB LH to G (A) and A;

6. Using AB obner bases; LH in terms of Gr¨ 7. Recognizing (non-)homogeneous N -Koszulity via AB LH ; 8. Gr¨obner basis and Rees algebra. Convention throughout this paper: Let K be a field and K ∗ = K − {0}. All algebras considered are associative Kalgebras with identity 1, and all modules, unless otherwise stated, are unitary left modules. Let R be a K-algebra. Whenever the phrase “an ideal of R” is used, it means a two-sided ideal of R. For S ⊂ R, we write hSi for the ideal of R generated by the subset S, and write hS] for the left ideal of R generated by S.

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The Γ-Graded Isomorphism GΓ (A) ∼ = AΓLH

L Let R = γ∈Γ Rγ be a Γ-graded K-algebra, where Γ is a totally ordered semigroup by a total ordering ≺. Then R has the Γ-grading filtration F R = {Fγ R}γ∈Γ with L S Fγ R = γ 0 ¹γ Rγ 0 , that is, F R satisfies R = γ∈Γ Fγ R and Fγ RFγ 0 R ⊆ Fγγ 0 R for all γ, γ 0 ∈ Γ. Let I be an arbitrary ideal of R and A = R/I the corresponding quotient algebra. Equipped with the Γ-filtration F A = {Fγ A}γ∈Γ induced by F R, that is, Fγ A = (Fγ R + I)/I, γ ∈ Γ, A has its associated Γ-graded algebra L GΓ (A) = γ∈Γ GΓ (A)γ with [ GΓ (A)γ = Fγ A/Fγ∗ A, where Fγ∗ A = Fγ 0 A. γ 0 ≺γ

In this section, we extend [24, Proposition 2.1] to the Γ-graded K-algebra isomorphism GΓ (A) ∼ = R/hLH(I)i (see the definition of LH(I) below). To begin with, note that each element f ∈ R can be written uniquely as a sum Ps of finitely many homogeneous elements, say f = i=1 rγi , rγi ∈ Rγi . Assuming γ1 Â γ2 Â · · · Â γs , we define the Γ-leading homogeneous element of f , denoted by LH(f ), to be rγ1 , that is, LH(f ) = rγ1 , and say that f is of degree γ1 , denoted by d(f ) = γ1 . Thus, for a subset S ⊂ R, we write LH(S) = {LH(f ) | f ∈ S} for the set of Γ-leading homogeneous elements of S. Let I be an arbitrary ideal of R. As LH(I) consists of homogeneous elements, the ideal hLH(I)i generated by LH(I) in R is Γ-graded, and consequently, the quotient algebra R/hLH(I)i is a Γ-graded K-algebra, namely M¡ ¢ R/hLH(I)i = Rγ + hLH(I)i /hLH(I)i. γ∈Γ

We call R/hLH(I)i the Γ-leading homogeneous algebra of the quotient algebra A = R/I and denote it by AΓLH , that is, AΓLH = R/hLH(I)i. 1.1. Theorem. With notation as fixed above, there is an isomorphism of Γgraded K-algebras GΓ (A) ∼ = AΓLH = R/hLH(I)i. Proof. First, note that Γ is ordered by the total ordering ≺ . By the definition of GΓ (A), for γ ∈ Γ, GΓ (A)γ = Fγ A/Fγ∗ A with Fγ A = (Fγ R + I)/I and, as a K-subspace, S [ [ Fγ 0 R + I Fγ∗ R + I γ 0 ≺γ Fγ 0 R + I ∗ 0 Fγ A = Fγ A = = = . I I I 0 0 γ ≺γ

γ ≺γ

It turns out that there are canonical isomorphisms of K-subspaces Rγ ⊕ Fγ∗ R ∼ Fγ R = −→ GΓ (A)γ , = ∗ ∗ (I ∩ Fγ R) + Fγ R (I ∩ Fγ R) + Fγ R

γ ∈ Γ,

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and consequently, we can extend the canonical epimorphisms of K-subspaces φγ : Rγ −→

Rγ ⊕ Fγ∗ R , (I ∩ Fγ R) + Fγ∗ R

γ ∈ Γ,

to define a Γ-graded K-algebra epimorphism φ : R −→ GΓ (A). We claim that Ker φ = hLH(I)i. To see this, noticing that hLH(I)i is a Γ-graded ideal, it is sufficient to prove the equalities Ker φγ = hLH(I)i ∩ Rγ ,

γ ∈ Γ.

Suppose rγ ∈ Ker φγ ⊂ Rγ . Then rγ ∈ (I ∩ Fγ R) + Fγ∗ R. If rγ 6= 0, then as a homogeneous element of degree γ, rγ = LH(f ) for some f ∈ I ∩ Fγ R. This shows that rγ ∈ hLH(I)i ∩ Rγ . Hence, Ker φγ ⊆ hLH(I)i ∩ Rγ . Conversely, suppose rγ ∈ hLH(I)i ∩ Rγ . Then as a homogeneous element of degree γ, rγ = Ps i=1 vi LH(fi )wi , where vi , wi are homogeneous elements of R and fi ∈ I. Write fi = LH(fi ) + fi0 such that d(fi0 ) ≺ d(fi ), i = 1, . . . , s. By the fact that Γ is an ordered semigroup with the total ordering ≺, we may see that the expression rγ =

s X

vi fi wi −

vi fi0 wi

i=1

i=1

Ps

s X

Ps

satisfies i=1 vi fi wi ∈ I ∩ Fγ R and i=1 vi fi0 wi ∈ Fγ∗ R. This shows that rγ ∈ (I ∩ Fγ R) + Fγ∗ R, i.e., rγ ∈ Ker φγ . Hence, hLH(I)i ∩ Rγ ⊆ Ker φγ . Summing up, we conclude the desired equalities Ker φγ = hLH(I)i ∩ Rγ for all γ ∈ Γ. ¤ Remark. Obviously, if I is a Γ-graded ideal of R, then A = R/I = GΓ (A) with respect to F A induced by the Γ-grading filtration F R of R. Let A = R/I be as before and L a left ideal of R such that I ⊂ L. If we equip M = R/L with the Γ-filtration F M = {Fγ M = (Fγ R + L)/L}γ∈Γ induced by F R, M becomes a Γ-filtered A-module with the associated Γ-graded G(A)-module L GΓ (M ) = γ∈Γ G(M )γ , where G(M )γ = Fγ M/Fγ∗ M and Fγ∗ M = (Fγ∗ R + L)/L for all γ ∈ Γ (the general definition of a Γ-filtered A-module is given in Section 4). Here we point out that the proof of Theorem 1.1 may be carried to deal with the A-module M = R/L directly so long as L is used in place of I and only left-hand side action is considered. We mention the result below but will not explore this module theory in detail in this paper. 1.2. Theorem. Let M = R/L be as fixed above. Then there is an isomorphism of Γ-graded GΓ (A)-modules: GΓ (M ) ∼ = R/hLH(L)], where hLH(L)] denotes the Γ-graded left ideal of R generated by LH(L).

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Γ-Standard Basis and Γ-PBW Isomorphism

L Let R = γ∈Γ Rγ be a Γ-graded K-algebra, where Γ is a totally ordered semigroup by a total ordering ≺, and let I be an arbitrary ideal of R. Consider the quotient algebra A = R/I and its Γ-filtration F A induced by the Γ-grading filtration F R of R, as defined in the last section. In this section, we establish an equivalence between the existence of a Γ-standard basis F of I and the existence of a Γ∼ = PBW isomorphism R/hLH(F )i −→ GΓ (A) with respect to the Γ-filtration F A (see Definitions 2.1 and 2.2 below). Notations are maintained as before. First, note that the Γ-grading filtration F R of R has two basic properties: (i) If f ∈ R with LH(f ) = rγ , then M f ∈ Fγ R − Fγ∗ R, where Fγ∗ R = Rγ 0 . γ 0 ≺γ

T

It turns out that γ∈Γ Fγ R = {0}, namely, the Γ-grading filtration F R is separated. (ii) GΓ (R) ∼ = R as Γ-graded algebras. Thus, for f ∈ R with LH(f ) = rγ 6= 0, if we denote the (nonzero) image of f in GΓ (R)γ = Fγ R/Fγ∗ R ∼ = Rγ by σ(f ), then σ(f ) = LH(f ) = rγ . Applying this fact to the ideal I of R considered, we have σ(I) = {σ(f ) | f ∈ I} = {LH(f ) | f ∈ I} = LH(I) and GΓ (I) = hσ(I)i = hLH(I)i. Consequently, we may adopt the same notion as given in [17] to specify a standard basis for I as follows. 2.1. Definition. Let F be a set of generators of the ideal I. With notation as above, if hLH(I)i = hLH(F )i, then we call F a Γ-standard basis of I. On the other hand, considering the quotient algebra A = R/I, from the proof of Theorem 1.1 we may see that if the ideal I is generated by the subset F , then since LH(F ) ⊆ LH(I) = Ker φ, there is the naturally induced commutative diagram of Γ-graded algebra homomorphisms 0 −→ hLH(F )i −→

R    φ y

−→ R/hLH(F )i .ρ

GΓ (A) in which Ker ρ = hLH(I)i/hLH(F )i. It turns out that hLH(I)i = hLH(F )i if and only if ρ is an isomorphism. Comparing with the classical PBW theorem,

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actually, having such an isomorphism amounts to have a version of PBW theorem for Γ-filtered algebras of the form A, that is, the following definition and theorem make sense. 2.2. Definition. The algebra A = R/I is said to have a Γ-PBW isomorphism if I has a set of generators F such that ρ

R/hLH(F )i −→ GΓ (A). ∼ =

2.3. Theorem. The algebra A = R/I has a Γ-PBW isomorphism if and only I has a Γ-standard basis. Remark. It is easy to see that Definition 2.2 covers the classical N-PBW isomorphism GN (U (g)) ∼ = K[x1 , . . . , xn ] for an n-dimensional K-Lie algebra g, the N -type PBW property discussed for N-filtered algebras in connection with (non-) homogeneous Koszulity (see [10], [7] and [13]), as well as the PBW property recognized for N-filtered and Z-filtered algebras by means of Gr¨obner bases (see [24] and [22]). Furthermore, we will see in the subsequent Section 3 and Section 8 that solving the Γ-PBW isomorphism problem may also lead to solution to the so-called PBW-deformation problem and the regular central extension problem. We will see in Section 3 that the next proposition, which is a modification of the equivalent characterization of a standard basis in [17], characterizes actually a Gr¨obner basis in a quite general setting. It is this result that enables us to gain insights into the possibility of realizing the PBW isomorphism of different types by means of Gr¨obner bases. 2.4. Proposition. Let A = R/I be as before and F a subset of the ideal I. The following two statements hold: (i) Suppose that F is a generating set of I having the property that each nonzero f ∈ I has a finite presentation f =

X

vij fj wij , where the vij , wij are homogeneous elements of R,

i,j

¡ ¢ fj ∈ F, vij fj wij 6= 0 and d LH(vij LH(fj )wij ) ¹ d(f ). Then hLH(I)i = hLH(F )i, i.e., F is a Γ-standard basis for I. (ii) If ≺ is a well-ordering on Γ and hLH(I)i = hLH(F )i, then F is a Γ-standard basis of I having the property mentioned in (i) above. Proof. (i) By the definition, if f ∈ R, f 6= 0 and LH(f ) ∈ Rγ , then d(f ) = d(LH(f )) = γ. Since Γ is an ordered semigroup by the total ordering ≺, by the assumption on the presentation of f , the Γ-leading homogeneous element LH(f )

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P 0 0 of f must have the form LH(f ) = vij LH(fj0 )wij with fj0 ∈ F , i.e., LH(f ) ∈ hLH(F )i. Hence, hLH(I)i = hLH(F )i. (ii) For f ∈ I, f 6= 0, suppose LH(f ) ∈ Rγ . By the assumption, the Γ-leading homogeneous element LH(f ) can be written as X LH(f ) = vij LH(fj )wij , i,j

in which vij , wij are homogeneous elements of R and fj ∈ F with vij LH(fj )wij 6= 0 and d(vij LH(fj )wij ) = d(LH(f )) = d(f ) = γ, j = 1, . . . , s. Thus, rewriting each fj as fj = LH(fj ) + fj0 such that d(fj0 ) ≺ d(f ), we have LH(f ) =

X

vij fj wij −

i,j

X

vij fj0 wij ,

i,j

¡P ¢ in which each vij fj wij 6= 0, d = d(LH(f )) = d(f ) = γ, and ij vij fj wij ¡P ¢ 0 d v f w ≺ γ. It turns out that the element ij ij j i,j f0 = f −

X

vij fj wij ∈ I

i,j

has d(f 0 ) ≺ d(f ) = γ. For f 0 , we may repeat the same procedure and get some element X f 00 = f 0 − v`j f` w`j ∈ I `,j 00

0

with d(f ) ≺ γ , where v`j , w`j are Γ-homogeneous elements of R, f` ∈ F , sat¡P ¢ 0 isfying each v`j f` w`j 6= 0 and d `,j v`j f` w`j = γ . Since ≺ is a well-ordering, after a finite number of repetitions, such reduction procedure of decreasing degrees P must stop to give us an expression f = i,j vij fj wij with the desired property. ¤ Corresponding to Theorem 1.2, similar statements of Proposition 2.4 hold for a left ideal. 2.5. Proposition. Let F be a subset of a left ideal L of R. The following two statements hold: (i) Suppose that F is a generating set of L having the property that each nonzero f ∈ L has a finite presentation X f = vij fj , where the vij are homogeneous elements of R, i,j ¡ ¢ fj ∈ F, vij fj 6= 0 and d LH(vij LH(fj )) ¹ d(f ).

Then hLH(L)] = hLH(F )]. (ii) In the case that ≺ is a well-ordering on Γ, if hLH(L)] = hLH(F )], then F is a generating set of L having the property mentioned in (i) above.

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Realizing Γ-PBW Isomorphism by Gr¨ obner Basis

By means of Gr¨obner bases, in this section, we realize the Γ-PBW isomorphism (as described in the last section) with respect to different filtered-graded structures. We begin with introducing the Gr¨obner basis theory for a class of algebras. Let R be a K-algebra. If R has a K-basis B satisfying ½ u · v = λw for some λ ∈ K ∗ , w ∈ B, (SMB) u, v ∈ B implies or u · v = 0, then we call B a skew multiplicative K-basis of R. The reason that we use the word “skew” in the definition above is that the class of algebras considered in this section contains not only ordered semigroup algebras, free algebras, commutative polynomial algebras, but also the skew polynomial algebra of n generators a1 , . . . , an subject to the relations aj ai − λji ai aj with λji ∈ K ∗ and 1 ≤ i < j ≤ n (if λji = q −2 for some q ∈ K ∗ , 1 ≤ i < j ≤ n, which is known the coordinate ring of the quantum affine n-space), path algebras defined by finite directed graphs, and exterior algebras, etc. It is known that such a class of algebras provides the most important structural basis in computational algebra. Let B be a skew multiplicative K-basis of the algebra R, and let ≺ be a total ordering on B. Adopting the commonly used terminology in computational algebra, we call an element u ∈ B a monomial; and for f ∈ R, say f=

s X

λi u i ,

λi ∈ K ∗ , ui ∈ B, u1 ≺ u2 ≺ · · · ≺ us ,

i=1

the leading monomial of f , denoted by LM(f ), is defined as LM(f ) = us ; the leading coefficient of f , denoted by LC(f ), is defined as LC(f ) = λs ; and the leading term of f , denoted by LT(f ), is defined as LT(f ) = LC(f )LM(f ) = λs us . Thus, for a subset S of R, the set of leading monomials of S is defined as LM(S) = {LM(f ) | f ∈ S}. Bearing the assumption (SMB) on B in mind, recall that a monomial ordering on R is a well-ordering ≺ on B satisfying the following conditions: (Mo1) If u ≺ v, then LM(uw) ≺ LM(vw) for w ∈ B such that uw 6= 0 and vw 6= 0. (Mo2) If u ≺ v, then LM(su) ≺ LM(sv) for s ∈ B such that su 6= 0 and sv 6= 0. (Mo3) If uw = λv, then v  u and v  w. Besides, if R has the identity element 1 and 1 ∈ B, then we require 1 ≺ u for all u ∈ B − {1}, and moreover, we insist that v, u, w 6= 1 in the axiom (Mo3). L Let R = γ∈Γ Rγ be a Γ-graded K-algebra, where Γ is an ordered semigroup by a total ordering < . If R has a skew multiplicative K-basis B consisting of

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homogeneous elements, and if ≺ is a well-ordering on B, we may define the ordering ≺gr for u, v ∈ B by the rule u ≺gr v ⇐⇒ d(u) < d(v) or d(u) = d(v) and u ≺ v, where d( ) is referred to the degree function on homogeneous elements of R. If ≺gr is a monomial ordering on B in the sense of satisfying the foregoing (Mo1)–(Mo3), then we call ≺gr a Γ-graded monomial ordering on R. A typical N-graded monomial ordering is the N-graded (reverse) lexicographic ordering on a free K-algebra KhX1 , . . . , Xn i, a commutative polynomial K-algebra K[x1 , . . . , xn ], and a path algebra KQ defined by a finite directed graph Q over K, where the N-gradation may be any weight N-gradation obtained by assigning a positive degree ni to each generator, for, in each case considered the “monomials” in the standard K-basis B are all N-homogeneous elements. If ≺ is a monomial ordering on R, then we call the pair (B, ≺) an admissible system of R. That is, the multiplication of R “preserves” the order of elements determined by ≺. If a K-algebra R has an admissible system (B, ≺), then by mimicking (see, e.g., [18]), R holds a Gr¨obner basis theory, that is, every ideal I of R has a (finite or infinite) Gr¨obner basis G in the sense that hLM(I)i = hLM(G)i. Below we use this Gr¨obner basis theory to reach the goal of this section. Realization of the B-PBW isomorphism Let R be a K-algebra with a skew multiplicative K-basis and (B, ≺) an admissible system of R. In this part we assume that R does not have divisors of zero (thus, path algebras and exterior algebras are excluded). L Note that R is B-graded by the foregoing (SMB), namely, R = u∈B Ru with Ru = Ku. In this case, we see that for f ∈ R the B-leading homogeneous element LH(f ) of f defined in Section 1 is the same as the leading term LT(f ) of f defined above, that is, LH(f ) = LT(f ) = LC(f )LM(f ), Thus, for an ideal I of R we have hLH(I)i = hLM(I)i. It turns out that the algebra A = R/I has its B-leading homogeneous algebra AB LH = R/hLH(I)i = R/hLM(I)i. Remark. The algebra R/hLM(I)i is usually called the associated monomial algebra of A in the literature (e.g., see [2], [15] and [16]). To avoid confusing notions in the general Γ-context of this paper, we still use AB LH to denote R/hLM(I)i and call it the B-leading homogeneous algebra of A, for, here we have Γ = B.

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Furthermore, since R has no divisors of zero, it has the B-grading filtration L F R = {Fu R}u∈B with Fu R = v¹u Rv , and hence the algebra A = R/I has the induced B-filtration F A. It follows that we may talk about the B-PBW isomorphism of A determined by a B-standard basis of I as described in Section 2. 3.1. Theorem. Let I be an ideal of R and A = R/I. With notation as fixed above, consider the B-filtration F A of A induced by F R and its associated Bgraded algebra GB (A). If G is a generating set of the ideal I, the following two statements are equivalent: (i) G is a Gr¨obner basis for I with respect to (B, ≺). (ii) G is a B-standard basis of I and hence G determines the B-PBW isomorphism ρ AB GB (A). LH = R/hLM(I)i = R/hLM(G)i −→ ∼ =

Proof. By the discussion on the relation between B-leading homogeneous elements and ≺-leading monomials made above, a Gr¨obner basis in R is obviously a Bstandard basis in R (see Definition 2.1). Also note that the proof of Theorem 1.1 works completely for Γ = B, the skew multiplicative K-basis of R. So the equivalence mentioned in the theorem is clear now. ¤ Realization of the Γ-PBW isomorphism L In this part we assume that R is a Γ-graded K-algebra, i.e., R = γ∈Γ Rγ , where Γ is an ordered semigroup by a total ordering <, and that (B, ≺gr ) is an admissible system of R in which B is a skew multiplicative K-basis of R consisting of Γ-homogeneous elements, and ≺gr is a Γ-graded monomial ordering on R as defined before. Since in this case R is also B-graded, we use the terminology “Γ-homogeneous element” with respect to the Γ-gradation of R to distinguish the “B-homogeneous element” with respect to the B-gradation of R. Moreover, noticing that in the last part the condition that R has no divisors of zero is only used to guarantee the existence of the B-grading filtration, now in the Γ-graded case it is clear that we may allow R to have divisors of zero (thus, for instance, path algebras and exterior algebras are included). Let I be an ideal of R and A = R/I. Considering the Γ-filtration F A of A induced by the Γ-grading filtration F R of R, our aim is to show that if G is a Gr¨obner basis of I with respect to (B, ≺gr ), then G is a Γ-standard basis of I and consequently G determines a Γ-PBW isomorphism of A in the sense of Definition 2.2. To see this, we first prove a useful result concerning the relation between a Gr¨obner basis G and its associated set of Γ-leading homogeneous elements LH(G) in R (see Section 1 for the definition). 3.2. Proposition. Let I be an ideal of R and LH(I) be the set of Γ-leading homogeneous elements of I. Put J = hLH(I)i. The following two statements hold:

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(i) LM(J) = LM(I). (ii) Let G be a generating set of I. Then G is a Gr¨obner basis of I with respect to (B, ≺gr ) if and only if LH(G), the set of Γ-leading homogeneous elements of G, is a Gr¨obner basis for the Γ-graded ideal J with respect to (B, ≺gr ). Proof. (i) First, note that ≺gr is a graded monomial ordering on B and every element of B is a Γ-homogeneous element. For f ∈ R, we have LM(f ) = LM(LH(f )),

(1)

and this turns out LM(I) = LM(LH(I)). Hence, LM(I) ⊂ LM(J). It remains to prove the inverse inclusion. Since J is a Γ-graded ideal of R, noticing the formula (1) above we only need to consider the B-leading monomials of Γhomogeneous elements. Let F ∈ J be a Γ-homogeneous element of degree γ. P Then F = i,j Gij LH(fi )Hij , where Gij , Hij are Γ-homogeneous elements of R and fi ∈ I such that d(Gij )d(fi )d(Hij ) = γ whenever Gij LH(fi )Hij 6= 0. Write fi = LH(fi ) + fi0 such that d(fi0 ) < d(fi ). Then X X Gij fi Hij = F + Gij fi0 Hij , i,j

in which d

¡P i,j

i,j

¢

Gij fi0 Hij < γ = d(F ). Hence, µX ¶ LM(F ) = LM Gij fi Hij ∈ LM(I). i,j

This shows that LM(J) ⊂ LM(I), and consequently, the desired equality follows. (ii) Note that the formula (1) given in the proof of (i) yields hLM(G)i = hLM(LH(G))i. By the equality obtained in (i), we have hLM(I)i = hLM(G)i if and only if hLM(J)i = hLM(LH(G))i. It follows that G is a Gr¨obner basis of I with respect to (B, ≺gr ) if and only if LH(G) is a Gr¨obner basis for the Γ-graded ideal J with respect to (B, ≺gr ), as desired. ¤ 3.3. Theorem. Let I and A = R/I be as fixed above, and let GΓ (A) be the associated Γ-graded algebra of A determined by the Γ-filtration F A. Then any Gr¨obner basis G of I with respect to (B, ≺gr ) is a Γ-standard basis of I, and thereby G determines the Γ-PBW isomorphism ρ

AΓLH = R/hLH(I)i = R/hLH(G)i −→ GΓ (A). ∼ =

Proof. First note that by Theorem 1.1, GΓ (A) ∼ = R/hLH(I)i = AΓLH . If G is a Gr¨obner basis of I, then by Proposition 3.2, LH(G), the set of Γ-leading homogeneous elements of G, is a Gr¨obner basis of the ideal hLH(I)i. It follows that

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167

hLH(I)i = hLH(G)i. Therefore, LH(G) is a Γ-standard basis by Definition 2.1. This proves the theorem. ¤ It is a good exercise to check directly that if G is a Gr¨obner basis of I, then G satisfies the condition of Proposition 2.4(i), and hence G is a Γ-standard basis for I. Realization of the N-PBW isomorphism The foregoing Theorem 3.3 applies immediately to quotient algebras of N-graded algebras, including those having positive weight N-gradation such as free algebras, commutative polynomial algebras, the skew polynomial algebra of n generators a1 , . . . , an subject to the relations aj ai − λji ai aj with λji ∈ K ∗ and 1 ≤ i < j ≤ n (a typical case is the coordinate ring of a quantum affine K-space), path algebras and exterior algebras, etc. In particular, the classical N-PBW isomorphism on enveloping algebras of Lie algebras and the N -type PBW property (see [10], [7] and [13]) may be realized by means of Gr¨obner bases in free algebras. L 3.4. Theorem. Let R = p∈N Rp be an N-graded K-algebra, I an ideal of R and A = R/I. If (B, ≺gr ) is an admissible system of R, where B is a skew multiplicative K-basis consisting of N-homogeneous elements of R and ≺gr is an N-graded monomial ordering on B, then any Gr¨obner basis G of I with respect to (B, ≺gr ) is an N-standard basis of I, and thereby G determines the N-PBW isomorphism ρ

AN GN (A), LH = R/hLH(I)i = R/hLH(G)i −→ ∼ =

where GN (A) is the associated N-graded algebra of A with respect to the Nfiltration F A induced by the N-grading filtration F R of R. Remark. The foregoing results 3.1–3.4 generalize [24, Proposition 2.4 and Theorem 3.2] or [21, Chapter III, Theorem 3.7] to a quite extensive context. A solution to the PBW-deformation problem Let the free K-algebra KhXi = KhX1 , . . . , Xn i of n generators be equipped with the natural N-gradation (i.e., each Xi has degree 1), G = {g1 , . . . , gs } a finite set of N-homogeneous elements of KhXi, and R = KhXi/hGi the N-graded quotient algebra. Recall from the literature (e.g., see [11] and [13]) that if f1 , . . . , fs ∈ KhXi satisfy d(fi ) < d(gi ), 1 ≤ i ≤ s, and I = hGi with G = {gi + fi | 1 ≤ i ≤ s}, then the quotient algebra A = KhXi/I is called a deformation of R. Furthermore, considering GN (A), the associated N-graded algebra of A with respect to the natural N-filtration F A induced by the natural N-grading filtration F KhXi, if there is an N-graded K-algebra isomorphism GN (A) ∼ = R, then A is said to be a PBWdeformation of R. Taking the N-leading homogeneous elements into account, it is

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clear that G = LH(G) ⊂ LH(I), which yields the following commutative diagram of N-graded algebra homomorphisms (as described in Section 2): 0 −→ hLH(G)i −→ KhXi  −→ KhXi/hLH(G)i   φ .ρ y GN (A) So by Theorem 3.4, the next result is clear now. 3.5. Theorem. With notation as above, if G is a Gr¨obner basis with respect to some admissible system (B, ≺gr ) of KhXi, where ≺gr is an N-graded monomial ordering with respect to the natural N-gradation of KhXi, then the algebra A = KhXi/I is a PBW-deformation of the N-graded algebra R = KhXi/hGi. In the last Section 8, we will see that the study of the deformation A of R is closely related to the study of the so-called regular central extension D of R, and e of A in case G is a Gr¨obner that D may be realized as the usual Rees algebra A basis.

4

Recognizing A via AΓLH

L Let us return to a general Γ-graded K-algebra R = γ∈Γ Rγ , where Γ is a totally ordered semigroup, and let I be an arbitrary ideal of R, A = R/I. With notation as before, if the Γ-filtration F A of A induced by the Γ-grading filtration F R of R is taken into account, then as in the classical N-filtered case, we are naturally concerned with those structural properties that may be lifted from GΓ (A) to A, where GΓ (A) is the associated Γ-graded algebra of A with respect to F A. At this point the good news is that Theorem 1.1 allows us to replace the role of GΓ (A) by the Γ-leading homogeneous algebra AΓLH = R/hLH(I)i of A, namely, the lifting procedure from GΓ (A) to A may be realized more effectively in terms of the structure of AΓLH : A = R/I ¡ ∧ µ ¡ ¡ lifting

AΓLH

¡ ¡ ¡ ∼ = = R/hLH(I)i > GΓ (A)

In this section, we will try to list as many as possible structural properties which can be transferred from AΓLH to A via the diagram above, but will not give detailed proof of them, for, due to the isomorphism AΓLH ∼ = GΓ (A), it is only a matter of verifying the corresponding lifting property from GΓ (A) to A as in

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169

the classical N-filtered case (the reader may refer to a pre-version of this paper arXiv:math.RA/0609583 for the tedious verification of each lifting property from GΓ (A) to A). Nevertheless, necessary notions and preparatory results concerning Γ-filtered structures are given in order to make the text readable. Convention and assumption throughout this section are made as follows. Γ denotes an ordered monoid by the well-ordering ≺. If γ0 is the identity element of Γ, we assume that γ0 is the smallest element in Γ. Fixing the Γ-graded K-algebra L R = γ∈Γ Rγ , the quotient algebra A = R/I determined by an arbitrary ideal I of R, the Γ-filtration F A of A induced by the Γ-grading filtration F R of R, and the associated Γ-graded algebra GΓ (A) of A, we always assume that 1 ∈ Fγ0 R (and hence 1 ∈ Fγ0 A = GΓ (A)γ0 ). Besides, all notations used in previous sections are maintained. We start with an easy lemma that may help to verify our results via Theorem 1.1. 4.1. Lemma. For a ∈ Fγ A − Fγ∗ A, writing σ(a) for the (nonzero) image of a in GΓ (A)γ = Fγ A/Fγ∗ A, the following properties hold: (i) If a, b ∈ A, then either σ(a)σ(b) = 0 or σ(a)σ(b) = σ(ab). (ii) For f ∈ R, σ(f ) = σ(LH(f )), where f , respectively LH(f ), denotes the image of f , respectively the image of LH(f ), in A. Our first result deals with K-basis, divisors of zeros, (semi-)primeness, and maximal orders (in the sense of [25, Chapter 5]). 4.2. Theorem. Let AΓLH = R/hLH(I)i be the Γ-leading homogeneous algebra of the algebra A = R/I. The following statements hold: (i) Suppose that {fi }i∈J is a subset of R such that the image of {LH(fi )}i∈J in AΓLH forms a K-basis for AΓLH , then the image of {fi }i∈J in A forms a K-basis for A. Hence, if AΓLH is finite dimensional over K, then so is A. (ii) If AΓLH is a domain, then so is A. (iii) If AΓLH is a (semi-)prime ring, then so is A. (iv) If AΓLH is a Noetherian domain and maximal order in its quotient ring, then so is A (the Noetherianity will follow from latter Theorem 4.7). Next, we focus on modules. Let the algebra A = R/I and its Γ-filtration F A be as fixed, and let M be a Γ-filtered A-module with Γ-filtration F M = {Fγ M }γ∈Γ , that is, each Fγ M is a K-subspace of M , Fγ 0 M ⊆ Fγ M whenever γ 0 ¹ γ, and S Fγ RFγ 0 M ⊆ Fγγ 0 M for all γ, γ 0 ∈ Γ. By putting Fγ∗ M = γ 0 ≺γ Fγ 0 M for γ ∈ Γ, the associated Γ-graded GΓ (A)-module of M is defined as GΓ (M ) =

M γ∈Γ

GΓ (M )γ with GΓ (M )γ = Fγ M/Fγ∗ M.

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Since ≺ is a well-ordering on Γ, for each element m ∈ M we define the degree of m, denoted by d(m), as d(m) = min{γ ∈ Γ | m ∈ Fγ M }. If m 6= 0 and d(m) = γ, then we write σ(m) for the corresponding nonzero homogeneous element of degree γ in GΓ (M )γ . Similar to Lamme 4.1, we record a useful property of the σ-elements as follows. 4.3. Lemma. For all a ∈ A and m ∈ M , either σ(a)σ(m) = 0 or σ(a)σ(m) = σ(am). Noticing Lemma 4.3 and the well-ordering property of ≺ on Γ, the following 4.4–4.6 are obtained. 4.4. Proposition. Let M be an A-module. (i) If F M = {Fγ M }γ∈Γ is a Γ-filtration of M , {ξi }i∈J ⊂ M with d(ξi ) = γi ∈ Γ P P such that GΓ (M ) = i∈J GΓ (A)σ(ξi ), then M = i∈J Aξi with Fγ M =

Xµ i∈J

X

¶ Fsi A ξi ,

γ ∈ Γ.

s¹γ, si γi =s

In particular, if GΓ (M ) is a finitely generated GΓ (A)-module, then M is a finitely generated A-module. (ii) If M is a finitely generated A-module, then M has a Γ-filtration F M = {Fγ M }γ∈Γ such that GΓ (M ) is a finitely generated GΓ (A)-module. Indeed, if Pn M = i=1 Aξi and {ξ1 , . . . , ξn } is a minimal set of generators for M , then the desired Γ-filtration F M consists of ¶ n µ X X Fγ M = Fsi A ξi , γ ∈ Γ, i=1

s¹γ, si γi =s

where γ1 , . . . , γn ∈ Γ are chosen arbitrarily. Let M and N be Γ-filtered A-modules with Γ-filtration F M and F N , respectively. An A-homomorphism ϕ : M → N is said to be a Γ-filtered A-homomorphism if ϕ(Fγ M ) ⊆ Fγ N for all γ ∈ Γ; ϕ is said to be strict if it satisfies ϕ(Fγ M ) = ϕ(M ) ∩ Fγ N,

γ ∈ Γ.

For instance, if M is a Γ-filtered A-module with Γ-filtration F M and H is a submodule of M , then with respect to the induced filtration F H and F (M/H), the canonical A-homomorphisms H ,→ M and M → M/H are strict Γ-filtered A-homomorphisms.

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171

Obviously, a Γ-filtered A-homomorphism ϕ induces a Γ-graded GΓ (A)-homo¡ ¢ morphism GΓ (ϕ) : GΓ (M ) → GΓ (N ) with GΓ (ϕ) GΓ (M )γ ⊆ GΓ (N )Γ , γ ∈ Γ. 4.5. Proposition. Let ϕ

(∗)

ψ

L −→ M −→ N

be a sequence of Γ-filtered A-modules and Γ-filtered A-homomorphisms satisfying ψ ◦ ϕ = 0. Then the following properties are equivalent: (i) The sequence (∗) is exact and ϕ and ψ are strict. (ii) The associated sequence of Γ-graded GΓ (A)-modules and Γ-graded GΓ (A)homomorphisms GΓ (∗)

GΓ (ϕ)

GΓ (ψ)

GΓ (L) −→ GΓ (M ) −→ GΓ (N )

is exact. 4.6. Corollary. The following statements hold: (i) Let ϕ : M → N be a Γ-filtered A-homomorphism. Then GΓ (ϕ) is injective, respectively surjective, if and only if ϕ is injective, respectively surjective, and ϕ is strict. (ii) Let N and W be submodules of a Γ-filtered A-module M with Γ-filtration F M = {Fγ M }γ∈Γ . Consider the Γ-filtration F N = {Fγ N = N ∩ Fγ M }γ∈Γ of N and the Γ-filtration F W = {Fγ W = W ∩ Fγ M }γ∈Γ induced by F M , respectively. If N ⊆ W , then GΓ (N ) ⊆ GΓ (W ); and if GΓ (N ) = GΓ (W ), then N = W . We summarize some immediate applications of previous results into a theorem. 4.7. Theorem. Let A = R/I and AΓLH = R/hLH(I)i be as in Theorem 4.2. The following statements hold: (i) Suppose that AΓLH is Γ-graded left Noetherian, that is, every Γ-graded left ideal of G(A) is finitely generated. Then A is left Noetherian. (ii) Suppose that AΓLH is Γ-graded left Artinian, that is, AΓLH satisfies the descending chain condition for Γ-graded left ideals. Then A is left Artinian. (iii) If AΓLH is a Γ-graded simple K-algebra, that is, AΓLH does not have nontrivial Γ-graded ideal, then A is a simple K-algebra. (iv) If the Krull dimension (in the sense of Gabriel and Rentschler) of AΓLH is well-defined, then the Krull dimension of A is defined and K.dim A ≤ K.dim AΓLH . (v) Let L be a left ideal of R containing I. If hLH(L)]/hLH(I)i is a Γ-graded maximal left ideal of AΓLH , then L/I is a maximal left ideal of A, that is, R/L is a simple A-module. (vi) If AΓLH is semi-simple (simple) Artinian, then A is semi-simple (simple) Artinian.

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Remark. Let A be a finitely generated K-algebra and let ϕ : K[X] → A be an epimorphism, where X is a free semigroup with free generators x1 , . . . , xn . If f ∈ K[X] and f denotes the dominant (i.e., maximal) word in the support of f with respect to the lexicographic ordering on X, then I = {g ∈ X | g = f for some f ∈ K[X] with ϕ(f ) = 0} is an ideal of the semigroup X. In [28] Jan Okninski proved that the Noetherian property, Artinian property, and the (semi-)primeness of the quotient semigroup X/I are equivalent to the same properties of the semigroup algebra K[X/I] and thereby may be lifted to A. With notation used in this paper, let us note that the algebra K[X/I] is nothing but K[X]/hLM(Ker ϕ)i. So our results transferring Noetherian property, Artinian property, and the (semi-)primeness are just extensions of Okninski’s results to general Γ-filtered algebras. Finally, we discuss the transfer of homological properties. To this end, we need some basics on Γ-graded free modules and Γ-graded projective modules. L Let R = γ∈Γ Rγ be as before. A Γ-graded R-module T is called a Γ-graded free R-module if it is a free R-module on a homogeneous R-basis {ei }i∈J , that is, L L T = i∈J Rei , and if deg(ei ) = γi , i ∈ J, then T = γ∈Γ Tγ with Tγ =

X

Rwi ei ,

γ ∈ Γ.

i∈J, wi γi =γ

L By the definition, to construct a Γ-graded free R-module T = i∈J Rei with the R-basis {ei }i∈J , it is sufficient to assign to each ei a chosen degree. L Given any Γ-graded R-module M = γ∈Γ Mγ , M has a generating set {mi }i∈J P consisting of homogeneous elements, i.e., M = i∈J Rmi . Suppose d(mi ) = γi , γi ∈ Γ, i ∈ J. Then it is easy to see that X Mγ = Rwi mi , γ ∈ Γ. i∈J, wi γi =γ

L L Thus, considering the Γ-graded free R-module T = i∈J Rei = γ∈Γ Tγ with d(ei ) = γi , the map ϕ : ei 7→ mi defines a Γ-graded R-epimorphism ϕ : T → M . Let T be a Γ-graded free R-module and P a Γ-graded R-module. If there is another Γ-graded R-module Q such that T = P ⊕ Q and Tγ = Pγ + Qγ ,

γ ∈ Γ,

then P is called a Γ-graded projective R-module. As in the classical N-graded case, one verifies that a Γ-graded R-module P is Γ-graded projective if and only if it is projective as an ungraded R-module. Now considering the Γ-filtered algebra A = R/I, we proceed to construct a Γfiltered free A-module L with a Γ-filtration F L such that its associated Γ-graded

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module GΓ (L) is a Γ-graded free GΓ (A)-module. Starting with a free A-module L L = i∈J Aei on the A-basis {ei }i∈J , if we choose arbitrarily γi ∈ Γ, i ∈ J, then L has the Γ-filtration F L = {Fγ L}γ∈Γ with ¶ Mµ X Fγ L = Fsi A ei , γ ∈ Γ. i∈J

s¹γ, si γi =s

4.8. Observation. Note that since Γ is a monoid with the identity element γ0 which is the smallest element in Γ, it is not difficult to see that in the construction of F L = {Fγ L}γ∈Γ above, for each i ∈ J, ei ∈ Fγi L − Fγ∗i L, that is, each ei is of degree γi . In what follows, if we say that L is a Γ-filtered free A-module, then it is certainly the type constructed above. 4.9. Proposition. The following statements hold: L (i) Let L = i∈J Aei be a Γ-filtered free A-module with Γ-filtration F L, then the associated Γ-graded GΓ (A)-module GΓ (L) of L is a Γ-graded free GΓ (A)L L Γ Γ module. More precisely, we have GΓ (L) = i∈J G (A)σ(ei ) = γ∈Γ G (L)γ with X GΓ (L)γ = GΓ (A)si σ(ei ), γ ∈ Γ. i∈J, si γi =γ

L

(ii) If L0 = i∈J GΓ (A)ηi is a Γ-graded free GΓ (A)-module with the GΓ (A)basis {ηi }i∈J consisting of homogeneous elements, then there is some Γ-filtered free A-module L such that L0 ∼ = GΓ (L) as Γ-graded GΓ (A)-modules. (iii) Let M be a Γ-filtered A-module with Γ-filtration F M . Then there is an exact sequence of Γ-filtered A-modules and strict Γ-filtered A-homomorphisms ι

ϕ

0 → N −→ L −→ M → 0, where L is a Γ-filtered free A-module with Γ-filtration F L, N is the kernel of the Γ-filtered A-epimorphism ϕ that has the Γ-filtration F N = {Fγ N = N ∩ Fγ L}γ∈Γ induced by F L, and ι is the inclusion map. (iv) If L is a Γ-filtered free A-module with Γ-filtration F L, N is a Γ-filtered A-module with Γ-filtration F N , and ϕ : GΓ (L) → GΓ (N ) is a Γ-graded GΓ (A)epimorphism, then ϕ = GΓ (ψ) for some strict Γ-filtered A-epimorphism ψ : L → N. 4.10. Proposition. Let P be a Γ-filtered A-module with Γ-filtration F P . The following statements hold: (i) If GΓ (P ) is a projective GΓ (A)-module, then P is a projective A-module. (ii) If GΓ (P ) is a Γ-graded free GΓ (A)-module, then P is a free A-module.

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4.11. Proposition. Let M be a Γ-filtered A-module with Γ-filtration F M , and let 0 → N 0 → L0n → · · · → L00 → GΓ (M ) → 0 (2) be an exact sequence of Γ-graded GΓ (A)-modules and Γ-graded GΓ (A)-homomorphisms, where the L0i are Γ-graded free GΓ (A)-modules. The following statements hold: (i) There exists an exact sequence of Γ-filtered A-modules and strict Γ-filtered A-homomorphisms 0 → N → Ln → · · · → L0 → M → 0

(3)

in which the Li are Γ-filtered free A-modules such that we have the isomorphism of chain complexes 0→

0 N    ∼ = y

→

L0n   ∼ = y

→ ··· →

L00   ∼ = y

→ G(M  )→0   = y

0 → GΓ (N ) → GΓ (Ln ) → · · · → GΓ (L0 ) → G(M ) → 0 (ii) If N 0 is a projective GΓ (A)-module, then N is a projective A-module. If N is a Γ-graded free GΓ (A)-module, then N is a free A-module. (iii) If all modules in the sequence (2) are finitely generated over GΓ (A), then all modules in the sequence (3) are finitely generated over A. 0

4.12. Corollary. Let M be an A-module with Γ-filtration F M . Then p.dimA M ≤ p.dimGΓ (A) GΓ (M ), where p.dim abbreviates the phrase “projective dimension”. In particular, if GΓ (M ) has a (finite or infinite Γ-graded) free resolution, then M has a (finite or infinite) free resolution. To deal with flat modules over a Γ-filtered K-algebra A, we need to define a Γ-filtration, respectively a Γ-gradation, for a tensor product of two Γ-filtered A-modules, respectively for a tensor product of two Γ-graded GΓ (A)-modules. Let M be a Γ-filtered left A-module with Γ-filtration F M , and let N be a Γ-filtered right A-module with Γ-filtration F N . Viewing N ⊗A M as a Z-module, we define the Γ-filtration F (N ⊗A M ) of N ⊗A M as ¯ © ª Fγ (N ⊗A M ) = Z-span x ⊗ y ¯ x ∈ Fv N, y ∈ Fw M, vw ¹ γ , γ ∈ Γ. The associated Γ-graded Z-module of N ⊗A M with respect to F (N ⊗A M ) is then L defined as GΓ (N ⊗A M ) = γ∈Γ GΓ (N ⊗A M )γ with GΓ (N ⊗A M )γ = Fγ (N ⊗A M )/Fγ∗ (N ⊗A M ), where Fγ∗ (N ⊗A M ) =

S γ 0 ≺γ

Fγ 0 (N ⊗ M ).

γ ∈ Γ,

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175

Let P be a Γ-graded left GΓ (A)-module, and let Q be a Γ-graded right GΓ (A)module. Viewing Q ⊗GΓ (A) P as a Z-module, it is Γ-graded by ¯ © ª (Q ⊗GΓ (A) P )γ = Z-span z ⊗ t ¯ z ∈ Qv , t ∈ Pw , vw = γ ,

γ ∈ Γ.

4.13. Lemma. Let M be a Γ-filtered left A-module with Γ-filtration F M , and let N be a Γ-filtered right A-module with Γ-filtration F N . By the construction made above, the following statements hold: (i) For xv ∈ GΓ (N )v represented by x ∈ Fv N , and y w ∈ GΓ (M )w represented by y ∈ Fw M , the mapping ϕ(M, N ) : GΓ (N ) ⊗GΓ (A) GΓ (M ) −→ GΓ (N ⊗A M ) defined by xv ⊗ y w 7→ (x ⊗ y)vw is an epimorphism of Γ-graded Z-modules. (ii) The canonical A-isomorphisms ∼ =

A ⊗A M −→ M

∼ =

and N ⊗A A −→ N

are strict Γ-filtered A-isomorphisms. (iii) The strict Γ-filtered A-isomorphisms in (ii) induce Γ-graded GΓ (A)-isomorphisms ∼ =

∼ =

GΓ (A ⊗A M ) −→ GΓ (M ) and GΓ (N ⊗A A) −→ GΓ (N ). (iv) The canonical GΓ (A)-isomorphisms ∼ =

∼ =

GΓ (A) ⊗GΓ (A) GΓ (M ) −→ GΓ (M ) and GΓ (N ) ⊗GΓ (A) GΓ (A) −→ GΓ (N ) are Γ-graded GΓ (A)-isomorphisms. 4.14. Proposition. The following statements hold: (i) Let M be a Γ-filtered left A-module with Γ-filtration F M . If GΓ (M ) is a flat GΓ (A)-module, then M is a flat A-module. (ii) Let N be an A-module with Γ-filtration F N . Then w.dimA N ≤ w.dimGΓ (A) GΓ (N ), where w.dim abbreviates the phrase “weak dimension”. Since every A-module can be endowed with a Γ-filtration, the foregoing results enable us to reach the following theorem. In the text mentioned below, gl.dim denotes the homological global dimension, and gl.w.dim denotes the global weak dimension.

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4.15. Theorem. Let A = R/I and AΓLH = R/hLH(I)i be as in Theorem 4.2. The following statements hold: (i) gl.dim A ≤ gl.dim AΓLH . (ii) If AΓLH is left hereditary, then A is left hereditary. (iii) gl.w.dim A ≤ gl.w.dim AΓLH . (iv) If AΓLH is a von Neumann regular ring, then so is A.

5

Γ A Working Chart from AB LH to G (A) and A

In this section, we first picture clearly the working strategy developed in the preceding sections, and then give an immediate illustration of this philosophy. L Let R = γ∈Γ Rγ be a Γ-graded K-algebra, where Γ is an ordered semigroup by a total ordering < , and let (B, ≺gr ) be an admissible system of R, where B is a skew multiplicative K-basis of R consisting of Γ-homogeneous elements, and ≺gr is a Γ-graded monomial ordering on R (see Section 3). Consider an arbitrary ideal I of R and the quotient algebra A = R/I. Then by Theorem 1.1, AΓLH = R/hLH(I)i ∼ = GΓ (A), where AΓLH is the Γ-leading homogeneous algebra of A defined with respect to the Γ-gradation of R, and GΓ (A) is the associated Γ-graded algebra of A defined by the Γ-filtration F Γ A induced by the Γ-grading filtration F Γ R of R. On the other hand, since R is also B-graded, it follows from Section 3 that the B-leading homogeneous algebra AB LH of A is well-defined with respect to the B-gradation of R and the monomial ordering ≺gr on B, i.e., AB LH = R/hLM(I)i, which is obviously Γ-B-graded by the choice of B. Furthermore, by Proposition 3.2(i) the following corollary is clear. 5.1. Corollary. With notation as fixed above, the algebras A and AΓLH have the same B-leading homogeneous algebra, that is, ¡ Γ ¢B AB LH = R/hLM(I)i = ALH LH . Noticing that AB LH is a kind of “monomial algebra” which may be studied more effectively by using combinatorial and computational methods, for every reason we may expect (and may see in the sequel parts), at a modest level, that certain nice properties of AB LH will be transferred directly through the routes of the diagram ∼ =

A< I @ @ @

@ @ @

AB LH

GΓ (A) ←− AΓLH µ ¡ ¡ ¡ ¡ ¡ ¡

Now suppose that the B-grading filtration F B R exists (for instance, if R has no divisors of zero). Then both A and AΓLH have the induced B-filtration F B A and

AΓLH and Gr¨obner Bases

177

F B AΓLH , respectively. Considering the associated B-graded algebras GB (A) and GB (AΓLH ) of A and AΓLH , respectively, if G is a Gr¨obner basis of I, then on the basis of Theorem 1.1, the results of Section 3 and the above Corollary 5.1, the following diagram may be established to indicate how to realize the results of Section 4 by means of the monomial ideal hLM(G)i in an algorithmic way. A = R/hGi µ ¡ I @ @ lifitng lifting ¡¡ @ ¡ @ ¡ @ ¡ @ ∼ R = B G (A) GΓ (A) ←− = AΓLH Γ−PBW hLH(G)i ∧ ∧ ∼ lifting = B−PBW AB LH =

R hLM(G)i

∼ = B−PBW

> GB (AΓLH )

To give an immediate illustration of the philosophy presented above, let R = K[a1 , . . . , an ] be a finitely generated K-algebra and let B be a skew multiplicative K-basis of R consisting of monomials of the form αn 1 aα i1 · · · ain ,

aij ∈ {a1 , . . . , an }, αj ∈ N.

For instance, R is a finitely generated free K-algebra, or a commutative polynomial K-algebra in n variables, or the skew polynomial algebra of n generators a1 , . . . , an subject to the relations aj ai − λji ai aj with λji ∈ K ∗ and 1 ≤ i < j ≤ n (including the coordinate ring of a quantum affine K-space), or a path algebra defined by a finite directed graph, or a finitely generated exterior algebra. Then by assigning to each ai a positive degree ni , R has the weight N-gradation {Rp }p∈N determined by B with the weight {ni }ni=1 . Thus, R is doubly N-B-graded. Note that under the N-B-graded structure of R, each monomial u ∈ B is clearly an N-B-homogeneous element. So suppose further that R has an admissible system (B, ≺gr ), where ≺gr is an N-graded monomial ordering on R with respect to the weight N-gradation of R. Then R holds a Gr¨obner basis theory. If I is an ideal of R, then it follows from the fundamental decomposition theorem in Gr¨obner basis theory concerning the K-vector space R that R = I ⊕ K-span(B − LM(I)) = hLM(I)i ⊕ K-span(B − LM(I)). Consequently, if we consider the N-filtration F N A of the quotient algebra A = R/I induced by the weight N-grading filtration F N R of R, then by the foregoing discussion, the next result is straightforward. 5.2. Proposition. Let A = R/I be as fixed above. Bearing AB LH = R/hLM(I)i

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∼ N and the isomorphism AN LH = R/LH(I) = G (A) in mind, the following statements hold: (i) The image of the set B − LM(I) in R/hLM(I)i, R/I, and R/hLH(I)i, respectively, serves to give a K-basis for each algebra listed. (Compare with Theorem 4.2(i).) (ii) A is finite dimensional over K if and only if GN (A) is finite dimensional over K, if and only if AB LH is finite dimensional over K, and in this case we have dimK A = dimK GN (A) = dimK AB LH = |B − LM(I)|. (iii) All three algebras A, GN (A) and AB LH have the same Hilbert function, and hence, they have the same growth, or equivalently, they have the same GK (Gelfand-Kirillov) dimension. (iv) If I is an N-graded ideal of R, then the N-graded K-algebra A = R/I and the N-graded K-algebra AB LH = R/hLM(I)i have the same Hilbert series. Hence, N in (iii) above, the N-graded algebras AB LH and G (A) have the same Hilbert series. (v) If a Gr¨obner basis G of I is computable in R, then the set B − LM(I), the Gelfand-Kirillov dimension, and the Hilbert series of the respective algebra considered in (i)–(iv) above may be obtained algorithmically.

6

Using AB obner Bases LH in Terms of Gr¨

Let the free K-algebra KhXi = KhX1 , . . . , Xn i generated by X = {X1 , . . . , Xn } be equipped with a positive weight N-gradation by assigning to each Xi a positive degree ni , 1 ≤ i ≤ n, and fix the admissible system (B, ≺gr ) for KhXi, where B is the multiplicative K-basis of KhXi consisting of all monomials (words in X1 , . . . , Xn ) and ≺gr is an N-graded monomial ordering on B with respect to the weight N-gradation of KhXi (see the definition in Section 3). By using several algorithmic results of [1, 2, 9, 14–16, 27, 32, 33] concerning monomial algebras, we demonstrate in this section how to realize the foregoing transfer principle for quotient algebras of KhXi in a computational way. The main tools used in our approach are Ufnarovski graphs and Anick resolutions determined by Gr¨obner bases. At this stage, the reader is reminded of the fact that any nonempty subset Ω of B itself is a Gr¨obner basis in KhXi, and more potentially, Ω stands for a family of Gr¨obner bases that have the same set of leading monomials. So our results of this and the next two sections also provide interesting families of algebras with better global structure properties. To be convenient, we fix notations and conventions for this section as follows. If I is an ideal of KhXi and A = KhXi/I is the corresponding quotient algebra, then bearing Theorem 1.1 in mind we will use the isomorphisms of graded algebras B ∼ B GN (A) ∼ = AN LH = KhXi/hLH(I)i and G (A) = ALH = KhXi/hLM(I)i freely wherever they are needed, where GN (A) is the associated N-graded algebra of A

AΓLH and Gr¨obner Bases

179

defined by the N-filtration F N A of A induced by the weight N-grading filtration F N KhXi of KhXi, AN LH is the N-leading homogeneous algebra of A determined by the set LH(I) of N-leading homogeneous elements of I with respect to the weight N-gradation of KhXi, GB (A) is the associated B-graded algebra of A defined by the B-filtration F B A of A induced by the B-grading filtration F B KhXi of KhXi, and AB LH is the B-leading homogeneous algebra of A determined by the set LM(I) of B-leading homogeneous elements of I with respect to the B-gradation of KhXi. The Ufnarovski graph Let Ω = {u1 , . . . , us } be a reduced subset of B in the sense that ui and uj are not divisible each other if i 6= j. For each ui ∈ Ω, say ui = Xiα11 · · · Xiαss with Xij ∈ X and αj ∈ N, we write l(ui ) = α1 + · · · + αs for the length of ui . Put ¯ © ª ` = max l(ui ) ¯ ui ∈ Ω . Then the Ufnarovski graph of Ω (in the sense of [32]), denoted by Γ(Ω), is defined as a directed graph, in which the set of vertices V is given by © ¯ ª V = vi ¯ vi ∈ B − hΩi, l(vi ) = ` − 1 , and the set of edges E contains the edge vi → vj if and only if there exist Xi , Xj ∈ X such that vi Xi = Xj vj ∈ B − hΩi. We say that a Gr¨obner basis G = {g1 , . . . , gs } in KhXi is LM-reduced if the set of leading monomials LM(G) = {LM(g1 ), . . . , LM(gs )} of G is reduced as defined above. The Ufnarovski graph of G is then defined to be the Ufnarovski graph Γ(LM(G)) of LM(G). Remark. To better understand the practical application of Γ(LM(G)) in the subsequent parts, it is essential to notice that the Ufnarovski graph is defined by using the length of the monomial (word) u ∈ B instead of using the degree of u as an N-homogeneous element in KhXi, though both notions coincide when each Xi is assigned to the degree 1. Let Ω = {u1 , . . . , us } be a reduced finite subset of monomials in KhXi. The first effective application of the graph Γ(Ω) was given by V. Ufnarovski in 1982 to determine the growth of the monomial algebra KhXi/hΩi. Theorem. [32] The following statements hold: (i) The growth of KhXi/hΩi is exponential if and only if there are two different cycles with a common vertex in the graph Γ(Ω). Otherwise, KhXi/hΩi has polynomial growth of degree d, or equivalently, the GK dimension of KhXi/hΩi is equal to d, where d is, among all routes of Γ(Ω), the largest number of distinct cycles occurring in a single route.

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(ii) KhXi/hΩi is a finite dimensional K-vector space if and only if Γ(Ω) does not contain any cycle. Example 1. Consider in the free K-algebra KhX1 , X2 , X3 , X4 i the subset of monomials Ω = {Xj Xi | 1 ≤ i < j ≤ 4}. Then ` = max{l(ui ) | ui ∈ Ω} = 2, V = {v1 = X1 , v2 = X2 , v3 = X3 , v4 = X4 }. It turns out that Γ(Ω) is presented by

©v

ª

©

ª

> v2 1 @ ¡ @ ¡ @¡ ¡@ ∨ ¡ @ ∨ ¡ ª R @ v4 < v3

Similarly, if in place of V we use successively Vijkl = {v1 = Xi , v2 = Xj , v3 = Xk , v4 = Xl },

1 ≤ i < j < k < l ≤ n,

we get the graph Γ(Ω) for the subset of monomials Ω = {Xj Xi | 1 ≤ i < j ≤ n} in the free K-algebra KhXi = KhX1 , . . . , Xn i. Let KhXi be equipped with a weight N-gradation, and consider the algebra A = KhXi/hGi defined by a Gr¨obner basis G = {gji | 1 ≤ i < j ≤ n} with respect to some N-graded monomial ordering ≺gr such that LM(gji ) = Xj Xi . Then it follows from Proposition 5.2 that the N ∼ N algebras A, AB LH = KhXi/hLM(G)i and G (A) = ALH = KhXi/hLH(G)i, all have GK dimension n. For a reduced finite Gr¨obner basis G in KhXi, the application of Γ(LM(G)) to the determination of the growth of A = KhXi/hGi was first observed in [16, Proposition 2], that is, the same statements of the above theorem can be mentioned for A in terms of Γ(LM(G)). Noetherianity First, let us see how the Ufnarovski graph can be used to determine special Noetherian algebras. 6.1. Theorem. Let I be an arbitrary ideal of KhXi and J = hΩi with Ω = {u1 , . . . , us } ⊂ B a reduced finite subset. Suppose that J ⊆ hLM(I)i (for instance, Ω ⊆ LM(I)) with respect to the fixed data (B, ≺gr ) of KhXi. If there is no edge entering (leaving) any cycle of the graph Γ(Ω), then all three algebras AB LH = N ∼ KhXi/hLM(I)i, A = KhXi/I and AN = KhXi/hLH(I)i G (A) are left = LH (right) Noetherian of GK dimension not exceeding 1.

AΓLH and Gr¨obner Bases

181

Proof. By [33] and [27], the finitely presented monomial algebra KhXi/J is left (right) Noetherian if and only if there is no edge entering (leaving) any cycle of the graph Γ(Ω). Hence, the Noetherianity of A and GN (A) follow from Corollary 5.1 and Theorem 4.7, while the statement on Gelfand-Kirillov dimension follows from [28, Proposition 7]. ¤ Example 2. Let KhXi = KhX1 , X2 i be the free K-algebra of two generators. As a small example, let us look at the monomial algebra S = KhXi/hΩi, where Ω = {X12 , X2 X1 }. It is easy to see that the Ufnarovski graph Γ(Ω) of Ω is of the form X1 > X2

ª

and there is no edge leaving the only cycle of Γ(Ω). Hence, S is right Noetherian of GK dimension 1. Consider an N-graded monomial ordering ≺gr on KhXi with respect to some weight N-gradation of KhXi, any algebra A = KhXi/I, and the associated N-graded algebra GN (A) ∼ = AN LH = KhXi/hLH(I)i of A with respect N to the N-filtration F A induced by the weight N-grading filtration F N KhXi. If X12 , X2 X1 ∈ hLM(I)i, then it follows from Theorem 6.1 that both A and GN (A) are right Noetherian of GK dimension not exceeding 1. For instance, consider the weight N-gradation of KhXi by assigning to d(X1 ) = n and d(X2 ) = 1, where n is an arbitrarily fixed positive integer, and let f (X2 ), g(X2 ) ∈ KhXi be two polynomials of X2 with degree ≤ 2n, n + 1, respectively. By setting the N-graded lexicographic ordering X2 ≺gr X1 , one may see that Theorem 6.1 applies to the algebra A = KhXi/I with I = hX12 + aX1 + f (X2 ), X2 X1 + bX1 + g(X2 ), V i, where a, b ∈ K and V is any subset of KhXi. Remark. Recall that an algebra is called weak Noetherian if it satisfies the ascending chain condition for ideals. If Ω ⊂ B is a reduced finite subset, then it was proved in [27] that the algebra KhXi/hΩi is weak Noetherian if and only if the Ufnarofski graph Γ(Ω) does not contain any cycle with edges both entering and leaving it. In a similar way one may get an analogue of Theorem 6.1 on the weak Noetherianity of A = KhXi/I and GN (A) ∼ = AN LH = KhXi/hLH(I)i. (Semi-)primeness and PI-property Let Ω = {u1 , . . . , us } be a reduced finite subset of monomials in B, and let Γ(Ω) be the Ufnarovski graph of Ω with ` = max{l(ui ) | ui ∈ Ω}. Assume l(ui ) ≥ 2, i = 1, . . . , s. Recall from [32] that any monomial Xi1 · · · Xis = u ∈ B − hΩi with l(u) = s ≥ ` − 1 is associated to a route in Γ(Ω): R(u) : v0 → v1 → · · · → vm , where m = s − ` + 1 and vj = Xij+1 Xij+2 · · · Xij+`−1 , 0 ≤ j ≤ m. Also recall from [32] and [14] that a vertex v of Γ(Ω) is called a cyclic vertex if it belongs to a cyclic

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route of Γ(Ω); and a monomial u ∈ B − hΩi, u 6= 1, is called a cyclic monomial if l(u) ≤ ` − 1 and u is a suffix of some cyclic vertex of Γ(Ω), or, if l(u) > ` − 1 and the associated route R(u) of u is a subroute of some cyclic route. 6.2. Theorem. Let G = {g1 , . . . , gs } be an LM-reduced finite Gr¨obner basis in KhXi with respect to the fixed data (B, ≺gr ), and let Γ(LM(G)) be the Ufnarovski graph of G. Consider the quotient algebra A = KhXi/hIi with I = hGi, and N ∼ N bear AB LH = KhXi/hLM(I)i, ALH = KhXi/hLH(I)i = G (A) in mind. With ` = max{l(ui ) | ui ∈ LM(G)}, if each monomial u ∈ B−hLM(G)i with 1 ≤ l(u) ≤ ` is cyclic, then the monomial algebra AB LH is semi-prime, and consequently, both N algebras A and G (A) are semi-prime. B Proof. By Section 3, KhXi/hLH(G)i = AN LH , KhXi/hLM(G)i = ALH ; and by [14, Theorem 2.21], the assumption implies that the monomial algebra AB LH is semi-prime (or Jacobson semi-simple). So the semi-primeness of A and GN (A) follows from Corollary 5.1 and Theorem 4.2. ¤

Let Ω = {u1 , . . . , us } be a finite subset of B. In [9] it was proved that the monomial algebra S = KhXi/hΩi satisfies a polynomial identity, i.e., S is a PIalgebra, if and only if it has the polynomial growth, if and only if in the Ufnarovski graph Γ(Ω) any two cycles do not have a common vertex. Although an algebra A = KhXi/I has the same growth as its B-leading homogeneous algebra AB LH = KhXi/hLM(I)i (Proposition 5.2), in general we do not know if it is always possible to transfer the PI-property of AB LH (if it has such property) to A. Nevertheless, in light of [14], [31] and [3], the following two results are obtained. 6.3. Theorem. Let G = {g1 , . . . , gs } be an LM-reduced finite Gr¨obner basis in KhXi with respect to the fixed (B, ≺gr ), and let Γ(LM(G)) be the Ufnarovski graph of G. Consider the quotient algebra A = KhXi/hIi with I = hGi, and N ∼ N bear AB LH = KhXi/hLM(I)i, ALH = KhXi/hLH(I)i = G (A) in mind. With ` = max{l(ui ) | ui ∈ LM(G)}, the following statements hold: (i) If in the graph Γ(LM(G)) any connected component contains at most one cycle, and there is at least one connected component contains exactly one cycle, N then all three algebras AB LH , A and G (A) are PI-algebras of GK dimension 1. (ii) Under the assumption of (i) above, if furthermore each monomial u ∈ N B −hLM(G)i with 1 ≤ l(u) ≤ ` is cyclic, then all three algebras AB LH , A and G (A) are Noetherian semi-prime PI-algebras of GK dimension 1 and Krull dimension 1. Proof. Note that Section 3 yields KhXi/hLH(G)i = AN LH and KhXi/hLM(G)i = B ALH . Under the assumption of (i) and (ii), it follows from the previously mentioned Ufnarovski theorem [14, Theorem 2.21], [9] and [31], the monomial algebra AB LH is a Noetherian semi-prime PI-algebra of GK dimension 1 and Krull dimension 1. Hence, by Corollary 5.1, Theorems 4.2 and 4.7 and [31], both A and GN (A) are

AΓLH and Gr¨obner Bases

183

Noetherian semi-prime PI-algebras of GK dimension 1 and Krull dimension 1. ¤ 6.4. Theorem. Let G = {g1 , . . . , gs } be an LM-reduced finite Gr¨obner basis in KhXi with respect to the fixed (B, ≺gr ), and let Γ(LM(G)) be the Ufnarovski graph of G. Consider the quotient algebra A = KhXi/hIi with I = hGi, and N ∼ N bear AB LH = KhXi/hLM(I)i, ALH = KhXi/hLH(I)i = G (A) in mind. With ` = max{l(ui ) | ui ∈ LM(G)}, if the following two conditions are satisfied: (a) any u ∈ B−hLM(G)i with l(u) < `−1 is a suffix of some vertex of Γ(LM(G)), (b) for any two vertices vi and vj of Γ(LM(G)), there exists a route from vi to vj and vice versa, then the monomial algebra AB LH is prime, and consequently, both algebras A and GN (A) are prime or Noetherian prime PI-algebras of GK dimension 1 and Krull dimension 1. Proof. By [14, Theorem 2.28], the assumption implies that the monomial algebra B AB LH is prime. It follows from [3] that ALH is either primitive or has GK dimension 1. Thus, by Corollary 5.1, Theorem 4.2 and [31], both A and GN (A) are prime or Noetherian prime PI-algebras of GK dimension 1 and Krull dimension 1. ¤ Example 3. Consider in KhXi = KhX1 , X2 , X3 i the subset of monomials Ω = {X1 X2 , X3 X1 , X22 }. Then ` = max{l(ui ) | ui ∈ Ω} = 2, V = {v1 = X1 , v2 = X2 , v3 = X3 }, and the Ufnarovski graph Γ(Ω) is presented by

©v

1 < @ @ @

@ @ @ R v3

¡ ¡ µ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ª ¡ ¡

v2

ª

By [14, Theorem 2.21], one checks that the monomial algebra KhXi/hΩi is a prime algebra. It follows from the above theorem that if G = {g1 , g2 , g3 } ⊂ KhXi is any Gr¨obner basis with respect to some graded monomial ordering ≺gr such that LM(g1 ) = X1 X2 , LM(g2 ) = X3 X1 and LM(g3 ) = X22 , then all three algebras A = KhXi/hGi, GN (A) ∼ = KhXi/hLH(G)i and GB (A) ∼ = KhXi/hLM(G)i are prime. Example 4. Consider in the free K-algebra KhXi = KhX1 , X2 i the subsets of monomials Ω1 = {X12 , X22 } and Ω2 = {X1 X2 , X2 X1 }. Then the Ufnarovski graph Γ(Ω1 ) of Ω1 is > X2 X1 <

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and the Ufnarovski graph Γ(Ω2 ) of Ω2 is

©X

1

X2

ª

It is easy to see that the conditions of Theorem 6.3 are satisfied. Hence, both monomials algebras KhXi/hΩ1 i and KhXi/hΩ2 i are Noetherian semi-prime PIalgebras of GK dimension 1 and Krull dimension 1 (indeed, K(X)/Ω1 is prime). Consequently, for any Gr¨obner basis G ⊂ KhXi with respect to some N-graded monomial ordering ≺gr such that LM(G) = Ω1 or LM(G) = Ω2 , all three algebras N ∼ N AB LH = KhXi/hLM(G)i, ALH = KhXi/hLH(G)i = G (A) and A = KhXi/hGi are Noetherian semi-prime PI-algebras of GK dimension 1 and Krull dimension 1. For instance, under the N-graded lexicographic ordering X1 ≺gr X2 , we may take, for a, b, c, d, e, f, g ∈ K, G = {g1 = X12 + aX1 + bX2 + c, g2 = X22 + dX1 X2 + eX1 + f X2 + g}. Finiteness of global dimension Let KhXi = KhX1 , . . . , Xn i be equipped with the augmentation map ε sending each Xi to zero. For any ideal I contained in the augmentation ideal (i.e., the kernel of ε), D.J. Anick constructed in [2] a free resolution of the trivial A-module K over the quotient algebra A = KhXi/I: δ

δ

δ

ε

n 1 0 · · · → Cn K ⊗K A −→ · · · → C1 K ⊗K A −→ C0 K ⊗K A −→ A −→ K → 0,

where each Cn is the set of all n-chains determined by LM(I). The Anick resolution has several efficient applications to the homological aspects of associative algebras (see [1] and [2]). To understand and use Anick’s resolution from a computational viewpoint, V. Ufnarovski constructed in [33] the graph of n-chains of a reduced (finite or infinite) subset of monomials Ω ⊂ B (also see [12]), that is a directed graph ΓC (Ω), in which the set of vertices V is defined as V = {1} ∪ X ∪ {all proper suffixes of u ∈ Ω}, and the set of edges E consists of all edges 1 −→ Xi for every Xi ∈ X and edges defined by the rule: for u, v ∈ V − {1}, u −→ v in E ⇐⇒ there is a unique w = Xi1 · · · Xim−1 Xim ∈ Ω ½ w, or such that uv = sw, s ∈ B and sXi1 · · · Xim−1 ∈ B\hΩi. For n ≥ −1, an n-chain in ΓC (Ω) is a word (monomial) which can be read out of the graph through a route of length n + 1 starting from 1. Writing Cn for the set

AΓLH and Gr¨obner Bases

185

of all n-chains in ΓC (Ω), it is clear that C−1 = {1}, C0 = X, and C1 = Ω. For an LM-reduced Gr¨obner basis G of KhXi, ΓC (LM(G)) is referred to the graph of n-chains of G. Remark. As with the Ufnarovski graph Γ(Ω) defined before, to better understand the practical application of the graph ΓC (Ω) of n-chains determined by Ω in the sequel parts, it is essential to notice that an n-chain is defined by a route of length n + 1 starting with 1 instead of by the degree of the N-homogeneous element (a monomial) u read out of that route. Example 5. Consider in the free K-algebra KhXi = KhX1 , . . . , Xn i any subset Ω ⊆ {Xj Xi | 1 ≤ i < j ≤ n}, or Ω ⊆ {Xi Xj | 1 ≤ i < j ≤ n}. Then it may be easily verified that the graph ΓC (Ω) does not contain any i-chain for i ≥ n. For instance, in the case that n = 3 and Ω = {X2 X1 , X3 X1 , X3 X2 }, we have V = {1} ∪ {X1 , X2 , X3 } and the graph ΓC (Ω) is presented by 1

A@HHH A @@ HH A HH A @@ HH A H j @ R A X3 X2 < A ¡ A ¡ A ¡ A ¡ A ¡ ¡ AU ∨ ª X1

6.5. Theorem. Let I be an arbitrary ideal of KhXi, A = KhXi/I, and bear N ∼ N AB LH = KhXi/hLM(I)i and ALH = KhXi/hLH(I)i = G (A) in mind. Consider the reduced monomial generating set Ω of hLM(I)i (it always exists!) with respect to the fixed (B, ≺gr ). If Ω ∩ X = ∅ and the graph ΓC (Ω) does not contain any d-chain, then gl.dim A ≤ gl.dim GN (A) ≤ gl.dim AB LH ≤ d. Proof. Consider the Anick resolution of K over the monomial algebra AB LH . After B tensoring the resolution over ALH with K, by [2, Lemma 3.3], we have the following isomorphisms of N-graded K-modules: B LH (K, K) ∼ C TorA = n−1 K, n

n = 1, 2, . . . ,

and thereby [1, Theorem 4] asserts that gl.dim AB LH ≤ d if and only if Ω does not have any d-chains. Hence, our theorem follows now from Corollary 5.1 and Theorem 4.15. ¤

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Note that if Ω ⊂ B is a reduced finite subset of monomials, then ΓC (Ω) is a finite directed graph. Thus, as illustrated by the next corollary, a finite Gr¨obner basis will make Theorem 6.5 work more effectively. 6.6. Corollary. Consider any subset Ω ⊆ {Xj Xi | 1 ≤ i < j ≤ n}, or Ω ⊆ {Xi Xj | 1 ≤ i < j ≤ n} in KhXi. If G is any Gr¨obner basis in KhXi with respect to the fixed (B, ≺gr ) such that LM(G) = Ω, then the K-algebras A = KhXi/hGi, B ∼ N AN LH = KhXi/hLH(G)i = G (A) and ALH = KhXi/hLM(G)i satisfy gl.dim A ≤ gl.dim GN (A) ≤ gl.dim AB LH ≤ n. Proof. This follows from Theorem 6.5 and the foregoing Example 5.

7

¤

Recognizing (Non-)homogeneous N -Koszulity via AB LH

In [22, Section 4], only very little about lifting classical Koszulity [30] was discussed. In this section, by using some computational results obtained in preceding sections, we focus on non-commutative algebras defined by relations and intend to demonstrate how to realize both the homogeneous and non-homogeneous Koszulity for the algebras considered. Moreover, as a dividend of the main result (Theorem 7.2), we also give some examples of (non-)homogeneous N -Koszul algebras for N ≥ 2. For related notions and notations we refer to previous sections. Let the free K-algebra KhXi = KhX1 , . . . , Xn i be equipped with the natural N-gradation, that is, each Xi is assigned to the degree 1, and let J be an N-graded ideal of KhXi, R = KhXi/J. The N-graded algebra R is said to be a homogeneous N -Koszul algebra in the sense of [7] if the following conditions are satisfied: (a) J is generated by N-homogeneous elements of degree N ≥ 2; (b) For any p ≥ 1, the N-graded K-vector space TorR p (K, K) is concentrated in degree ζ(p), or in other words, ¡

¢ TorR p (K, K) q = 0 for q 6= ζ(p),

where ζ : N → N is the jump map defined by ½ mN if p = 2m (i.e., p is even), ζ(p) = mN + 1 if p = 2m + 1 (i.e., p is odd). Clearly, in the case that N = 2, the above definition recovers classical quadratic Koszul algebras [30]. In what follows, we let (B, ≺gr ) be an admissible system of KhXi, where B is the standard K-basis of KhXi and ≺gr is an N-graded monomial ordering on B with respect to the natural N-gradation of KhXi.

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7.1. Proposition. Let J be an N-graded ideal of KhXi, R = KhXi/J, Ω the reduced monomial generating set of hLM(I)i with respect to the fixed (B, ≺gr ), and ΓC (Ω) the graph of n-chains of Ω as defined in Section 6. For N ≥ 2, let ζ : N → N be the jump map associated to N . If the N-graded K-module Ci−1 K spanned by all (i − 1)-chains from ΓC (Ω) is generated in degree ζ(i) (i.e., every monomial given by an (i−1)-chain is an N-homogeneous element of degree ζ(i)), B i = 1, 2, . . . , then the B-leading homogeneous algebra RLH = KhXi/hLM(I)i of R is homogeneous N -Koszul and thereby R is homogeneous N -Koszul. Proof. Note that J is not necessarily a monomial ideal. The assertion may follow immediately from [2, Lemma 3.4]. To see this clearly, consider the Anick resolution of K over R constructed in terms of n-chains from the graph ΓC (Ω): δ

δ

δ

ε

2 1 0 · · · → C2 K ⊗K R −→ C1 K ⊗K R −→ C0 K ⊗K R −→ R −→ K → 0.

After tensoring the resolution over R with K, we have Ci−1 K ⊗K R ⊗R K ∼ Ker (δi−1 ⊗ 1K ) Ci−1 K ⊆ . = Im(δi ⊗ 1K ) Im(δi ⊗ 1K ) Im(δi ⊗ 1K ) ¡ ¢ RB By [1, Lemma 3.3], the degree-q component of Ci−1 K is just Tori LH (K, K) q . So the proof is done. ¤ TorR i (K, K) =

Consider the natural N-grading filtration F N KhXi of KhXi. For a K-subspace P ⊂ FNN KhXi − FNN −1 KhXi, N ≥ 2, let the algebra A = KhXi/hP i be equipped with the N-filtration F N A induced by F N KhXi. The algebra A is said to be a non-homogeneous N -Koszul algebra in the sense of [29] and [7] if the following two conditions are satisfied: (a) The N-graded algebra KhXi/hLH(P )i is homogeneous N -Koszul as defined before, where LH(P ) is the set of N-leading homogeneous elements of P with respect to the natural N-gradation of KhXi; (b) A has the N -type PBW property in the sense of [7], that is, the N-graded K-algebra homomorphism ρ appearing in the commutative diagram π

KhXi  −→ KhXi/hLH(P )i   φ .ρ y GN (A) is an isomorphism, where GN (A) is the associated N-graded algebra of A defined by F N A. (One is referred to the proof of Theorem 1.1 for understanding this diagram, where π is the canonical epimorphism.) 7.2. Theorem. With notation as before, for N ≥ 2, let G ⊂ FNN KhXi − FNN −1 KhXi

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be an LM-reduced Gr¨obner basis with respect to the fixed (B, ≺gr ), and consider the N-filtered algebra A = KhXi/I with the N-filtration F N A induced by N F N KhXi, where I = hGi. Bearing AB LH = KhXi/hLM(I)i, ALH = KhXi/hLH(I)i in mind, the following statements hold: N (i) If AB LH is homogeneous N -Koszul, then ALH is homogeneous N -Koszul, and N hence G (A), the associated N-graded algebra of A defined by F N A, is homogeneous N -Koszul. (ii) If GN (A) is homogeneous N -Koszul, then A is non-homogeneous N -Koszul. (iii) Put Ω = LM(G) and consider the directed graph ΓC (Ω) of n-chains of Ω. Let ζ : N → N be the jump map associated to N . If the N-graded K-module Ci−1 K spanned by all (i − 1)-chains of Ω is generated in degree ζ(i), i = 1, 2, . . . , then N AB LH and G (A) are homogeneous N -Koszul, and thereby A is non-homogeneous N -Koszul. Proof. Note that by Theorem 1.1 and Corollary 5.1 we have ¡ N ¢B ∼ N AN ALH LH = AB LH = G (A) and LH = KhXi/hLM(G)i. If the condition of (iii) is satisfied, then (i) follows from Proposition 7.1. Moreover, by Section 3 we have ρ

AN GN (A), LH = KhXi/hLM(I)i = KhXi/hLM(G)i −→ ∼ =

showing that A has the N -type PBW property. So under the assumption of (iii), the homogeneous N -Koszulity of GN (A) follows from (i) and thereby A is nonhomogeneous N -Koszul by definition. ¤ 7.3. Corollary. Let G ⊂ F2N KhXi − F1N KhXi be a Gr¨obner basis and A = KhXi/I with I = hGi. Then both the algebras AB LH = KhXi/hLM(I)i and AN = KhXi/hLH(I)i are homogeneous 2-Koszul. Hence, GN (A) is homogeneous LH 2-Koszul, and consequently, A is non-homogeneous 2-Koszul. Proof. By [2] and [19], any quadratic Gr¨obner basis in KhXi defines a (classical) homogeneous 2-Koszul algebra. Since G is a Gr¨obner basis consisting of elements from F2N KhXi, the set LH(G) of N-leading homogeneous elements of G forms a quadratic Gr¨obner basis by Proposition 3.2. Hence, the corollary follows now from Theorems 3.3 and 7.2. ¤ Theorem 7.2 also guarantees the existence of (non-)homogeneous N -Koszul algebras for every N ≥ 2. 7.4. Proposition. For every positive integer N ≥ 2, there is a (non-monomial) homogeneous N -Koszul algebra and a non-homogeneous N -Koszul algebra. Proof. Let KhXi = KhX1 , X2 i be the free K-algebra of two generators, and consider the N-graded lexicographic ordering X1 ≺gr X2 with respect to the natural

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N-gradation of KhXi. For any positive integer n, if g = X2n X1 − qX1 X2n + F with q ∈ K and F ∈ KhXi such that LM(F ) ≺gr X2n X1 , then it is easy to check that {g} forms a Gr¨obner basis for the principal ideal I = hgi. Put Ω = {LM(g) = X2n X1 }. Then it is straightforward to see that the graph ΓC (Ω) of n-chains of Ω is presented by X2n−1 X1 % X1

←− -

X2 %

X1

X2 -

1 (n = 1)

% 1 (n ≥ 2)

such that Ci−1

  {X1 , X2 } i = 1, = {X2n X1 } i = 2,  ∅ i ≥ 3.

Moreover, for N = n + 1 we have ζ(1) = 1, ζ(2) = N = n + 1, ζ(3) = N + 1 = n + 2. If we consider the algebra A = KhXi/I, then AB LH = KhXi/hLM(I)i = n B KhXi/hX2 X1 i. It follows from Theorem 7.2 that ALH and AN LH = KhXi/hLH(I)i = KhXi/hLH(g)i ∼ = GN (A) are homogeneous N -Koszul, and consequently, A is non-homogeneous N -Koszul in the case that F 6= 0 with total degree ≤ n. ¤ N By Theorem 6.5 we may also see that all three algebras AB LH , ALH and A constructed in the proof of the last proposition have global dimension ≤ 2.

8

Gr¨ obner Basis and Rees Algebra

If A is a Z-filtered K-algebra with Z-filtration F A = {Fn A}n∈Z , then it is wellknown from the literature that in addition to having its associated Z-graded e= K-algebra GZ (A), A is also closely related to another Z-graded K-algebra A L e e e n∈Z An with An = Fn A, where the multiplication of A is induced by Fn AFm A ⊆ Fn+m A for all n, m ∈ Z, which is called the Rees algebra of A with respect to F A. Let the free K-algebra KhXi = KhX1 , . . . , Xn i of n generators be equipped with a fixed weight N-gradation by assigning to each Xi a positive degree ni > 0, 1 ≤ i ≤ n, and let (B, ≺gr ) be an admissible system of KhXi, where ≺gr is an N-graded monomial ordering on the standard K-basis B of KhXi. Consider an arbitrary ideal I of KhXi and the corresponding quotient algebra A = KhXi/I, which is equipped with the N-filtration F A induced by the N-grading filtration F KhXi of KhXi. In this section, firstly we show that if the ideal I is generated e by a Gr¨obner basis G, then many structural properties of the Rees algebra A of A may be obtained in terms of G and the non-central homogenized Gr¨obner

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basis Ge of G; and secondly we clarify the relation between Rees algebras, regular central extensions and PBW-deformations in terms of Gr¨obner bases. Notations and conventions used in previous sections are maintained. To reach the goal of the current section, it is necessary to recall some basic results concerning Rees algebras (see [21] and [24]). Let KhXi[t] be the polynomial ring in one commuting variable t over KhXi L which is equipped with the mixed N-gradation, that is, KhXi[t] = p∈Z KhXi[t]p with ¯ ½ X ¾ ¯ KhXi[t]p = Fi tj ¯¯ Fi ∈ KhXii , j ≥ 0 , p ∈ N, i+j=p

where KhXii is the degree-i homogeneous component of the weight N-gradation Pp of KhXi. Recall that if f = i=0 Fi is an element of KhXi with Fi ∈ KhXii and Fp 6= 0, then the homogenization of f (with respect to t) is the degree-p P Pp homogeneous element f ∗ = i=0 Fi tp−i in KhXi[t]; and if F = i,j Fi tj is an element of KhXi[t], then the dehomogenization of F (with respect to t) is the P element F∗ = i Fi . Also note that any N-graded monomial ordering ≺gr on the K-basis B of KhXi extends to an N-graded monomial ordering ≺t-gr on the K-basis B ∗ = {tr w | w ∈ B, r ∈ N} of KhXi[t], that is, tr1 w1 ≺t-gr tr2 w2 if and only if w1 ≺gr w2 , or w1 = w2 and r1 < r2 . Hence, KhXi[t] holds a Gr¨obner basis theory with respect to the admissible system (B ∗ , ≺t-gr ). 8.1. Proposition. [24, Proposition 2.1.4 and Theorem 2.3.2] Let I be an ideal of KhXi and A = KhXi/I. Writing hI ∗ i for the N-graded ideal generated by the set I ∗ = {f ∗ | f ∈ I} of homogenized elements of I in KhXi[t], the following statements hold: (i) G is a Gr¨obner basis of I with respect to an admissible system (B, ≺gr ) if and only if G ∗ = {g ∗ | g ∈ G} is a Gr¨obner basis for hI ∗ i with respect to the admissible system (B ∗ , ≺t-gr ). e is the Rees algebra of A defined by the N-filtration F A induced by the (ii) If A weight N-grading filtration F KhXi of KhXi, then there is an N-graded K-algebra e∼ isomorphism A = KhXi[t]/hI ∗ i. Consequently, if G is a Gr¨obner basis as in (i) e∼ above, then A = KhXi[t]/hG ∗ i. For the reason of distinguishing between the Gr¨obner basis G ∗ presented in Proposition 8.1 and the Gr¨obner basis Ge to appear below, we call G ∗ the central homogenization of G with respect to the commuting variable t, or G ∗ is a central homogenized Gr¨obner basis with respect to t. Next, we give a detailed verification of [21, Chapter III, Corollary 3.8(i)]. Let (B, ≺gr ) be the admissible system of KhXi as before, i.e., ≺gr is chosen with respect

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to the fixed weight N-gradation of KhXi. If f ∈ KhXi has the linear presentation by elements of B: X f = LC(f )LM(f ) + λi wi with LM(f ) ∈ B ∩ KhXip , wi ∈ B ∩ KhXiqi , i

then the homogenization of f in KhXi[t] with respect to t is given by X f ∗ = LC(f )LM(f ) + λi tp−qi wi , i

which corresponds to a homogeneous element in the free algebra KhX, T i = KhX1 , . . . , Xn , T i, i.e., the element X fe = LC(f )LM(f ) + λi T p−qi wi . i

Assigning to T the degree 1 in KhX, T i and using the fixed weight of X in KhXi, we get the weight N-gradation of KhX, T i which extends the weight N-gradation of KhXi. Consequently, writing Be for the standard K-basis of KhX, T i, we may extend ≺gr to an N-graded monomial ordering ≺ T -gr on Be such that T ≺ T -gr Xi , 1 ≤ i ≤ n, and hence LM(f ) = LM(fe).

(4)

If I is an ideal of KhXi generated by the subset S, then we put Ie = {fe | f ∈ I } ∪ {Xi T − T Xi | 1 ≤ i ≤ n}, Se = {fe | f ∈ S} ∪ {Xi T − T Xi | 1 ≤ i ≤ n}. 8.2. Proposition. With the preparation made above, suppose that G is a Gr¨obner basis of the ideal I in KhXi with respect to the admissible system (B, ≺gr ), and e in KhX, T i let A = KhXi/I. Then Ge is a Gr¨obner basis for the N-graded ideal hIi ∼ e e e with respect to the admissible system (B, ≺ T -gr ), and hence A = KhX, T i/hGi, e where A is the Rees algebra of A defined by the N-filtration F A induced by the weight N-grading filtration F KhXi of KhXi. Proof. Firstly, by Proposition 8.1, G ∗ is a Gr¨obner basis for the ideal hI ∗ i in e ∼ KhXi[t]. Hence, B ∗ − hLM(G ∗ )i provides a K-basis for the Rees algebra A = ∗ KhXi[t]/hI i. Furthermore, note that if f ∈ KhXi and LH(f ) is the N-leading homogeneous part of f , then LM(f ) = LM(LH(f )) w.r.t. ≺gr on B, LM(f ∗ ) = LM(f ) w.r.t. ≺t-gr on B ∗ , where f ∗ is the homogenization of f in KhXi[t] with respect to t. It follows that B ∗ − hLM(G ∗ )i = {tr w | w ∈ B − hLM(G)i, r ∈ N}.

(5)

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Next, using the definition of Ie and I ∗ , it is straightforward that the canonical K-algebra epimorphism ϕ : KhX, T i → KhXi[t], with ϕ(Xi ) = Xi , 1 ≤ i ≤ n, and ϕ(T ) = t, induces the K-algebra isomorphism ϕ e −→ KhXi[t]/hI ∗ i KhX, T i/hIi ∼ =

which yields immediately the isomorphism ϕ e −→ KhXi[t]/hG ∗ i. KhX, T i/hGi ∼ =

Hence, by the foregoing (5) and the K-isomorphism ϕ, the set (B ∗ − hLM(G ∗ )i) e = {T r w | w ∈ B − hLM(G)i, r ∈ N} e Moreover, by the gives rise to a K-basis for the quotient algebra KhX, T i/hGi. formula (4) given above, we obtain e = {LM(g), Xi T | g ∈ G, 1 ≤ i ≤ n}. LM(G) e the remainder F G of F Thus, implementing the division by Ge for each F ∈ hIi, has a presentation X e G F = λj T rj wj with λj ∈ K ∗ , T rj wj ∈ (B ∗ − hLM(G ∗ )i) e . j

But by the assertion on the subset (B ∗ − hLM(G ∗ )i) e made above, no element of e e Therefore, Ge is a e other than 0. So F G = 0 for all F ∈ hGi. such form is in hIi e e Gr¨obner basis for the ideal hIi with respect to (B, ≺ T -gr ). ¤ We call the Gr¨obner basis Ge obtained in Proposition 8.2 the non-central homogenization of G in KhX, T i with respect to T , or Ge is a non-central homogenized Gr¨obner basis with respect to T . Let the notations be as in the foregoing Propositions 8.1 and 8.2. We are now ready to derive the results recognizing certain structure properties of the Rees e of A = KhXi/I in terms of both the Gr¨obner bases G and G. e algebra A 8.3. Theorem. Let G = {g1 , . . . , gs } be an LM-reduced finite Gr¨obner basis of I in KhXi, and let Γ(LM(G)) be the Ufnarovski graph of G as defined in Section 6. The following statements hold: e has exponential growth if and only if the graph Γ(LM(G)) has two differ(i) A e has polynomial growth of degree ent cycles with a common vertex. Otherwise, A e d+1, or equivalently, GK.dim A = d+1, where d is, among all routes of Γ(LM(G)), the largest number of distinct cycles occurring in a single route. (ii) With ` = max{l(ui ) | ui ∈ LM(G)}, if each monomial u ∈ B − hLM(G)i e is semi-prime. with 1 ≤ l(u) ≤ ` is cyclic, then A (iii) With ` = max{l(ui ) | ui ∈ LM(G)}, if the following two conditions are satisfied:

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(a) any u ∈ B−hLM(G)i with l(u) < `−1 is a suffix of some vertex of Γ(LM(G)); (b) for any two vertices vi and vj of Γ(LM(G)), there exists a route from vi to vj and vice versa, e is prime or Noetherian prime of GK dimension 2. then A e1 = F1 A represented Proof. (i) Let z be the homogeneous element of degree 1 in A by the identity element 1 of A (note that 1 ∈ F0 A by our fixed convention). Then it is straightforward that z is not a divisor of zero and it is contained in the center e moreover, the localization of A e at the multiplicative subset {Z k | k ∈ N} of A; −1 is isomorphic to A[t, t ], the ring of Laurent series over A. Thus, by [25, Chape = GK.dim A + 1. Hence, ter 8, Proposition 2.13 and Corollary 2.15], GK.dim A the assertion follows from Proposition 5.2 and the Ufnarovski theorem [32] (see Section 6). (ii) & (iii) Let the element z be as indicated in the proof of (i) above. Note e say a = ap + ap−1 + ap−2 + · · · + a0 with aj ∈ A ej , then for some that if a ∈ A, 0 e a ∈ A, a = (ap + ap−1 z + ap−2 z 2 + · · · + a0 z p ) + (1 − z)a0 . So by sending z to 1 in A, it is easy to verify (or see a more general argument in e the next section) that A ∼ − zi and the ideal h1 − zi does not contain any = A/h1 e Thus, the (semi-)primeness of A implies the nonzero homogeneous element of A. e e (exercise). N-graded (semi-)primeness of A and hence the (semi-)primeness of A Therefore, the assertion follows now from Theorems 6.2 and 6.4. ¤ e may also be recognized in The next result shows that the Noetherianity of A an algorithmic way in certain cases. 8.4. Theorem. Let I and A = KhXi/I be as before, and let J = hΩi with Ω = {u1 , . . . , us } ⊂ B be a reduced finite subset. Suppose that J ⊆ hLM(I)i (for instance, Ω ⊆ LM(I)). If there is no edge entering (leaving) any cycle of the e is left (right) Noetherian of GK dimension not Ufnarovski graph Γ(Ω), then A exceeding 2. Proof. By Theorem 6.1, the associated N-graded algebra GN (A) is left (right) Noetherian of GK dimension not exceeding 1. Since the topology defined on A by the N-filtration F A is complete, it follows from [23, Chapter II, Proposition 1.2.3] e is left (right) Noetherian. Finally, the above Theorem 8.3 implies that the that A e is not exceeding 2. GK dimension of A ¤ Recall from [23, Chapter I, Corollary 7.6] that if R is a Z-filtered ring with e is left and right Noetherian, then filtration F R such that R e ≤ {1 + gl.dim R, 1 + gl.dim GZ (R)} gl.dim R and the equality holds if gl.dim GZ (R) is finite.

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e of A = KhXi/I, we Concerning the global dimension of the Rees algebra A have the following computational result without the assumption on Noetherianity e of A. 8.5. Theorem. Let I and A = KhXi/I be as before. If I is generated by an LM-reduced Gr¨obner basis G such that the graph ΓC (LM(G)) of n-chains does e ≤ d + 1. not contain any d-chain, then gl.dim A Proof. Considering the non-central homogenized Gr¨obner basis Ge of G in KhX, T i, it follows from the proof of Proposition 8.2 that e = {LM(g), Xi T | g ∈ G, 1 ≤ i ≤ n}. LM(G) Thus, it is easy to see that the graph of n-chains ΓC (LM(G)) of G is a subgraph e of G, e and that of the graph of n-chains ΓC (LM(G)) e e (a) the graph ΓC (LM(G)) of G has no edge of the form T → v for all v ∈ Ve , e where Ve is the set of vertices of ΓC (LM(G)); e contains the edge (b) if v ∈ Ve is of the form v = sXj , s ∈ B, then ΓC (LM(G)) v → T; e is of the form (c) any (d+1)-chain in ΓC (LM(G)) 1 → Xi → v1 → v2 → · · · → vd−1 → T, where 1 → Xi → v1 → v2 → · · · → vd−1 is a d-chain in the graph ΓC (LM(G)). e≤d+1 Therefore, if ΓC (LM(G)) does not contain any d-chain, then gl.dim A by Theorem 6.5. ¤ Combining Example 1 in Section 6, it follows from Corollary 6.6 and the above Theorem 8.5 (in particular, its proof) that the following corollary is obtained. 8.6. Corollary. Let the free K-algebra KhXi = KhX1 , . . . , Xn i be equipped with a weight N-gradation Ω ⊆ {Xj Xi | 1 ≤ i < j ≤ n} or Ω ⊆ {Xi Xj | 1 ≤ i < j ≤ n}, and let G be any Gr¨obner basis in KhXi with respect to some N-graded monomial e of the algebra ordering ≺gr such that LM(G) = Ω. Consider the Rees algebra A A = KhXi/hGi with respect to the N-filtration F A induced by the weight Ngrading filtration F KhXi of KhXi. Then e ≤ n + 1, and (i) gl.dim A e ≤ n + 1, and the equality holds if Ω = {Xj Xi | 1 ≤ i < j ≤ n} (ii) GK.dim A or Ω = {Xi Xj | 1 ≤ i < j ≤ n}. Concerning the Koszulity of Rees algebras, by the well-known fact that any quadratic Gr¨obner basis in KhXi defines a (classical) homogeneous 2-Koszul algebra (see [2] and [19]), Proposition 8.2 makes the next result clear.

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8.7. Theorem. Let the free K-algebra KhXi = KhX1 , . . . , Xn i be equipped with the natural N-filtration F KhXi, and let G ⊂ F2 KhXi − F1 KhXi be a Gr¨obner basis with respect to some admissible system (B, ≺gr ), where ≺gr is an N-graded monomial ordering on the standard K-basis B of KhXi. Then the Rees algebra e of the algebra A = KhXi/hGi with respect to the natural N-filtration F A is A homogeneous 2-Koszul. Finally, we clarify the relation between Rees algebras, PBW-deformations and regular central extensions (cf. [10], [13], [20], [11]). We introduce the principle of treatment in a little more general setting. Let A e=L e be a Z-filtered K-algebra with Z-filtration F A = {Fn A}n∈Z , and let A n∈Z An e with An = Fn A be the Rees algebra of A. Note that the identity element 1 of A is contained in F0 A by our fixed convention. If we write z for the homogeneous e1 = F1 A represented by 1, then it is clear that z is element of degree 1 in A e and z is not a divisor of zero. Furthermore, note that contained in the center of A e ej , then for some a0 ∈ A, e if a ∈ A, say a = ap + ap−1 + ap−2 + · · · + a0 with aj ∈ A a = (ap + ap−1 z + ap−2 z 2 + · · · + a0 z p ) + (1 − z)a0 . Hence, it is straightforward that the maps defined by z 7→ 1, respectively z 7→ 0, yield two Z-graded K-algebra isomorphisms: e e A e∼ A/h1 − zi ∼ = A, respectively A/z = GZ (A),

(6)

where GZ (A) is the associated Z-graded algebra of A with respect to F A. L Turning to an opposite direction, let D = n∈Z Dn be a Z-graded K-algebra and suppose that D has a central element z ∈ D1 , i.e., z is a homogeneous element of degree 1 and z is contained in the center of D. Then we have the Z-filtered Kalgebra A = D/h1−zi with the Z-filtration F A induced by the Z-grading filtration F D of D, that is, F A = {Fn A}n∈Z with µM ¶. h1 − zi. Fn A = Di + h1 − zi i≤n

Inspired by the isomorphisms in (6) above, the proposition given below answers when D serves as the Rees algebra of A with respect to F A, or in other words, e replaced by D. when an analogue of the foregoing (6) holds with A 8.8. Proposition. With notation as above, the following statements are equivalent: (i) z is not a divisor of zero in D. (ii) h1 − zi ∩ Dn = {0} for all n ∈ Z. e (iii) D ∼ = A. (iv) D/zD ∼ = GZ (A).

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Proof. Since z is a central homogeneous element of degree 1, if d ∈ D, say Pp d = i=1 dj with dj ∈ Dj , then d = (dp + dp−1 z + dp−2 z 2 + · · · + d0 z p ) + (1 − z)d0 for some d0 ∈ D. Considering the maps defined by z 7→ 1, respectively z 7→ 0, it is an easy exercise to complete the proof. Otherwise, one is referred to [21, Chapter I, Section 4] for finding similar argument. ¤ Now let KhXi = KhX1 , . . . , Xn i be the free K-algebra generated by X = {X1 , . . . , Xn }, and let G = {g1 , . . . , gs } be a finite set of N-homogeneous elements with respect to the natural N-gradation of KhXi. For any fixed f1 , . . . , fs ∈ KhXi satisfying d(fi ) < d(gi ), 1 ≤ i ≤ s, put G = {gi + fi | 1 ≤ i ≤ s}, and let KhXi[t] be the polynomial K-algebra in the commuting variable t over KhXi. Then we have the following three algebras: R = KhXi/hGi,

A = KhXi/hGi,

D = KhXi[t]/hG ∗ i,

where G ∗ is the central homogenization of G in KhXi[t] with respect to t (as described before). Recall that A is called a deformation of the N-graded algebra R (see Section 3), and also note that with respect to the N-gradation of D induced by the mixed N-gradation of KhXi[t], the image t of t in D is an N-homogeneous element of degree 1 and t is contained in the center of D. In the literature (e.g., see [20] and [11]), the N-graded algebra D is called a central extension of R associated to A; and if furthermore t is not a divisor of zero, then D is called a regular central extension. 8.9. Proposition. With notation as abopve, the following statements are equivalent: (i) D is a regular central extension of R associated to A. e where the latter is the Rees algebra of A with respect to the natural (ii) D ∼ = A, N-filtration F A. (iii) G is an N-standard basis for the ideal hGi in the sense of Section 2. (iv) A is a PBW-deformation of R, i.e., R ∼ = GN (A) with respect to the natural N-filtration F A. Proof. Putting I = hGi, note that G ∗ ⊂ I ∗ . By referring to Theorem 2.3, the equivalence follows now from Proposition 8.8. ¤ Next, combining Proposition 8.1, the following result is clear. 8.10. Theorem. With notation as before, if G is a Gr¨obner basis with respect to some admissible system (B, ≺gr ) of KhXi, where ≺gr is an N-graded monomial

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ordering on the standard K-basis B of KhXi, then D is a regular central extension of R associated to A. We finish this paper by examining a comprehensive example that provides a class of algebras similar to the down-up algebra in the sense of [4] and [5]. Example 6. Let the free K-algebra KhXi = KhX1 , X2 i be equipped with a weight N-gradation such that d(X1 ) = n1 and d(X2 ) = n2 , and let G = {g1 , g2 } be any Gr¨obner basis with respect to some N-graded monomial ordering ≺gr on the standard basis B of KhXi such that LM(g1 ) = X12 X2 and LM(g2 ) = X1 X22 . Consider the algebra A = KhXi/hGi which is equipped with the N-filtration F A induced by the weight N-grading filtration F KhXi. With notation as before, and e ∼ e as well as AB = bearing GN (A) ∼ = KhXi/hLH(G)i, A = KhX1 , X2 , T i/hGi, LH KhXi/hLM(G)i in mind, the following statements hold: N e (i) GK.dim AB LH = GK.dim A = GK.dim G (A) = 3, and GK.dim A = 4. B N e are semi-prime. (ii) ALH is Jacobson semi-simple. Hence, A, G (A) and A e ≤ 4. (iii) gl.dim A ≤ gl.dim GN (A) ≤ gl.dim AB ≤ 3, and gl.dim A LH (iv) If n1 = n2 = 1 and g1 = X12 X2 + H1 + F1 ,

g2 = X1 X22 + H2 + F1 ,

where H1 , H2 are homogeneous elements of degree 3, while both F1 , F2 have total e degree < 3, then A is a PBW-deformation of AN LH , and A is a regular central N extension of ALH associated to A. N (v) If n1 = n2 = 1, then AB LH and G are homogeneous 3-Koszul, and hence A is non-homogeneous 3-Koszul. e have Hilbert series (vi) If n1 = n2 = 1, then GN (A) and A HGN (A) (t) =

1 1 − 2t + 2t3 − t4

and HAe(t) =

1 , 1 − 3t + 2t2 + 2t3 − 3t4 + t5

respectively. In particular, all results mentioned above hold for the down-up algebra A(α,β,γ) (in the sense of [4] and [5]) which is defined by the relations g1 = X12 X2 − αX1 X2 X1 − βX2 X12 − γX1 , g2 = X1 X22 − αX2 X1 X2 − βX22 X1 − γX2 .

α, β, γ ∈ K.

Proof. Put Ω = {X12 X2 , X1 X22 }. The Ufnarovski graph Γ(Ω) of Ω is presented by > X2 X1 X1 X2 < > X12 ∧

ª

X22

ª

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It turns out that the assertions (i) and (ii) hold. Further, let us look at the graph of n-chains ΓC (Ω) of Ω: X22 % X1 % & 1 X1 X2 & . X2 which shows that Ci−1

 {X1 , X2 }    {X1 X22 , X12 X2 } = 2 2   {X1 X2 }  ∅

i = 1, i = 2, i = 3, i ≥ 4.

Moreover, note that for N = 3, under the jump map we have ζ(1) = 1, ζ(2) = 3 and ζ(3) = 4. Since AB LH = KhXi/hLM(G)i, it follows that (iii) and (v) hold. The assertion (iv) follows from the fact that G and Ge are Gr¨obner bases in KhX1 , X2 i ei−1 denotes the set of (i−1)-chains and KhX1 , X2 , T i, respectively. For i ≥ 1, let C of e = {X12 X2 , X1 X22 , X1 T, X2 T }. LM(G) Then it is straightforward that   {X1 , X2 , T }    2 2   {X1 T, X2 T, X1 X2 , X1 X2 } 2 2 2 2 e Ci−1 = {X1 X2 , X1 X2 T, X1 X2 T }    {X12 X22 T }    ∅

i = 1, i = 2, i = 3, i = 4, i ≥ 5.

ei−1 So the last assertion (vi) is derived by using [2, formula (16)] and the Ci , C obtained above. Finally, the conclusion on down-up algebras is obtained by checking the fact that the set of defining relations {g1 , g2 } of A = A(α, β, γ) is a Gr¨obner basis with respect to the N-graded lexicographic ordering X2 ≺grlex X1 . ¤ Remark. Considering the down-up algebra A = A(α, β, γ), in the case that β 6= 0, a direct proof for the homogeneous 3-Koszulity of GN (A) and the nonhomogeneous 3-Koszulity of A was given by R. Berger and V. Ginzburg in [7]. Also, it was proved that A is a Noetherian domain if and only if β 6= 0; moreover, if it is the case, A is Auslander regular of global dimension 3 (E. Kirkman, I.M. Musson and D.S. Passman, Proc. Amer. Math. Soc. 11 (127) (1999) 3161–3167); and in the case that β = 0, it was proved that the non-Noetherian A is prime of global dimension 3, and unless α = γ = 0 it is never primitive (E. Kirkman, J. Kuzmanovich, Comm. Algebra 11 (28) (2000) 5255–5268).

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