California Polytechnic State University San Luis Obispo Mathematics Department Senior Project
Introduction to Stanley-Reisner Theory
Supervisor: Dr. Benjamin Richert
Author: Chad Duna
June 10, 2013
Approval Page
Title:
Introduction to Stanley-Reisner Theory
Author:
Chad K. Duna
Date Submitted:
June 2013
Senior Project Advisor
Signature
Mathematics Department Chair
Signature
i
Contents 1
Simplicial Complexes
2
Stanley-Reisner Rings 2.1 Concerning Graded Rings . . 2.2 Hilbert Series . . . . . . . . . 2.2.1 Technical Results . . . 2.2.2 f-vectors and h-vectors
3
1
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
2 2 4 5 6
Applications of Stanley-Reisner Theory 8 3.1 Dimension of a Stanley-Reisner Ring . . . . . . . . . . . . . . . . . . . . . 9 3.2 Connections to Algebraic Geometry . . . . . . . . . . . . . . . . . . . . . 10
References
12
ii
Introduction to Stanley-Reisner Theory Chad Duna
A neat tool for studying algebraic varieties is that of Stanley-Reisner rings. A Stanley Reisner ring is a quotient ring of the form RI where R is a polynomial ring k[x1 , x2 , ..., xn ] (k a field) and I is a square-free monomial ideal of R. There is a one-to-one correspondence between the set of square-free monomial ideals of the polynomial ring R = k[x1 , x2 , ..., xn ] and the set of abstract simplicial complexes on n vertices. Much can be learned about the ring RI by looking at the combinatorial structure of the abstract simplicial complex corresponding to I. One warning to the reader: As is fairly common in algebra, a fair amount notational abuse will take place in this paper. Don’t let it bother you too much, as long you understand what we mean.
1 Simplicial Complexes Definition: Given a finite set V , an abstract simplicial complex on V is a subset ∆ of the power set of V, such that if Y ∈ ∆ and X ⊂ Y , then X ∈ ∆. The elements of ∆ are called faces. The dimension of a face X ∈ ∆ is |X| − 1. The dimension of ∆, denoted dim(∆), is the dimension of the largest face in ∆. It is common when referring to members of a simplicial complex to do away with set notation. For example, if {a, b, c} ∈ ∆ this would normally be written as abc ∈ ∆. Below is a drawing of the simplicial complex ∆ = {∅, a, b, c, d, ab, bc, cd, bd, bcd} on the vertex set V = {a, b, c, d}.
In this example, dim(∆) = 2. Throughout the paper, we will be referring back to this simplicial complex for examples. Now that we are familiar with simplicial complexes, we look to associate them with square-free monomial ideals.
1
Definition: An ideal I of the polynomial ring k[x1 , x2 , ..., xn ] is a monomial ideal if I is generated by monomials. I is called a square-free monomial ideal if I is a monomial ideal with each minimal generator of the form xα1 1 xα2 2 ...xαnn where each αi is either 0 or 1.
2 Stanley-Reisner Rings Definition: Let V = {1, ..., n}, and let ∆ be an abstact simplicial complex on V . The Stanley-Reisner Ideal of ∆, I∆ , is:
/ ∆). I∆ = (xi1 xi2 ... xir |{i1 , i2 , ..., ir } ∈
That is, I∆ is generated by the nonfaces of ∆. Furthermore, the minimal nonfaces of ∆ form a minimal set of generators of I∆ . Example: Using ∆ = {∅, a, b, c, d, ab, bc, cd, bd, bcd} from a previous example, I∆ = (ad, ac). Definition: Given an abstract simplicial complex ∆ on a set of n vertices, the Stanley-Reisner Ring, k[∆], is:
k[∆] =
k[x1 , ..., xn ] I∆
As we will show soon, k[∆] is a graded ring.
2.1 Concerning Graded Rings Definition: A ring R is said to be N-graded if R =
L i≥0
holds: 1. Each Ri is an additive group. 2. If a ∈ Ri and b ∈ Rj , then ab ∈ Ri+j . An element a ∈ Ri is called homogeneous of degree i.
2
Ri such that the following
In this paper, we will only be dealing with N-graded rings and so most of the time they will just be referred to as graded. Every ring R can be regarded as a graded ring with R0 = R and Ri = 0 for all i > 0. Example: Perhaps the simplest non-trivial example of a graded ring is a polynomial ring in one indeterminate graded by degree. Let R be any ring, and let Ri = {zxi |z ∈ R}. It is easy to check that this is a grading of R[x]. This can be extended to polynomial rings with more than one indeterminate using the idea of total n Q i xm degree. Given a monomial i ∈ R[x1 , ..., xn ], its total degree is defined to be i=1
m1 + ... + mn . We will call a polynomial homogeneous if each monomial in it has the same total degree. For example, in the ring k[x, y] the polynomial x2 + xy + y 2 is homogeneous while the polynomial x3 + y 2 is not. We can grade R[x1 , ..., xn ] in the following way:
R[x1 , ..., xn ] =
M
Ri , where Ri is the group of all homogeneous polynomials of degree i.
i≥0
To see explicitly how a Stanley-Reisner ring is graded, a discussion of homogeneous and graded ideals is necessary. L
Ri is homogeneous if I is generated L by homogeneous elements of R. I is called graded if I = Ii , where Ii = Ri ∩ I.
Definition: An ideal, I, in a graded ring R =
i≥0
i≥0
L
Proposition: Let I be a finitely generated ideal in the graded ring R =
Ri . If I is
i≥0
homogenous, then I is graded. Proof: Suppose I is homogeneous. Let I = (ai1 , ..., aik ), where aij ∈ Rij for 1 ≤ j ≤ k. Suppose f ∈ I. Then f = s1 ai1 + ... + sk aik for some s1 , ...sk ∈ R. Let Sj = {l ∈ N| sj k S P has a homogeneous part in Rl } and let S = Sj . Then for each 1 ≤ j ≤ k, sj = rjl j=1
l∈S
k P k P P P where each rjl ∈ Rjl (many of the rjl will be 0). So f = ( rjl )aij = rjl aij . j=1 l∈S
j=1 l∈S
Since R is graded, for every 1 ≤ j ≤ k and l ∈ S, rjl aij ∈ Rjl +ij and because I is an ideal,Lrjl aij ∈ I. So for every aij ∈ (Rjl +ij ∩ I). Therefore L 1 ≤ j ≤ k and l ∈ S, rjlL f∈ (Ri ∩ I). Trivially, (Ri ∩ I) ⊂ I. Thus I = (Ri ∩ I). I.e. I is graded. i≥0
i≥0
i≥0
Corollary: I∆ is a graded ideal. Proposition: If I =
L
Ii is a graded ideal in the ring R =
i≥0
L i≥0
3
Ri . Then
R I
=
L i≥0
Ri Ii .
Proof: L Ri Before we begin, it is worth observation that it is perhaps not obvious that Ii is even a ring. But it is not hard to check this. Multiplication in this ring can be i≥0
done by multiplying elements as if they were in R and then taking the equivalence class L Ri of each homogeneous part in the product. Define the function φ : R → by Ii i≥0
φ(ri ) = ri for each ri ∈ Ri where ri denotes the equivalence class of ri in RIii . Extend φ linearly. It is clear that φ is surjective. Suppose r ∈ R such that φ(r) = 0. Then 0 = φ(r) = φ(ri1 + ... + rin ) = φ(ri1 ) + ... + φ(rin ) = ri1 + ... + rin where rij is the homogeneous part of r belonging to Rij . Then rij = 0 for all 1 ≤ j ≤ n. That is, rij ∈ Iij . Therefore r ∈ I. Now, suppose s ∈ I. Then s = si1 + ... + sin where sij is the homogeneous part of s belonging to Iij . So φ(s) = φ(si1 + ... + sin ) = φ(si1 ) + ... + φ(sin ) = si1 + ... + siL n = 0. So, s ∈ ker(φ). Ri Thus I = ker(φ). By the First Isomorphism Theorem, RI = Ii . i≥0
Corollary: k[∆] is a graded ring. Using the proposition, we can identify the structure of k[∆]. Let k[∆]i denote the ith graded piece of k[∆]. Then k[∆]i is made up of all degree i homogeneous polynomials consisting of monomials which are not divisible by a minimal nonface. In our running example, k[∆]1 is the k-span of the set {a, b, c, d} and k[∆]2 is the k-span of the set {a2 , b2 , c2 , d2 , ab, bd, bc, cd}.
2.2 Hilbert Series A useful tool in study graded rings is that of the Hilbert series which encodes information about the “size” of the graded pieces. In the case of a Stanley-Reisner ring, the Hilbert series will always be a rational function and properties such as degree and dimension of the ring can be extracted directly from the form of the rational function. To begin discussing Hilbert Series, we will first define the Hilbert function. Definition: Given R =
L
Ri a finitely generated graded ring over a field k (each
i≥0
graded piece is a finite dimensional k-vector space and k ⊂ R0 ), define the Hilbert function of R, HR : (N ∪ {0}) → (N ∪ {0}) by HR (n) = dimk Rn where dimk Rn denotes the dimension of Rn as a k-vector space. Example: Hk[∆] (1) = 4 and Hk[∆] (2) = 8. Hk[∆] (n) will always be the number of monomials in the nth graded piece of k[∆]. Definition: Given R =
L
Ri a finitely generated graded ring over a field k, define the
i≥0
4
Hilbert Series of R to be: HSR (t) =
X
HR (n)tn .
n≥0
In general the Hilbert series can be difficult to calculate by hand. Luckily in the case of Stanley-Reisner rings, the Hilbert Series to compute. The goal is to show that P is teasy ( 1−t )|f | . To do so, we need some framework the Hilbert Series of k[∆] is equal to f ∈∆
and terminology. 2.2.1
Technical Results
In the following, V = {1, 2, 3, ..., n} is a vertex set and ∆ is a simplicial complex on V . Given a monomial m ∈ k[x1 , ..., xn ] define the support of m, denoted Q by supp(m), to be f xi . In a brazen supp(m) = {i ∈ V | xi divides m}. Given a subset f of V let x = i∈f
and sexy abuse of notation we will adopt the use of k[f ] to denote the polynomial ring with indeterminates indexed by the elements of f . For example, if f = {1, 3, 5} then k[f ] = k[x1 , x3 , x5 ]. Finally, given a face f ∈ ∆, Tf = {rxf | r ∈ (k[f ] − {0})}. We are looking to determine the dimension of the graded pieces of k[∆] which amounts to counting monomials. The usefulness of Tf is that when run across the faces of ∆, the Tf partition k[∆]. Thus the counting problem is reduced to counting monomials in each of the Tf . To make things easier for ourselves, we will get rid of the unnecessary parts of Tf . Define Tf∗ to be the collection of monomials in Tf and define k[∆]∗ to be the collection of monomials of positive degree in k[∆]. Notice that the monomial 1 did not make the cut for k[∆]∗ . The reason to do this is to avoid unnecessary complexity in the following argument. To account for 1 missing, we will also be omiiting the empty face of ∆ from discussion. The counting works out nicely since the Hilbert Series for k[∅] is 1 and therefore only counts the monomial 1. Proposition: Given a simplicial complex ∆ on the vertex set V = {1, ..., n}, k[∆]∗ is partitioned by {Tf∗ }f ∈(∆−∅) . Proof: Suppose m ∈ (Tf∗ ∩ Tf∗0 ) with f 6= f 0 . By definition of Tf , supp(m) = f . By the same token, supp(m) = f 0 . Thus, f = f 0 a contradiction because the faces of ∆ are squarefree. Therefore if f 6= f 0 , then (Tf∗ ∩ Tf∗0 ) = ∅. Suppose f ∈ (∆ − {∅}) and m = rxf ∈ Tf ∗. If m = 0 in k[∆], then m is divisible by some minimal generator, s, of I∆ since m is a monomial and I∆ is a monomial ideal. As I∆ is square-free, s = xsupp(s) . Since s | m, supp(s) ⊂ supp(m) = f . Because f is a face, it follows that supp(s) is also a face. This is a contradiction, is a S since∗ s being∗ a generator of I∆ means supp(s) non-face. This shows that Tf ⊂ k[∆] . Now, suppose m ∈ k[∆]∗ . Then f ∈(∆−∅)
m = rxsupp(m) for some r ∈ (k[supp(m)] − {0}). If supp(m) is not a face of ∆ then xsupp(m) ∈ I∆ . But then m = rxsupp(m) ∈ I∆ . This is a contradiction since m 6= 0 in ∗ k[∆]. Thus supp(m) ∈ ∆. I.e. m ∈ Tsupp(m) . Finally it is proven that 5
S f ∈(∆−∅)
Tf∗ = k[∆]∗ .
With this mighty beast slain, we may now easily compute the Hilbert Series for Stanley-Reisner rings. 1 d Note that the coefficient of ti in the power series of ( 1−t ) counts the number of degree i monomials in the polynomial ring k[x1 , ..., xd ]. Therefore the coefficient of ti−|f | in 1 |f | ) is the number of degree i − |f | monomials in k[f ]. Thus the number of degree i ( 1−t t |f | monomials in Tf∗ is the coefficient of ti in ( 1−t ) . Since the Tf∗ ’s partition k[∆]∗ , the P t |f | coefficient of ti in ( 1−t ) , is the number of degree i monomials in k[∆]∗ . Now f ∈(∆−∅)
t |∅| note that Hk[∆] (0) = 1 = ( 1−t ) and we arrive at:
HSk[∆] (t) =
X
Hk[∆] (n)tn =
X
(
f ∈∆
n≥0
t |f | ) 1−t
This formula is much nicer to deal with, however we can do better. To do so, we will introduce some more concepts related to the simplicial complex.
2.2.2
f-vectors and h-vectors
Let ∆ be a simplicial complex on the vertex set V = {1, ..., n}. The f-vector of ∆ is the sequence (f0 , f1 , f2 , ...) where fi is the number of degree i − 1 faces in ∆. Since V is finite, the sequence will zero out eventually. It is customary to just write the non-zero part of the f-vector. f0 is always 1, because the empty set is the only degree −1 face of ∆. Example: Let’s go back to our running example with ∆ =
The f-vector of ∆ is (1, 4, 4, 1). Remember that degree 0 faces are points, degree 1 faces are lines, and so on. Another important idea is the h-vector of ∆. Let d = dim(∆) + 1. Then define the
6
f-polynomial by f∆ (x) =
d P
fi xd−i . The h-vector is (h0 , h1 , ...) where hi is the
i=0
coefficient of xd−i in f∆ (x − 1). This is a tedious process to do by hand, however there is a much more interesting way to go about computing the h-vector [4]. Set up a pyramid with the f-vector running along the right side and 1’s running along the left side as done below.
Now proceed to fill in the the pyramid down to below the line in the following manner. Fill in an entry with the entry above and to the right minus the entry above and to the left. The h-vector will be what is produced below the line.
Thus the h-vector of ∆ is (1, 1, −1). Earlier we saw that the Hilbert Series of k[∆] was equal to
P
t |f | ( 1−t ) . By collecting
f ∈∆
faces of the same dimension, the formula becomes:
HSk[∆] (t) =
X i≥0
This formula can be simplified as follows:
7
fi
t 1−t
i
X i≥0
fi
t 1−t
i
−i
t(1 − t) dim(∆)+1 fi = t(1 − t) i≥0 dim(∆)+1 dim(∆)+1−i X 1 t fi = −1 t 1−t i≥0 dim(∆)+1 t 1 = f∆ −1 1−t t dim(∆)+1 dim(∆)+1 X t = hi (t−1 )dim(∆)+1−i 1−t X
1−t t
i=0
=
t 1−t
dim(∆)+1 dim(∆)+1 X i=0
dim(∆)+1 P
=
hi ti t−(dim(∆)+1)
hi ti
i=0
(1 − t)dim(∆)+1
.
Therefore the Hilbert Series for k[∆] is:
HSk[∆] (t) =
X
fi
i≥0
t 1−t
dim(∆)+1 P
i =
(1
hi ti i=0 − t)dim(∆)+1
(Equation 1).
Example: Using the running example, HSk[∆] (t) = 1 + 4(
t t 2 t 3 1 + t − t2 ) + 4( ) +( ) = . 1−t 1−t 1−t (1 − t)3
To see a more complete treatment of f-vectors and h-vectors see [1] [5].
3 Applications of Stanley-Reisner Theory As alluded to before, Stanley-Reisner Theory is more than just a cool thing to think about, it can be used to study rings and algebraic varieties. The dimension of a ring and the degree of an algebraic variety can be found by looking at the corresponding simplicial complex or its Hilbert Series.
8
3.1 Dimension of a Stanley-Reisner Ring Definition: A chain of prime ideals P0 ( P1 ( ... ( Pn is said to have length n. The dimension of a ring R is the supremum of the lengths of all chains of prime ideals in R (this is possibly infinite). It turns out that the dimension of k[∆] is the order of the pole at t = 1 in HSk[∆] (t) [4]. We should verify that dim(∆)+1 X hi 6= 0 i=0
to confirm that the order of the pole in Equation 1 is truly the exponent of the denominator (since this would verify that the numerator is non-zero at 1). Recall that dim(∆)+1
X
f∆ (t − 1) =
hi tdim(∆)+1−i .
i=0
Then
f∆ (
1 − t dim(∆)+1 1 )t = f∆ ( − 1)tdim(∆)+1 t t dim(∆)+1−i dim(∆)+1 X 1 dim(∆)+1 =t hi t i=0
dim(∆)+1
=
X
hi ti .
i=0
Thus dim(∆)+1
X i=0
dim(∆)+1
hi =
X
hi 1i
i=0
1 − 1 dim(∆)+1 )1 1 = f∆ (0)
= f∆ (
= fdim(∆)+1 . Since fdim(∆)+1 is the number of maximal faces of ∆, it is not zero. Therefore the order of the pole at t = 1 in HSk[∆] (t) is the exponent in the denominator. That is, the dimension of k[∆] is dim(∆) + 1.
9
3.2 Connections to Algebraic Geometry For those unfamiliar with the term algebraic variety, I will provide a quick description of the basic notions surrounding the topic. For proofs, justifications, and a deeper discourse on the topic interested readers can see [2]. We will be working in An , affine n-space over an algebraicly closed field k. Given a subset S of k[x1 , ..., xn ] define the zero set of S by: Z(S) = {x ∈ An | f (x) = 0 for all f ∈ S}.
A subset V of An such that V = Z(S) for some S ⊂ k[x1 , ..., xn ] is called an algebraic set. In algebraic geometry, the Zariski Topology is an important concept. The Zariski topology on An is defined by declaring the closed sets to be the algebraic sets in An . It is a fun exercise to verify that this is in fact a topology. An algebraic set V is irreducible if it cannot be written as the union of two proper algebraic subsets. Irreducible algebraic sets are called algebraic varieties. We will see that as a consequence of Hilbert’s Nullstellensatz, algebraic varieties are in a one-to-one correspondence with radical ideals of k[x1 , ..., xn ]. An ideal is radical if f m ∈ I for some m ≥ 1 implies f ∈ I. Given an ideal I of k[x1 , ..., xn ], the radical of I is defined to be: √
I = {f ∈ k[x1 , ..., xn ]|f m ∈ I for some m ≥ 1}.
Given a subset V of An , denote the set of polynomials in k[x1 , ..., xn ] which vanish at all points of V by I(V ). It is easy not difficult to show that I(V ) is an ideal of k[x1 , ..., xn ]. Hilbert’s Nullstellensatz: If I is an ideal of k[x1 , ..., xn ] then √ I(Z(I)) = I . Hilbert’s Nullstellensatz, or commonly just called the Nullstellensatz is a very important theorem in algebraic geometry. One of its consequences is that varieties in An are in a one-to-one correspondence with the radical ideals of k[x1 , ..., xn ]. This lovely fact allows us to trade in an algebraic variety V for its corresponding coordinate ring k[x1 , ..., xn ] . I(V )
This is where Stanley-Reisner Theory gets into the mix. I(V ) is assured to be finitely generated since k[x1 , ..., xn ] is Noetherian. However it is possibly not monomial. To fix 10
this problem fix a monomial ordering and take the initial ideal, in(I(V )), of I(V ). This is the ideal generated by the leading terms of each generator of I(V ). If in(I(V )) is squarefree we can continue on by regarding it as I∆ for some simplicial complex ∆. Otherwise, in(I(V )) can be “made” square-free through the process of polarization. The general idea behind polarization is that you get rid of powers of variable by introducing new variables. For instance, if you want to make x2 square-free. You could adjoin a variable x0 to the ring. Then x2 can be regarded as xx0 . You can regain the original ring by modding out by x − x0 . After polarization, you are left with a square-free monomial ideal which as said above can be associated with a simplicial complex. Remarkably even after all the changes, the Stanley-Reisner ring contains a lot of information about our original variety. For instance the dimension and degree of the variety can be obtained from either the simplicial complex or the Hilbert series of the Stanley-Reisner ring [3][4]. Example: Let I be the ideal (x21 + x1 x2 , x1 x4 ) in k[x1 , x2 , x4 ] (k is an algebraicly closed field). If x21 + x1 x2 = 0, then x1 = 0 with x2 and x4 free, or x1 6= 0 with x2 = −x1 and x4 free. The solution to x1 x4 = 0 is the union of the planes x1 = 0 and x4 = 0. Then the variety V = Z(x21 + x1 x2 , x1 x4 ) is the intersection of the two solution sets. That is, V is the union of the plane x1 = 0 and the line x1 + x2 = 0 with x4 = 0. Let’s look at this variety and associated ideal using Stanley-Reisner theory. First, compute in(I). Using the lexographic monomial ordering, in(I) = (x21 , x1 x4 ). This ideal is not square-free, so polarize it. Polarization results in the ideal (x1 x0 , x1 x4 ) ⊆ k[x1 , x2 , x0 , x4 ]. This square-free ideal corresponds to a simplicial complex, namely the complex that we have been using as an example throughout the paper. Thus the Hilbert Series for k[∆] is the Hilbert Series for V . HSV (t) =
1 + t − t2 . (1 − t)3
The dimension of V is the order of the pole at t = 1 or equivalently deg(∆) + 1, that is dim(V ) = 3. Another property that can be measured with the Hilbert series is the degree of V . To obtain the degree of V either sum the coefficients of the numerator of the Hilbert series or count the number of maximal faces of ∆ (recall that these quantities are equal). The definition of the degree of a variety would require a lot more definitions to be made and is therefore left to be investigated by an interested reader.
11
References [1] Winfried Bruns and J¨ urgen Herzog. Cohen-Macaulay rings, volume 39 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, 1993. [2] David Eisenbud. Commutative algebra with a view toward algebraic geometry, volume 150 of Graduate Texts in Mathematics. Springer-Verlag, New York, 1995. [3] E. Miller and B. Sturmfels. Combinatorial Commutative Algebra. Graduate Texts in Mathematics. Springer, 2004. [4] Victor Reiner. Informal seminar on stanley-reisner theory. 2002. [5] Richard P. Stanley. Combinatorics and commutative algebra, volume 41 of Progress in Mathematics. Birkh¨ auser Boston Inc., Boston, MA, second edition, 1996.
12
