Big Cohen-Macaulay Modules over Mixed Characteristic Local Rings Jun-ichi NISHIMURA [email protected] Osaka Electro-Communication University

2014.10.12

Flow of Presentation 1

Homological Conjectures and Modifications

2

Integral Closures vs.Tight Closures

3

Structure Theorem for Mixed Characteristic Complete Local Rings

4

Hochster’s Observation

5

Flenner’s Bertini Theorem

6

Base Change 1

7

Frobenius Maps

8

Base Change 2

9

The First Step of Induction and Inductive Assumptions

10

Inductive Step 1: The Case when ℓ ≥ 2 and rℓ ≥ 2

11

Inductive Step 2: The Case when ℓ ≥ 2 and rℓ = 1 Jun-ichi NISHIMURA (OECU)

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Conjectures

Homological Conjectures

Homological Conjectures

Big Cohen-Macaulay Modules Conjecture If A is a local ring and x1 , . . . , xd is a system of parameters, then there exists an A-module M (not necessarily of finite type) such that x1 , . . . , xd is an M -sequence. That is, (x1 , . . . , xi )M : xi+1 A = (x1 , . . . , xi )M for 0 ≤ i < d, and

Jun-ichi NISHIMURA (OECU)

(x1 , . . . , xd )M ̸= M.

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Conjectures

Homological Conjectures

Monomial Conjecture Let A be a local ring and let x1 , . . . , xd be a system of parameters. Then, t+1 xt1 · · · xtd ̸∈ (xt+1 1 , . . . , xd )A for every positive integer t.

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Conjectures

Homological Conjectures

Direct Summand Conjecture Let R be a regular local ring and let S be a module-finite extension algebra of R. Then R is a direct summand of S as R-module.

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Conjectures

Homological Conjectures

Improved New Intersection Conjecture Let F : 0 → Fn → Fn−1 → · · · → F0 → 0 be a complex of finitely generated free A-modules such that F is not exact and that ℓA (Hi (F)) < ∞ for any i ̸= 0, and there exists x ∈ H0 (F) \ mH0 (F) such that ℓA (Ax) < ∞. Then n ≥ dim A.

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Conjectures

Homological Conjectures

New Intersection Conjecture Let F : 0 → Fn → Fn−1 → · · · → F0 → 0 be a complex of finitely generated free A-modules such that F is not exact and that ℓA (Hi (F)) < ∞ for any i. Then n ≥ dim A.

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Conjectures

Homological Conjectures

Intersection Conjecture If M ̸= 0, N are A-modules of finite type such that M ⊗A N has finite length, then dim N ≤ pdA M.

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Conjectures

Homological Conjectures

Homological Height Conjecture Let A → B be a homomorphism of Noetherian rings, and let M be an A-module of finite type and finite projective dimension. Let I = AnnA M and let Q be a minimal prime of IB. Then htQ ≤ pdA M.

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Conjectures

Homological Conjectures

Zerodivisor Conjecture Let A be a local ring and let M be an A-module of finite type and finite projective dimension. If x ∈ A is not a zerodivisor on M , then x is not a zerodivisor on A.

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Conjectures

Homological Conjectures

Bass’ Conjecture If a local ring A possesses a nonzero module T of finite type and finite injective dimension, then A is Cohen-Macaulay.

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Conjectures

Diagram of Conjectures

Diagram of Conjectures Big Cohen-Macaulay Modules ↓ Direct Summand ↔ Monomial ↔

Improved New Intersection ↓ New Intersection ↓

Homological Height ↔ Intersection ↙ Zerodivisor Jun-ichi NISHIMURA (OECU)

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↘ Bass 2014.10.12

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Conjectures

Historical Theorems

Historical Theorems

Theorem 1.1 (Peskine–Szpiro 1971, P. Roberts) New Intersection Conjecture holds if A is a local ring either of positive characteristic p or of essentially finite type over a field.

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Conjectures

Historical Theorems

Theorem 1.2 (Hochster 1973) Let A be a d-dimensional local ring of positive characteristic p, and let x1 , . . . , xd be any system of parameters for A. Then there exists a big Cohen-Macaulay module M for x1 , . . . , xd .

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Conjectures

Historical Theorems

Theorem 1.3 (Hochster 1973) Let A be a d-dimensional local ring such that Ared contains a field, and let x1 , . . . , xd be any system of parameters for A. Then there exists a big Cohen-Macaulay module M for x1 , . . . , xd .

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Conjectures

Historical Theorems

Theorem 1.4 (P. Roberts 1987) New Intersection Conjecture holds for any local ring.

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Conjectures

Historical Theorems

Theorem 1.5 (Heitmann 2002) Direct Summand Conjecture holds for any local ring of dimension at most 3.

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Conjectures

Historical Theorems

Theorem 1.6 (Hochster 2002) Big Cohen-Macaulay Conjecture holds for any local ring of dimension at most 3.

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Conjectures

Propositions and Theorem

Propositions and Theorem In this talk, we explain the following.

Proposition 1.7 Let A be a mixed characteristic d-dimensional local ring and let x1 , p, x3 , . . . , xd be a system of parameters for A. Then there exists a big Cohen-Macaulay module M for x1 , p, x3 , . . . , xd .

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Conjectures

Propositions and Theorem

Consequently,

Proposition 1.8 A mixed characteristic local ring has a big Cohen-Macaulay module.

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Conjectures

Propositions and Theorem

Combining Theorems 1.2, 1.3 and the above, we get

Theorem 1.9 Let A be a d-dimensional local ring and let x1 , . . . , xd be a system of parameters for A. Then there exists a big Cohen-Macaulay module M for x1 , . . . , xd .

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Conjectures

Modifications

Modifications Let A be a d-dimensional local ring. Take a system of parameters x1 , . . . , xd for A, that is abbreviated to x. Let M be an A-module with α ∈ M . Suppose that x1 m1 + · · · + xr+1 mr+1 = 0.

We then refer to ρ := (m1 , . . . , mr+1 ) ∈ M r+1 as a type r relation for x on M .

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Conjectures

If we put

   v :=  

mr+1 xr .. .

Modifications

    ∈ M ⊕ Ar , 

x1 we have canonical maps / M → M ⊕ Ar → (M ⊕ Ar ) Av =: M ′ . Let α′ be the image of α in M ′ . That is,   α  / 0   ∈ (M ⊕ Ar ) Av. . α′ :=   ..  0 Jun-ichi NISHIMURA (OECU)

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Conjectures

Modifications

Then we have a map (M, α) → (M ′ , α′ ), a map M → M ′ that takes α to α′ . We call (M ′ , α′ ) a first modification of (M, α) with respect to a type r relation ρ for x. In general, we may have a sequence M : (M, α) =: (M0 , α0 ) → (M1 , α1 ) → · · · → (Mt , αt ) where (Mℓ , αℓ ) is a modification of (Mℓ−1 , αℓ−1 ) with respect to a relation ρℓ on Mℓ−1 of type rℓ for x. We then say that (Mt , αt ) is a t th modification of (M, α) of type r := (r1 , . . . , rt ). Jun-ichi NISHIMURA (OECU)

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Conjectures

Modifications

Further, (Mt , αt ) is called a degenerate modification of (M, α) with respect to the system of parameters x1 , . . . , xd , when αt ∈ (x1 , . . . , xd )Mt . With notation above, we remark the following. Lemma 1.10 Let A be a d-dimensional local ring and let x1 , . . . , xd be a system of parameters for A, that is abbreviated to x. Then the following two conditions are equivalent: (1) A possesses an x-regular module M . (2) Every modification (Mt , αt ) of (A, 1) does not degenerate.

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Conjectures

Outline of Proof of Proposition 1.7

Outline of Proof of Proposition 1.7 1 2

3 4 5 6 7 8 9 10 11

Integral Closures vs.Tight Closures Structure Theorem for Mixed Characteristic Complete Local Rings Hochster’s Observation Flenner’s Bertini Theorem Base Change 1 Frobenius Maps Base Change 2 The First Step of Induction and Inductive Assumptions Inductive Step 1: The Case when ℓ ≥ 2 and rℓ ≥ 2 Inductive Step 2: The Case when ℓ ≥ 2 and rℓ = 1 Flow Chart of Induction (in construction)

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Integral Closures vs.Tight Closures

Integral Closures vs.Tight Closures

In this section, we gather a few basic facts on the integral closure of ideals in a ring and the tight closure of ideals in a Noetherian ring of prime characteristic p > 0.

For the details, we refer the reader to [7] and [16].

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Integral Closures vs.Tight Closures

Integral Closures

Integral Closures Definition 2.1 (Integral Closure) Let I be an ideal of a ring A. An element x ∈ A is said to be integral over I if there exist an integer n and elements ai ∈ I i , i = 1, . . . , n, such that xn + a1 xn−1 + a2 xn−2 + · · · + an−1 x + an = 0. Such an equation is called an equation of integral dependence of x over I (of degree n). The set of all elements that are integral over I is called the integral ¯ closure of I, and is denoted I.

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Integral Closures vs.Tight Closures

Integral Closures

Theorem 2.2 (Basic Properties) Let A be a ring with x ∈ A and letI, I1 , I2 be ideals. Then I¯ is an integrally closed ideal, that is, I¯ = I¯ if I1 ⊂ I2 , then I¯1 ⊂ I¯2 x ∈ I¯ iff the image of x in A/P is in the integral closure of (I + P )/P for every minimal prime P of A

(1) (2) (3)

When A is Noetherian, let I, J be ideals of A, then I¯J¯ ⊂ IJ x ∈ I¯ iff there exists c ∈ A◦ such that cxn ∈ I n for all large n Jun-ichi NISHIMURA (OECU)

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Integral Closures vs.Tight Closures

Integral Closures

Remark 2.3 Let I be an ideal of a ring R. Let A and B be R-algebras. Take a ∈ A. Assume that a is integral over IA that has an equation of integral dependence over IA of degree n. Suppose that there exists an R-algebra homomorphism ψ¯ : A/I ν A → B/I ν B with ν ≧ n. ¯ a) ∈ B/I ν B is in the integral closure of Then b ∈ B such that ¯b ≡ ψ(¯ IB.

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Integral Closures vs.Tight Closures

Integral Closures

Proposition 2.4 (Contraction of Integral Closure) Let A ⊂ B be an integral extension of rings. Let I be an ideal in A. Then ¯ IB ∩ A = I.

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Integral Closures vs.Tight Closures

Tight Closures

Tight Closures

Definition 2.5 (Tight Closure) Let A be a Noetherian ring of characteristic p and let I be an ideal. If A is reduced or if I has positive height, then x ∈ A is in I ∗ , the tight closure of I if and only if there exists c ∈ A◦ such that cxq ∈ I [q] for all q = pe .

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Integral Closures vs.Tight Closures

Tight Closures

Theorem 2.6 (Basic Properties) Let A be a Noetherian ring of characteristic p with x ∈ A and let I, I1 , I2 be ideals. Then we have the following: I ∗ is a tightly closed ideal, that is, (I ∗ )∗ = I ∗ if I1 ⊂ I2 , then I1∗ ⊂ I2∗ x ∈ I ∗ iff the image of x in A/P is in the tight closure of (I + P )/P for every minimal prime P of A let I, J be ideals of A, then I ∗ J ∗ ⊂ (IJ)∗

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Integral Closures vs.Tight Closures

Tight Closures

Proof of Theorem 2.6 (3) We remember the proof of Theorem 2.6 (3). Let P1 , . . . , Ps be the minimal primes of A. If c′i ∈ A/Pi is nonzero we can always lift c′i to an element ci ∈ A◦ by using the Prime Avoidance theorem. Suppose that c′i ∈ A/Pi is nonzero and such that c′i xqi ∈ Ii for all large q, where xi (respectively Ii ) represent the images of x (respectively I) in A/Pi . [q]

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Integral Closures vs.Tight Closures

Tight Closures

Take a lifting ci ∈ A◦ of c′i . Then ci xq ∈ I [q] + Pi for every i. Choose elements ti in all the minimal primes except Pi and set c :=

s ∑

ci ti

(5)

i

It is easy to check that c ∈ A◦ . ′

Choose q ′ ≫ 0 so that N [q ] = 0 where N is the nilradical. ′

′

′

Then cxq ∈ I [q] + N , and so cq xqq ∈ I [qq ] , which proves that x ∈ I ∗. Jun-ichi NISHIMURA (OECU)

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Integral Closures vs.Tight Closures

Tight Closures

Theorem 2.7 (Tight Closure from Contractions) Let A ⊂ B be a module-finite extension of Noetherian domains of characteristic p. Let I be an ideal in A. Then (IB)∗ ∩ A ⊂ I ∗ .

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Integral Closures vs.Tight Closures

Tight Closures

Theorem 2.8 Let A be an F -finite reduced ring of characteristic p. Let c be any nonzero element of A such that Ac is regular. Then c has a power which is a test element.

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Integral Closures vs.Tight Closures

Tight Closures

Theorem 2.9 (Persistence of Tight Closure) Let ψ : A → B be a homomorphism of Noetherian rings of characteristic p. Let I be an ideal of A and let a ∈ A be an element in I ∗ . Assume either that A is essentially of finite type over an excellent local ring, or that Ared is F -finite. Then ψ(a) is in the tight closure of ψ(I)B.

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Integral Closures vs.Tight Closures

Tight Closures

Theorem 2.10 (Colon Capturing) Let (A, m, k) be a (d − 1)-dimensional complete local domain of characteristic p with a system of parameters x1 , x3 , . . . , xd . Let I and J be any two ideals of the subring R = k[[x1 , x3 , . . . , xd ]] ⊂ A. Then (IA)∗ :A JA ⊂ ((I :R J)A)∗ (IA)∗ ∩ (JA)∗ ⊂ ((I ∩ J)A)∗

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Integral Closures vs.Tight Closures

Tight Closures

Remark 2.11 With R in Theorem 2.10, let A be a Noetherian ring which is module -finite and torsion free over R. Take γ ∈ R such that γx1 , x3 , . . . , xd form a system of parameters for R

(1)

Then, by Theorem 2.10 and Definition 2.5, we have (x1 q , x3 q , . . . , xr q )(γ)∗ :A xr+1 q ⊂ ((x1 q , x3 q , . . . , xr q )γ)∗

(2)

whenever 3 ≤ r < d, and x1 q (γ)∗ :A x3 q ⊂ (x1 q γ)∗

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(3)

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Integral Closures vs.Tight Closures

Test Element

Test Element Let A be a reduced Noetherian ring of characteristic p > 0 and let P1 , . . . , Ps be the minimal primes of A. Suppose that cN ∈ A is a common test element of A/Pi for each i

(1)

Choose elements ti ∈ A in all the minimal primes except Pi and let s ∑

ti =: γ

(2)

i=1

Then, by Theorem 2.6 (5) ∑ ∑ cN ti = cN ti = cN γ is a test element for A i Jun-ichi NISHIMURA (OECU)

(3)

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Integral Closures vs.Tight Closures

Test Element

Lemma 2.12 (Chinese Remainder Theorem) With notation and assumptions above, let I := (y1 , . . . , yr ) be an ideal of A. Suppose that a ∈ I ∗ . Then cN γa = y1 b1 + · · · + yr br

(4)

with b1 , . . . , br ∈ (γA)∗ .

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Integral Closures vs.Tight Closures

Test Element

Proof of Lemma 2.12 Because cN ∈ A is a common test element of A/Pi , we have cN a ≡ y1 b1i + · · · + yr bri

(mod Pi ).

The choice of ti implies ti cN a = ti (y1 bi1 + · · · + yr bir ). By assumption (2), we get ∑ ∑ γcN a = ti cN a = ti (y1 b1i + · · · + yr bri ) i

with bj :=

∑

i

= y1 b1 + · · · + yr br ti bji . Thus, bj ≡ γbji (mod Pi ) for each i.

i

Therefore, bj ∈ (γA)∗ by Theorem 2.6 (3). Jun-ichi NISHIMURA (OECU)

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Integral Closures vs.Tight Closures

Test Element

Theorem 2.13 (Tight Closure Brian¸con–Skoda Theorem) Let A be a Noetherian ring of characteristic p. Let I be an ideal generated by ℓ elements. Then for all n > 0, I n+ℓ−1 ⊂ (I n )∗ ⊂ I n

(1)

Hence, if I is a principal ideal I n = (I n )∗

(2)

for all n > 0.

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Structure Theorem

Structure Theorem for Mixed Characteristic Complete Local Rings

Structure Theorem for Mixed Characteristic Complete Local Rings In this section, we gather a few basic facts on the p-adic representation of elements in a ring. We recall Structure Theorem for Mixed Characteristic Complete Local Rings.

For the details, we refer the reader to [10].

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Structure Theorem

p-adic Representation and n-th Witt polynomial

n-th Witt polynomial

Let wn denote the n-th Witt polynomial, that is n

n−1

wn (X0 , X1 , . . . , Xn ) := X0p + pX1p

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+ · · · + pn Xn

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Structure Theorem

p-adic Representation and n-th Witt polynomial

p-adic Representation Lemma 3.1 Let Φ designate any polynomial in two variables with integral coefficients, which covers arithmetic operations such as addition, multiplication, etc. Then, there exist polynomials φ0 (X0 ; X0′ ), . . . , φn (X0 , . . . , Xn ; X0′ , . . . , Xn′ ) with integral coefficients such that Φ(wn (X0 , . . . , Xn ), wn (X0′ , . . . , Xn′ )) = wn (φ0 (X0 ; X0′ ), . . . , φn (X0 , . . . , Xn ; X0′ , . . . , Xn′ )) = φ0 (X0 ; X0′ )

pn

pn−1

+ pφ1 (X0 , X1 ; X0′ , X1′ )1

(2)

+ ···

· · · + pn φn (X0 , . . . , Xn ; X0′ , . . . , Xn′ ) Jun-ichi NISHIMURA (OECU)

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Structure Theorem

p-adic Representation and n-th Witt polynomial

Lemma 3.2 Let A be a ring and a, b ∈ A. Suppose that for ℓ > 0 a ≡ b (mod pℓ A)

(3)

Then k

k

a p ≡ bp

(mod pk+ℓ A)

(4)

Hence, by sending any element a of each residue class a ¯ ∈ A/pA to k pk its p -th power a , we get a canonical map τA,k : A/pA → A/pk+1 A

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(5)

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Structure Theorem

p-adic Representation and n-th Witt polynomial

Moreover, Lemma 3.3 The maps τA,n , . . . , τA,0 induce TA,n : (A/pA)n+1 → A/pn+1 A

(6)

with (¯ a0 , a ¯1 , . . . , a ¯n ) 7→ wn (a0 , a1 , . . . , an ) n

= a0p + pa1p

n−1

+ · · · + pn an

(7)

TA,n is injective when p is not a zero-divisor and when A/pA is reduced. Jun-ichi NISHIMURA (OECU)

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Structure Theorem

p-adic Representation and n-th Witt polynomial

Lemma 3.4 Let A be a ring and a, z1 , z3 , . . . , zr ∈ A that satisfy a ≡ z1 b 1 + z3 b 3 + · · · + zr b r

(mod pA)

(8)

Suppose that ¯b1 , ¯b3 , . . . , ¯br ∈ J¯∗ for an ideal J¯ ⊂ A/pA. Then n

n

n

n

n

n

a p ≡ z1p b1p + z3p b3p + · · · + zrp brp n−1

+ pφ1 (b) p

n

+ · · · + pn φn (b)

(9)

(mod pn+1 A) with φ1 (b)

pn−1

n n , . . . , φn (b) ∈ (J¯∗ ) p ⊂ (J¯p )∗ .

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Structure Theorem

J¯∗ -Representation of Length n

J¯∗-representation of length n With notation as in Lemma 3.2, take γ ∈ A. n n−i i Let J := γ p A and let J¯1/p := γ¯ p A¯ where A¯ := A/pA. We say that a ∈ A has a J¯∗ -representation of length n

( ∗) ¯ . when the residue class of a in A/pn+1 A is contained in Im TA,n J That is, n

n−1

a ≡ a0p + pa1p

+ · · · + pn an

(mod pn+1 A)

(1)

¯ ∗ ⊂ (A/pA)n+1 , where with (¯ a0 , a ¯1 , . . . , a ¯n ) ∈ J ¯ ∗ : = (J¯1/pn )∗ × (J¯1/pn−1 )∗ × · · · × J¯∗ J n ¯ ∗ × (¯ ¯ ∗ × · · · × (¯ ¯∗ = (¯ γ A) γ p A) γ p A) Jun-ichi NISHIMURA (OECU)

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(2)

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Hochster’s Observation

Hochster’s Observation

Now we recall Hochster’s Observation [5, p.22] on an equational description (for degeneracy) of modifications.

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Hochster’s Observation

Notation and Assumptions

Notation and Assumptions Let (A, m, k) be a mixed characteristic d-dimensional complete normal local domain with an algebraically closed residue field k. Fix a system of parameters x1 , p, x3 , . . . , xd and a Witt ring (W, pW, k). Then, we get a d-dimensional complete regular local ring R := W [[x1 , x3 , . . . , xd ]]

(1)

that makes A a finite extension of R.

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Hochster’s Observation

Notation and Assumptions

Further, we have a (d + h)-dimensional complete regular local ring S := W [[x1 , x3 , . . . , xd , xd+1 , . . . , xd+h ]] ⊃ R

(2)

and a prime ideal of height h P := (f1 , . . . , fr )S

(3)

A := S/P ⊃ R

(4)

that express

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Hochster’s Observation

Equational Description

Equational Description With notation and assumptions above, we want to describe in an explicit way what a tth modification of (A, a) of type r := (r1 , . . . , rt ) with respect to a system of parameters x1 , p, x3 , . . . , xd looks like and what it means if such a modification degenerates. In working with vectors, it will be convenient to identify a vector of length r with the vector of length r + r′ whose last r′ entries are 0. It will also be convenient to define x2 = p and r0 = 1.

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Hochster’s Observation

Equational Description

Let M : (A, a) = (M0 , α0 ) → (M1 , α1 ) → · · · → (Mt , αt )

(1)

be the sequence of modifications considered. For each ℓ, we may identify Mℓ = A ⊗S Lℓ = S/P ⊗S Lℓ where Lℓ :=

ℓ ⊕ µ=0

Jun-ichi NISHIMURA (OECU)

S rµ

/ ℓ ∑

SVλ

(2)

λ=1

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Hochster’s Observation



Here

  Vλ :=  

with

(λ−1) Urλ +1

∈

λ−1 ⊕

Equational Description

(λ−1)

Urλ +1 xrλ .. . x1

 λ  ⊕  S rµ ∈  µ=0

(3)

S rµ that satisfies

µ=0 rλ ∑

(λ−1)

xi Ui

(λ−1)

+ xrλ +1 Urλ +1 =

λ−1 ∑

s(λ−1) Vµ + µ

µ=1

i=1

r ∑

(λ−1)

fj W j

j=1

for suitable choices of the vectors (λ−1) Ui ,

(λ−1) Wj

∈

λ−1 ⊕

S rµ and s(λ−1) ∈ S. µ

µ=0 Jun-ichi NISHIMURA (OECU)

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Hochster’s Observation

Equational Description

Then, by taking s ∈ S that represents a = s¯ ∈ S/P , the condition αt ∈ (x1 , p, x3 , . . . , xd )Mt is then expressed by the existence of vectors Ui , Wj ∈

t ⊕

S rµ and elements sλ ∈ S

µ=0

such that 

 s d t r ∑ ∑  0  ∑  . + x U = s V + fj W j i i λ λ  ..  i=1 j=1 λ=1 0

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Flenner’s Bertini Theorem

Notation and Assumptions

Flenner’s Bertini Theorem With notation and assumptions as in 4.1, let U := R \ pR.

Because AU is a one-dimensional regular semi-local ring, we may assume that I := (f1 , . . . , fh )S is an ideal of height h and that there exists u ∈ U that makes uP ⊂ I.

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Flenner’s Bertini Theorem

That is, ufk =

Notation and Assumptions

h ∑

ujk fj

(1)

j=1

with ujk ∈ S for k = h + 1, . . . , r. For a given γ ∈ R, put c := uγx1

(2)

c, p, x3 , . . . , xd form a system of parameters for R

(3)

Further we assume that

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Flenner’s Bertini Theorem

p-adic Representation and f˜0

p-adic Representation and f˜0 With notation and assumption above, fix an ϵ ∈ N. By taking pϵ th root ξi of xi , let ˜ := W [[ξ1 , ξ3 , . . . , ξd ]], R⊂R S ⊂ S˜ := W [[ξ1 , ξ3 , . . . , ξd , ξd+1 , . . . , ξd+h ]]. Given an n ∈ N. If ϵ above is chosen large enough, we have the p-adic representation for any s ∈ S: n

n−1

s = s˜0p + p˜ s1p Jun-ichi NISHIMURA (OECU)

+ · · · + pn s˜n with s˜i ∈ S˜

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Flenner’s Bertini Theorem

p-adic Representation and f˜0

In particular, if we choose an m ∈ N such that 2m ≤ n, we get n

n−1

fj = f˜j0p + pf˜j1p

n−m

p + · · · + pm f˜jm pn−m−1 + pm+1 f˜jm+1 + · · · + pn f˜jn .

Letting pn−m−1 r˜j := f˜jm+1 + · · · + pn−m−1 f˜jn ,

we express n n−1 pn−m fj = f˜j0p + pf˜j1p + · · · + pm f˜jm + pm+1 r˜j

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Flenner’s Bertini Theorem

p-adic Representation and f˜0

Take an indeterminate π and let ˜ T := S[[π]] ⊃ W [[π]]

Fix an µ ∈ N and let

(3)

Q := pµ .

Put f˜0 := π Q − p ∈ W [[π]]

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Flenner’s Bertini Theorem

p-adic Representation and f˜0

Because p = π Q − f˜0 , we get n n−1 pn−m fj = f˜j0p + (π Q − f˜0 )f˜j1p + · · · + (π Q − f˜0 )m f˜jm + pm+1 r˜j n n−1 pn−m = f˜j0p + π Q f˜j1p + · · · + π mQ f˜jm + σj f˜0 + pm+1 r˜j

where n−1 n−2 σj := −f˜j1p + (−2π Q + f˜0 )f˜j2p + . . . ( m ) ( ) ∑ m pn−m ··· + π (m−k)Q f˜0k−1 f˜jm (−1)k . k k=1

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Flenner’s Bertini Theorem

p-adic Representation and f˜0

Hence we have m

m−1

p p fj := φj0 + pφj1

+ · · · + pm φjm + σj f˜0 + pm+1 r˜j

with n−m−1 n−m m m pn−2m ∈ T, + π Q/p f˜j1p + · · · + π mQ/p f˜jm φj0 := f˜j0p ( n−m ) n−m−1 m m pn−2m φji := φi f˜j0p ; π Q/p f˜j1p ; . . . ; π mQ/p f˜jm ∈ T.

Remark that φji above satisfies m π Q/p φji

(5)

for i = 1, . . . , m. Jun-ichi NISHIMURA (OECU)

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Flenner’s Bertini Theorem

Let

p-adic Representation and f˜0

pm pm−1 + pφj1 + · · · + pm φjm + pm+1 r˜j . f˜j := φj0

Then we get fj = f˜j + σj f˜0

(6)

Consequently, the relations above imply that uf˜k =

h ∑

ujk f˜j + u0k f˜0

(7)

j=1

with u0k :=

h ∑

ujk σj − uσk

j=1

for k = h + 1, . . . , r. Jun-ichi NISHIMURA (OECU)

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Flenner’s Bertini Theorem

p-adic Representation and f˜0

Let I0 := (f˜0 , f1 , . . . , fh )S[[π]] ⊃ (f1 , . . . , fh )S[[π]] = IS[[π]], P0 := (f˜0 , f1 , . . . , fr )S[[π]] ⊃ (f1 , . . . , fr )S[[π]] = P S[[π]]. Then by (6), we have I0 T = (f˜0 , f˜1 , . . . , f˜h )T and P0 T = (f˜0 , f˜1 , . . . , f˜r )T

(8)

Finally, we have the rings as follows ˜ S˜ ⊂ T /P0 T =: A˜ R ⊂ A = S/P ⊂ S/P

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Flenner’s Bertini Theorem

Bertini Theorem

Bertini Theorem With notation and assumptions above, for any ν ∈ N, choose ν0 and νj ∈ N (j = 1, . . . , h) such that ν < ν0 < νj . Take c in 5.1 (2). Let g0 := f˜0 − cν0 π

(1)

= π − c π − p ∈ R[[π]] Q

ν0

and, for j = 1, . . . , h, let gj := f˜j + cνj τj (cf. (9) below) Jun-ichi NISHIMURA (OECU)

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Flenner’s Bertini Theorem

Bertini Theorem

Put P := (g1 , . . . , gh )Tc ∩ T

(3)

P0 := (g0 , g1 , . . . , gh )Tc ∩ T ⊃ P

(4)

and

Flenner’s Bertini Theorem shows that τj s above can be chosen so that P and P0 are prime ideals of T and further:

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Flenner’s Bertini Theorem

Bertini Theorem

Theorem 5.3 (Bertini Theorem) With notation and assumptions above, let E be the integral closure of T /P in its quotient field Q(T /P) and let ( ) B := E/g0 E ⊂ E/g0 E c . Then the local domains E and B ⊃ T /P0 satisfy: c, π is a sub-system of parameters in E¯ = E/pE ( ) ˜ c makes Ec = T /P c an etale R[[π]] c -algebra ( ) ˜ c -algebra c makes Bc = T /P0 c an etale R

(5) (6) (7)

˜ There exists an R-algebra homomorphism ˜ ν A˜ → B/cν B ψ˜ : A/c Jun-ichi NISHIMURA (OECU)

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Flenner’s Bertini Theorem

Outline of Proof

Outline of Proof Flenner’s Proof of Bertini Theorem [3, Satz (2.1)] shows that, if we choose a sequence of elements p p t˜1 , . . . , t˜h ∈ T [p] := W [[ξ1 , ξ3 , . . . , ξd , π, ξd+1 , . . . , ξd+h ]]

and sequences of natural numbers ν1 , . . . , νh with νj > ν0 and suitable µ1 , . . . , µh , we get ( µj ) gj := f˜j + cνj t˜j ξd+j + xp1 = f˜j + cνj τj

(9)

that make ˜ ˜ (T /(p, g1 , . . . , gh ))c etale over (R[[π]]/p R[[π]]) c. Jun-ichi NISHIMURA (OECU)

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Flenner’s Bertini Theorem

Outline of Proof

For k ≥ h + 1, put } 1 {∑ νj ν0 ˜ gk := fk + ujk c τj − u0k c π u j=1 h

(10)

Then by 5.2 (7), we have ugk =

h ∑

ujk gj

(11)

j=0

Hence, gj ∈ P0 for j = 0, 1, . . . , r. Therefore, E and B fulfil the conditions (5), (6), (7) and (8), because (P0 , cν )T ⊂ (P0 , cν ) by (1), (2) and (10). Jun-ichi NISHIMURA (OECU)

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Flenner’s Bertini Theorem

Remarks

Remarks With notation and assumption above, suppose that A¯ := A/pA is reduced and that any element v of a finite set Υ ⊂ A has p-adic representation 5.2 (1) which further satisfies the following: n

n−1

v = v0p + pv1p

+ · · · + p n vn

(1)

n−i ¯ ∗. with v¯ip ∈ (¯ γ κ A) Take (any) elements ω0 , ω1 , . . . , ωn ∈ B that satisfy

˜ i ) ∈ B/cν B ωi ≡ ψ(v for i = 0, 1, . . . , n and let n

n−1

ω := ω0p + pω1p Jun-ichi NISHIMURA (OECU)

+ · · · + pn ωn .

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Flenner’s Bertini Theorem

Remarks

Then

Remark 5.4 By Theorem 2.6 (4), we can choose ν large enough such that any ˜ ˜ element ψ(v) ∈ ψ(Υ) satisfies the following: ˜ ψ(v) ≡ ω ∈ B/cν B pn

pn−1

ω1 ω ¯=ω ¯ 0 + p¯

(2) + · · · + pn ω ¯n

(3)

n−i ¯ := B/pB with ω ¯ ∗. where ω ¯, ω ¯0, ω ¯1, . . . , ω ¯n ∈ B ¯ ip ∈ (¯ γ κ B)

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Flenner’s Bertini Theorem

Remarks

With notation and assumption above, take pϵˇth root ζi of ξi and let ˇ := W [[ζ1 , ζ3 , . . . , ζd ]] R (1) Then ˇ ⊃ W [[ξ1 , ξ3 , . . . , ξd ]] = R ˜ R and ¯ˇ := R/p ˇ R ˇ = k[[ζ1 , ζ3 , . . . , ζd ]] R ¯˜ ˜ R ˜ =: R. ⊃ k[[ξ1 , ξ3 , . . . , ξd ]] = R/p Let E¯ := E/pE ˇ ⊗˜ E Eˇ := R R ¯ ¯ˇ ⊗ ¯ E¯ ˇ Eˇ = R Eˇ := E/p ˜ R

ˇ := R ˇ ⊗˜ B B R ¯ ¯ˇ ⊗ ¯ B ˇ ˇ ˇ=R ¯ B := B/pB ˜ R Jun-ichi NISHIMURA (OECU)

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Flenner’s Bertini Theorem

Remarks

Then

Remark 5.5 With notation above, we have ˇ ⊗ ˜ Ec is an etale R[[π]] ˇ Eˇc = R c -algebra R ¯ˇ ⊗ ¯ E¯ is an etale R[[π]] ¯ˇ E¯ˇc = R c c -algebra ˜ R ( ) ˇc = E/g ˇ 0 Eˇ ˇ c -algebra B is an etale R c ¯ˇ = (E/¯ ¯ˇ g E¯ˇ ) is an etale R ¯ˇ -algebra B c 0 c

(2) (3) (4) (5)

c

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Flenner’s Bertini Theorem

Remarks

( ) ¯ Let Eˇ be the integral closure of E¯ˇ in Q E¯ˇ , the total quotient ring ¯ˇ Then of E. (¯ ) ¯ˇ ¯ ( ¯ˇ ¯) ˇ g0 E¯ˇ , E/¯ g0 Eˇ ⊂ E/¯ g0 Eˇ c = E/¯ c ¯ˇ because c and g0 form a regular sequence in E. Hence Remark 5.6 With notation above, we have ¯ˇ ¯ E/¯ g0 Eˇ is reduced

(1)

¯ Because E¯ˇ ⊂ Eˇ is a finite extension, ¯ g¯0 Eˇ ∩ E¯ˇ is the integral closure of the ideal g¯0 E¯ˇ Jun-ichi NISHIMURA (OECU)

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Flenner’s Bertini Theorem

Remarks

Finally, remark that Remark 5.7 With notation above, we have (¯ ) ( ¯) ˇ g0 E¯ˇ : Q R ˇ ] is bounded for any ϵˇ [Q E/¯ ( ) (¯ ) ˇ [Q E¯ˇ : Q R[[π]] ] is bounded for any ϵˇ

(1) (2)

¯ˇ ¯ ¯ˇ That is, both the number of minimal prime ideals P/¯ g0 Eˇ of E/¯ g0 E¯ˇ ( ) ¯ˇ ¯ˇ ¯ˇ ] are bounded and the degree of extension [Q(E/ P) : Q R independently for any large ϵˇ. Also, both the number of minimal prime ideals Pˇ¯ of E¯ˇ and its degree ¯ˇ P¯ˇ ) : Q(R[[π]])] ¯ˇ of extension [Q(E/ are bounded independently for any large ϵˇ. Jun-ichi NISHIMURA (OECU)

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Base Change 1

Base Change 1

Base Change 1 Let (A, m, k) and (B, n, k) be d-dimensional local rings with (common) system of parameters x := x1 , x2 , . . . , xd . Suppose that for some ν > 0, there exists a ring homomorphism ψ¯ : A/xν1 A → B/xν1 B that maps x¯i to x¯i . Let M be an A-module with α ∈ M and let N be a B-module with β ∈ N. Assume that, for ν0 ≤ ν, we have an A/xν1 A-module homomorphism ¯ ϕ¯ : M/xν10 M → N/xν10 N with α ¯ 7→ β. Jun-ichi NISHIMURA (OECU)

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Base Change 1

Base Change 1

Let ρ := (m1 , . . . , mr+1 ) ∈ M r+1 be a type r relation for x: x1 m1 + · · · + xr+1 mr+1 = 0 . By taking ni ∈ N such that ¯m n ¯ i ≡ ϕ( ¯ i ) on N/xν10 N , we have: x1 n ¯ 1 + x2 n ¯ 2 + · · · + xr+1 n ¯ r+1 ≡ 0 . That is, x1 n1 + x2 n2 + · · · + xr+1 nr+1 = xν10 n with n ∈ N . Put

Jun-ichi NISHIMURA (OECU)

n∗1 := n1 − xν10 −1 n. Big Cohen-Macaulay Modules

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Base Change 1

Base Change 1

Then we get a type r relation ρ∗ := (n∗1 , n2 , . . . , nr+1 ) ∈ N r+1 for x on N : x1 n∗1 + x2 n2 + · · · + xr+1 nr+1 = 0 ∗ ¯m where n ¯ ∗ ≡ ϕ( ¯ 1 ) on N/xν N for ν ∗ ≥ ν0 − 1. 1

1

When r ≥ 1, put    v˜ :=  

nr+1 xr .. .

    ∈ N ⊕ Br 

x1 and let v˜ := n∗1 ∈ N when r = 0. Jun-ichi NISHIMURA (OECU)

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Base Change 1

Base Change 1

Then we have canonical maps / N → N ⊕ B r → (N ⊕ B r ) B v˜ =: N ′ . Let β ′ be the image of β in N ′ . That is,   β  / 0   ∈ (N ⊕ B r ) B v˜ . . β ′ :=   ..  0 Then we have a map (N, β) → (N ′ , β ′ ), a map N → N ′ that takes β to β ′ . We call (N ′ , β ′ ) a first modification of (N, β) with respect to a type r relation ρ∗ for x. Jun-ichi NISHIMURA (OECU)

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Base Change 1

Base Change 1

By construction, for ν0 ≥ ν1 ≥ ν0 − 1, we have an A/xν1 A-module homomorphism ϕ¯′ : M ′ /xν11 M ′ → N ′ /xν11 N ′ with α¯′ 7→ β¯′ that makes the following diagram commutative

(M/xν10 M, α ¯ ) −−−→ (M ′ /xν11 M ′ , α¯′ )     ¯′ ϕ¯y yϕ ¯ −−−→ (N ′ /xν1 N ′ , β¯′ ) . (N/xν10 N, β) 1

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Base Change 1

Base Change 1

Consequently, assume that we have a sequence M : (M, α) =: (M0 , α0 ) → (M1 , α1 ) → · · · → (Mt , αt ) where (Mℓ , αℓ ) is a modification of (Mℓ−1 , αℓ−1 ) with respect to a relation ρℓ on Mℓ−1 of type rℓ for x. Then, by the argument above, whenever ν0 > t, we get a sequence N : (N, β) =: (N0 , β0 ) → (N1 , β1 ) → · · · → (Nt , βt ) where (Nℓ , βℓ ) is a modification of (Nℓ−1 , βℓ−1 ) with respect to a relation ρ∗ℓ on Nℓ−1 of type rℓ for x.

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Base Change 1

Base Change 1

Further, we have A/xν1 A-module homomorphisms ϕ¯ℓ : (Mℓ /xν1ℓ Mℓ , α ¯ ℓ ) → (Nℓ /xν1ℓ Nℓ , β¯ℓ ) with νℓ−1 ≥ νℓ ≥ ν0 − ℓ, that make the following diagram commutative ν

(Mℓ−1 /x1ℓ−1 Mℓ−1 , α ¯ ℓ−1 ) −−−→ (Mℓ /xν1ℓ Mℓ , α ¯ℓ)    ¯ ϕ¯ℓ−1 y yϕℓ ν (Nℓ−1 /x1ℓ−1 Nℓ−1 , β¯ℓ−1 ) −−−→ (Nℓ /xν1ℓ Nℓ , β¯ℓ ).

Hence, if (Mt , αt ) is a degenerate modification of (M, α) with respect to x, (Nt , βt ) is a degenerate modification of (N, β) with respect to x. Jun-ichi NISHIMURA (OECU)

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Frobenius Maps

Frobenius Maps

Frobenius Maps

In this section, we gather a few basic facts on Frobenius maps.

For the details, we refer the reader to [10] and [13].

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Frobenius Maps

Frobenius Maps

With notation and assumptions in Section 5, let FW : W → W be the Frobenius map of a Witt ring (W, pW, k) that lifts the Frobenius map Fk : k → k of the residue field k of characteristic p. Because ˜ = W [[ξ1 , ξ3 , . . . , ξd ]] R is a formal power series ring over W with indeterminates ξ1 , ξ3 , . . . , ξd , FW has an extension ˜→R ˜ FR˜ : R by mapping ξi to ξi p for i = 1, 3, . . . , d. Jun-ichi NISHIMURA (OECU)

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Frobenius Maps

Frobenius Maps

With c in 5.1 (2), FR˜ has the canonical extension ˜c → R ˜ F (c) FR˜c : R ˜ R that induces ˜ c /pn+1 R ˜c → R ˜ F (c) /pn+1 R ˜ F (c) FR˜c ,n : R ˜ ˜ R R for any n ∈ N0 . ˜ c -algebra, Because Bc in Theorem 5.3 (Bertini Theorem) is an etale R the Frobenius map FBc /pBc : Bc /pBc → Bc /pBc has the extension FBc ,n : Bc /pn+1 Bc → BFR˜ (c) /pn+1 BFR˜ (c) which satisfies the following commutative diagrams: Jun-ichi NISHIMURA (OECU)

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Frobenius Maps

FB

Frobenius Maps

c Bc /pn+1 Bc −−− → BFR˜ (c) /pn+1 BFR˜ (c) x x     ,n

˜ c /pn+1 R ˜ c −−−→ R ˜ F (c) /pn+1 R ˜ F (c) R ˜ ˜ R R FR ˜ c ,n

and

FB

c Bc /pn+1 Bc −−− → BFR˜ (c) /pn+1 BFR˜ (c)     y y ,n

Bc /pn Bc −−−−−→ BFR˜ (c) /pn BFR˜ (c) FBc ,n−1

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Frobenius Maps

Frobenius Maps

˜ generated by Hence, with the FR˜ -stable multiplicative set ∆ of R { e } FR˜ (c) e∈N0 , FR˜ is extended to the endomorphism FB∆∗ of B∆∗ , the pB∆ -adic completion of B∆ , that makes the following diagram commutative:

FB ∗

B∆∗ −−−∆→ x  

B∆∗ x  

˜ −−−→ R ˜. R FR ˜

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Definition of De

Frobenius Maps

Definition of De With notation and assumptions above, take e ∈ N and let ˜ De := the derived normal ring of R[B, FB∆∗ (B), . . . , FBe∆∗ (B)]. Then De is a finite B-algebra both via the inclusion map ι and via φe , the canonical homomorphism induced by FBe∆∗ ι

B −−−→ De x φe 

(1)

B Jun-ichi NISHIMURA (OECU)

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Frobenius Maps

Remarks

Remarks Remark 7.3 With notation above, let P1 , . . . , Pse be the minimal primes of pDe . That is, pDe = P1 ∩ · · · ∩ Pse . Because the set of quotient fields ΦB := {Q(De /Pi ) | e ∈ N, i = 1, . . . , se }

(1)

is a finite set, when eB is large enough, we have canonical isomorphisms ( ) ∼ = DeB / Pi ∩ DeB ,→ De /Pi

(2)

for any e > eB and for any i = 1, . . . , se . Jun-ichi NISHIMURA (OECU)

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Frobenius Maps

Remarks

Hence, putting ¯ e := De /pDe and P¯i := Pi /pDe D we find an N ∈ N that satisfies: ¯ e /P¯i c¯N is a common test element of D

(1)

for any i = 1, . . . , se and for any e (cf.Theorem 2.8, Remark 7.3 (2)). Choose elements ¯e t¯i ∈ D in all the minimal primes except P¯i and assume further that se ∑

t¯i = γ¯

(2)

i=1

Jun-ichi NISHIMURA (OECU)

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Frobenius Maps

Remarks

Then

Remark 7.4 With notation and assumptions above, ∑ ∑ c¯N t¯i = c¯N t¯i = c¯N γ¯ i

(3)

i

¯ e. is a test element for D

Jun-ichi NISHIMURA (OECU)

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Frobenius Maps

Remarks

With notation and assumptions above, for r ≥ 2, suppose that e e e e x¯p1 d¯1 + x¯p3 d¯3 + · · · + x¯pr d¯r + x¯pr+1 d¯r+1 = 0

(1)

¯ e )∗ . with d¯1 , d¯3 , . . . , d¯r ∈ (¯ γ0 D Here γ0 ∈ R that satisfies γ0 , p, x3 , . . . , xd form a system of parameters for R (cf. 5.1 (2)).

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Frobenius Maps

Remarks

Then

Remark 7.5 By Theorem 2.10 (Colon Capturing) , we get e e e e ¯ e )∗ (¯ d¯r+1 ∈ (¯ γ0 D xp1 , x¯p3 , . . . , x¯pr ) :D¯ e x¯pr+1 e

e

(2)

⊂ ((¯ γ0 )(¯ xp1 , x¯p3 , . . . , x¯pr ))∗

Jun-ichi NISHIMURA (OECU)

e

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Frobenius Maps

Remarks

Thus, we get Lemma 7.6 With notation and assumptions above, for r ≥ 2, suppose that e e e e x¯p1 d¯1 + x¯p3 d¯3 + · · · + x¯pr d¯r + x¯pr+1 d¯r+1 = 0

(1)

¯ e )∗ . Here γ0 ∈ R that satisfies with d¯1 , d¯3 , . . . , d¯r ∈ (¯ γ0 D γ0 , p, x3 , . . . , xd form a system of parameters for R.

(2)

Then e e e c¯N γ¯ d¯r+1 = γ¯0 (¯ xp1 ω ¯ 1 + x¯p3 ω ¯ 3 + · · · + x¯pr ω ¯r ) e

e

e

= x¯p1 γ¯0 ω ¯ 1 + x¯p3 γ¯0 ω ¯ 3 + · · · + x¯pr γ¯0 ω ¯r

(3)

¯ e )∗ (cf. Lemma 2.12 (4)). with ω ¯1, ω ¯3, . . . , ω ¯ r ∈ (¯ γD Jun-ichi NISHIMURA (OECU)

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Base Change 2

Base Change 2

Base Change 2 Let (B, n, k), (De , ne , k) be d-dimensional mixed characteristic local rings with system of parameters x1 , p(= x2 ), x3 , . . . , xd , that is abbreviated to x.

Suppose that we have a ring homomorphism φ e : B → De e

that maps xi to xpi for i = 1, 3, . . . , d (cf. 7.2 (1)).

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Base Change 2

Base Change 2

If N is a B-module with β ∈ N . Then De ⊗B N becomes a De -module and we have a canonical homomorphism φe ⊗ 1N : (N, β) → (De ⊗B N, 1 ⊗ β) with β 7→ 1 ⊗ β.

Let

ρ∗ := (n∗1 , n2 , n3 , . . . , nr+1 ) ∈ N r+1

be a type r relation for x on N x1 n∗1 + pn2 + x3 n3 + · · · + xr+1 nr+1 = 0.

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Base Change 2

Base Change 2

Then, we get a type r relation De ⊗B ρ∗ := (1 ⊗ n∗1 , 1 ⊗ n2 , 1 ⊗ n3 , . . . , 1 ⊗ nr+1 ) ∈ (De ⊗B N )r+1 e

e

e

for xp1 , p, xp3 , . . . , xpd , abbreviated to xp , on De ⊗B N . e

Namely, e

e

e

xp1 ⊗ n∗1 + p ⊗ n2 + xp3 ⊗ n3 + · · · + xpr+1 ⊗ nr+1 = 0.

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Base Change 2

Base Change 2

When r ≥ 2, put    1 ⊗ v˜ :=  

1 ⊗ nr+1 e xpr .. . xp1

e

    ∈ (De ⊗B N ) ⊕ Der . 

When r = 1, let ( 1 ⊗ v˜ := and put

1 ⊗ n2 e xp1

) ∈ (De ⊗B N ) ⊕ De

1 ⊗ v˜ := 1 ⊗ n∗1 ∈ De ⊗B N

when r = 0. Jun-ichi NISHIMURA (OECU)

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Base Change 2

Base Change 2

Then we have canonical maps De ⊗B N → (De ⊗B N ) ⊕ Der

/ → ((De ⊗B N ) ⊕ Der ) De (1 ⊗ v˜) = De ⊗B N ′ .

Let 1 ⊗ β ′ be the image of 1 ⊗ β in De ⊗B N ′ . That is,   1⊗β  0  /  ∈ ((De ⊗B N ) ⊕ Der ) De (1 ⊗ v˜). . 1 ⊗ β′ =  .  .  0 We may call (De ⊗B N ′ , 1 ⊗ β ′ ) a first modification of e (De ⊗B N, 1 ⊗ β) with respect to a type r relation De ⊗B ρ∗ for xp . Jun-ichi NISHIMURA (OECU)

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Base Change 2

Base Change 2

By construction, we have a B-module homomorphism φe ⊗ 1N ′ : (N ′ , β ′ ) → (De ⊗B N ′ , 1 ⊗ β ′ ) that makes the following diagram commutative

(N, β)   φe ⊗ 1 N y

−−−→

(N ′ , β ′ )  φ ⊗ 1 ′ y e N

(De ⊗B N, 1 ⊗ β) −−−→ (De ⊗B N ′ , 1 ⊗ β ′ ).

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Base Change 2

Base Change 2

Consequently, assume that we have a sequence N : (N0 , β0 ) → (N1 , β1 ) → · · · → (Nt , βt ) where (Nℓ , βℓ ) is a modification of (Nℓ−1 , βℓ−1 ) with respect to a relation ρ∗ℓ on Nℓ−1 of type rℓ for x. Then, by the argument above, we get a sequence De ⊗B N : (De ⊗B N0 , 1 ⊗ β0 ) → · · · → (De ⊗B Nt , 1 ⊗ βt ) where (De ⊗B Nℓ , 1 ⊗ βℓ ) is a modification of (De ⊗B Nℓ−1 , 1 ⊗ βℓ−1 ) e with respect to a relation De ⊗B ρ∗ℓ on De ⊗B Nℓ−1 of type rℓ for xp .

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Base Change 2

Base Change 2

Further, we have B-module homomorphisms φe ⊗ 1Nℓ : (Nℓ , βℓ ) → (De ⊗B Nℓ , 1 ⊗ βℓ ) that make the following diagram commutative (Nℓ−1 , βℓ−1 )   φe ⊗ 1Nℓ−1 y

−−−→

(Nℓ , βℓ )  φe ⊗ 1 Nℓ y

(De ⊗B Nℓ−1 , 1 ⊗ βℓ−1 ) −−−→ (De ⊗B Nℓ , 1 ⊗ βℓ ). Hence, if (Nt , βt ) is a degenerate modification of (N, β) with respect to x1 , p, x3 , . . . , xd , (De ⊗B Nt , 1 ⊗ βt ) is a degenerate modification e e e of (De ⊗B N, 1 ⊗ β) with respect to xp1 , p, xp3 , . . . , xpd . Jun-ichi NISHIMURA (OECU)

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First Step of Induction

Notation and Assumptions

Notation and Assumptions We prove Proposition 1.7. Suppose that the assertion of Proposition 1.7 is false. Then, there exists a d-dimensional mixed characteristic local ring (A, m, k) which has no big Cohen-Macaulay module. We may assume that A is a complete normal local domain of dim A = d ≥ 3 with a system of parameters x1 , p, x3 , . . . , xd , that is abbreviated to x, and that the residue field k is algebraically closed of characteristic p > 0.

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First Step of Induction

Notation and Assumptions

Because A has no big Cohen-Macaulay module, there exist an integer t and a degenerate sequence of modifications of type r = (r1 , . . . , rt ) with respect to x: M : (A, 1) =: (M0 , α0 ) → (M1 , α1 ) → · · · → (Mt , αt )

(1)

such that αt ∈ (x1 , p, x3 , . . . , xd )Mt .

We work with this sequence and obtain a contradiction.

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First Step of Induction

The case when ℓ = 1

The case when ℓ = 1 With notation and assumptions above, we prove Proposition 1.7 by inductive steps. Let ℓ = 1. Because A is normal, we may assume r1 ≥ 2. Suppose that we have a type r1 relation on A: x1 a1 + pa2 + · · · + xr1 ar1 + xr1 +1 ar1 +1 = 0

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First Step of Induction

The case when ℓ = 1

Then, by the argument in 5.1, we get a type r1 relation on B: x1 b∗1 + pb2 + · · · + xr1 br1 + xr1 +1 br1 +1 = 0 and we put

   v˜1 =  

br1 +1 xr1 .. .

(3)

    ∈ B ⊕ B r1 

(4)

x1 Take sufficiently large ϵ1 . Then by 5.2 (1), we have n1

n1 −1

br1 +1 = b(rp 1 +1)0 + pb(rp 1 +1)1 + · · · + pn1 b(r1 +1)n1

(5)

with b(r1 +1)i ∈ B for i = 0, 1, . . . , n1 . Jun-ichi NISHIMURA (OECU)

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First Step of Induction

q1 and Φ1

q1 and Φ1 Let c1 := u1 x1

(1)

(cf. 5.1 (2)). Assume that K1 ∈ N satisfies K1 K1 ¯ 1 c¯1 ̸∈ (xK 1 , x3 , . . . , xd )R

(2)

Take eB and N1 in 7.3. Then we can find e1 ∈ N (e1 > eB ) enough large that satisfies the following and let 4K1 N1 pn1 < pe1 =: q1 (3) Jun-ichi NISHIMURA (OECU)

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First Step of Induction

q1 and Φ1

Then we have a type r1 relation on De1 with respect to the system of parameters x1q1 , p, x3q1 , . . . , xdq1 : x1q1 φe1 (b∗1 ) + pφe1 (b2 ) + · · · + xrq11 φe1 (br1 ) + xrq11+1 φe1 (br1 +1 ) = 0 ¯ e1 with respect to the system And we have a type r1 − 1 relation on D q1 q1 q1 of parameters x1 , x3 , . . . , xd : x1q1 φ¯e1 (b∗1 ) + x3q1 φ¯e1 (b3 ) + · · · + xrq11 φ¯e1 (br1 ) + xrq11+1 φ¯e1 (br1 +1 ) = 0 We remark that n1

n

φ¯e1 (br1 +1 ) = φ¯e1 (b(rp 1 +1)0 ) = φ¯e1 (b(r1 +1)0 ) p 1 .

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First Step of Induction

q1 and Φ1

By Theorem 2.8, Remark 2.11 and 2.12 (3), we may further assume that 1 ¯ e1 c¯N ¯1 is a test element for D (4) 1 γ Then, by Lemma 7.6 n

p 1 1 (cN φe1 (b(r1 +1)0 ) p 1 γ1 )

n1

n1

n1

n1

p p p + x3q1 ω13 + · · · + xrq11 ω1r ≡ x1q1 ω11 1 n1 −1

n1 −2

p p + p2 η12 + pη11 (mod pn1 +1 De1 )

+ · · · + pn1 η1n1

with n1

n1

n1

n1 −1

p p p p ω ¯ 11 ,ω ¯ 13 , ..., ω ¯ 1r , η¯11 1

Jun-ichi NISHIMURA (OECU)

n1 −2

p , η¯12

n1 ¯ e1 )∗ , . . . , η¯1n1 ∈ (¯ γ1p D

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First Step of Induction

q1 and Φ1

Hence n

p 1 1 (cN φe1 (br1 +1 ) 1 γ1 ) n1

n1

n1

p p p = x1q1 ω11 + x3q1 ω13 + · · · + xrq11 ω1r 1 { } N1 pn1 pn1 −1 n1 + (c1 γ1 ) pφe1 (b(r1 +1)1 ) + · · · + p φe1 (b(r1 +1)n1 ) n1 −1

p + pη11

n1 −2

p + p2 η12

+ · · · + pn1 η1n1 + pn1 +1 η1(n1 +1)

Let Φ1 : De1 ⊕ Der11 → De1 be a De1 -homomorphism given by ( 1 pn1 ) pn1 pn1 pn1 Φ1 := (cN − ω1r . . . − ω13 − η˜1 − ω11 1 γ1 ) 1

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First Step of Induction

where n1

p 1 η˜1 := (cN 1 γ1 )

{ } n −1 φe1 (b(r1 +1)1 ) p 1 + · · · + pn1 −1 φe1 (b(r1 +1)n1 ) n1 −1

p + η11

Then Φ1 kills

n1 −2

p + pη12

    1 ⊗ v˜1 =   

Jun-ichi NISHIMURA (OECU)

q1 and Φ1

+ · · · + pn1 −1 η1n1 + pn1 η1(n1 +1)

φe1 (br1 +1 ) xrq11 .. . p x1q1

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      

(6)

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First Step of Induction

ϕ1 and J1

ϕ1 and J1 Because De1 ⊗B N1 = De1 ⊕ Der11 /De1 (1 ⊗ v˜1 ) , Φ1 induces a De1 -homomorphism ϕ1 : De1 ⊗B N1 → De1 . Thus, we have the following diagram ι

−−−→

De 1



De1 ⊕ Der11 −−−→ De1  Φ y 1

De1 −−−−−−n→ N

(c1 1 γ1 ) p

Jun-ichi NISHIMURA (OECU)

1

De 1

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⊗B N1  ϕ y 1

De1

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First Step of Induction

ϕ1 and J1

Then n1

n

n1

n1

p p p p 1 1 Im ϕ1 = ((cN , ω1r , . . . , ω13 , η˜1 , ω11 1 γ1 ) 1

)

De1

(1)

Because a1 := ϕ1 (1 ⊗ β1 )  β0 (  0 = ϕ1 1 ⊗   ... 0

   

)

( = ϕ1



 1 )  0  pn1 1  .  = (cN 1 γ1 )  ..  0

(2)

we have n1

1p a ¯1 = c¯N γ¯1p 1

n1

n n n1 n1 1 N1 p 1 1 N1 p 1 ¯ ̸∈ (xK , xK , . . . , xdK1 N1 p )¯ γ1p R 1 3

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First Step of Induction

Let

ϕ1 and J1

n1

J1 := γ1p De1

(4)

Then ) ( N1 pn1 pn1 pn1 ¯ pn1 ˜ ¯ 13 , η˜1 , ω ¯ 11 De 1 Im ϕ¯1 = (¯ c1 γ¯1 ) , ω ¯ 1r1 , . . . , ω pn1 ¯ ∗ ⊂ (¯ γ1 De1 ) = J¯1∗

(5)

Hence, by 9.3 (3), 9.4 (3) and 9.4 (4) ( q /4 q /4 )∗ q /4 a ¯1 ̸∈ (x1 1 , x3 1 , . . . , xd 1 )J¯1 .

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First Step of Induction

Inductive Notation

Inductive Notation Applying ϕ1 to the top r0 + r1 = 1 + r1 components of each vectors, we get the following diagram: De1 ⊗B N1 −−−→ De1   ϕ1 y De 1

−−−→

⊗B N2 −−−→ · · · −−−→ De1 ⊗B Nt   ϕ1 ⊕1 r2 ϕ1 ⊕1 r2 +···+rt  De D y y e1 1

M11

−−−→ · · · −−−→

M1 (t−1)

Let n

p 1 1 (De1 , (cN ) =: (A1 , a1 ) = (M10 , α10 ) 1 γ1 )

and, for 1 ≤ ℓ ≤ t − 1, put / M1ℓ := (M1(ℓ−1) ⊕ Der1ℓ+1 ) De1 (ϕ1 ⊕ 1Dr2 +···+rℓ+1 (1 ⊗ v˜ℓ+1 )) e1

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First Step of Induction

Inductive Notation

Then we get a new degenerate sequence of modifications of type r1 = (r11 , . . . , r1(t−1) ) = (r2 , . . . , rt ): M1 : (M10 , α10 ) → (M11 , α11 ) → · · · → (M1(t−1) , α1(t−1) )

(1)

in which (M1ℓ , α1ℓ ) is a modification of (M1(ℓ−1) , α1(ℓ−1) ) for a relation ϕ1 ⊕ 1Dr2 +···+rℓ+1 (1 ⊗ ρ∗ℓ+1 ) on M1(ℓ−1) of type r1ℓ = rℓ+1 e1

with respect to the system of parameters x1q1 , p, x3q1 , . . . , xdq1 of A1 . We remark further that n1 γ1p A¯1 )∗ Im ϕ¯1 ⊂ J¯1∗ = (¯ ( q /4 q /4 )∗ q /4 a ¯1 ̸∈ (x1 1 , x3 1 , . . . , xd 1 )J¯1

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First Step of Induction

Notation and Inductive Assumptions

Notation and Inductive Assumptions Suppose that ℓ ≥ 2. Take eℓ−1 ∈ N such that e0 = 0, e1 < · · · < eℓ−2 < eℓ−1 . Let qℓ−1 := pe1 · · · peℓ−1 = pe1 +···+eℓ−1

(1)

We want to describe in an explicit way what a (t − ℓ + 1)th modification of (Aℓ−1 , aℓ−1 ) of type rℓ−1 = (r(ℓ−1)1 , . . . , r(ℓ−1)(t−ℓ+1) ) = (rℓ , . . . , rt ) q

q

q

with respect to a system of parameters x1 ℓ−1 , p, x3 ℓ−1 , . . . , xd ℓ−1 looks like. Jun-ichi NISHIMURA (OECU)

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First Step of Induction

Notation and Inductive Assumptions

Namely, applying ϕℓ−1 to the top r(ℓ−2)0 + r(ℓ−2)1 = 1 + rℓ−1 components of each vectors, we get the following diagram: Deℓ−1 ⊗Bℓ−2 N(ℓ−2)1 −−−→   ϕℓ−1 y Deℓ−1

Deℓ−1 ⊗Bℓ−2 N(ℓ−2)2 −−−→ · · ·  ϕℓ−1 ⊕1 r(ℓ−2)2 De y ℓ−1

−−−→

M(ℓ−1)1

−−−→ · · ·

· · · −−−→

Deℓ−1 ⊗Bℓ−2 N(ℓ−2)(t−ℓ+2)  ϕℓ−1 ⊕1 r(ℓ−2)2 +···+r(ℓ−2)(t−ℓ+2)  y De ℓ−1 · · · −−−→

Jun-ichi NISHIMURA (OECU)

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M(ℓ−1)(t−ℓ+1)

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First Step of Induction

Notation and Inductive Assumptions

Let (Deℓ−1 , ϕℓ−1 (1 ⊗ β(ℓ−2)1 )) =: (Aℓ−1 , aℓ−1 ) = (M(ℓ−1)0 , α(ℓ−1)0 )

(2)

(cf. 9.4 (2)) and, for 1 ≤ m ≤ t − ℓ + 1, put / ℓ+m−1 M(ℓ−1)m := (M(ℓ−1)(m−1) ⊕ Derℓ−1 ) Deℓ−1 (ϕℓ−1 ⊕ 1Drℓ +···+rℓ+m−1 (1 ⊗ v˜ℓ+m−1 ))

(3)

eℓ−1

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First Step of Induction

Notation and Inductive Assumptions

Then we get a new degenerate sequence of modifications of type rℓ−1 : Mℓ−1 : (M(ℓ−1)0 , α(ℓ−1)0 ) → (M(ℓ−1)1 , α(ℓ−1)1 ) → . . . · · · → (M(ℓ−1)(t−ℓ+1) , α(ℓ−1)(t−ℓ+1) ) in which (M(ℓ−1)m , α(ℓ−1)m ) is a modification of (M(ℓ−1)(m−1) , α(ℓ−1)(m−1) ) for a relation ϕℓ−1 ⊕ 1Drℓ +···+rℓ+m−1 (1 ⊗ ρ∗ℓ+m−1 ) eℓ−1

on M(ℓ−1)(m−1) of type r(ℓ−1)m = rℓ+m−1 with respect to the system q q q of parameters x1 ℓ−1 , p, x3 ℓ−1 , . . . , xd ℓ−1 of Aℓ−1 .

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First Step of Induction

Let

Notation and Inductive Assumptions

nℓ−1

p Jℓ−1 := γℓ−1 φeℓ−1 (Jℓ−2 )Deℓ−1

(4)

Then, on A¯ℓ−1 = Aℓ−1 /pAℓ−1 : nℓ−1

p ¯ e )∗ = J¯∗ Im ϕ¯ℓ−1 ⊂ (¯ γℓ−1 φ¯eℓ−1 (J¯ℓ−2 )D ℓ−1 ℓ−1

(5)

and ( q /4ℓ−1 qℓ−1 /4ℓ−1 )∗ q /4ℓ−1 ¯ a ¯ℓ−1 ̸∈ (x1 ℓ−1 , x3 , . . . , xd ℓ−1 )Jℓ−1

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The Case when ℓ ≥ 2 and rℓ ≥ 2

Assumption and Notation

Assumption and Notation Assume that rℓ ≥ 2. We have a type rℓ relation on Aℓ−1 : q

x1 ℓ−1 a(ℓ−1)1 + pa(ℓ−1)2 + · · · q · · · + xrqℓℓ−1 a(ℓ−1)rℓ + xrℓℓ−1 +1 a(ℓ−1)(rℓ +1) = 0 (1) With notation in 6.1, we get a type rℓ relation on Bℓ−1 : x1 ℓ−1 b∗(ℓ−1)1 + pb(ℓ−1)2 + · · · q

q

· · · + xrqℓℓ−1 b(ℓ−1)rℓ + xrℓℓ−1 +1 b(ℓ−1)(rℓ +1) = 0 (2)

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The Case when ℓ ≥ 2 and rℓ ≥ 2

Assumption and Notation

and we put     v˜ℓ =   

b(ℓ−1)(rℓ +1) q xrℓℓ−1 .. . p q x1 ℓ−1

    rℓ  ∈ Bℓ−1 ⊕ Bℓ−1  

(3)

Taking sufficiently large ϵℓ (cf.5.2 (1)), we have b(ℓ−1)(rℓ +1) = nℓ

nℓ −1

p p b(ℓ−1)(r + pb(ℓ−1)(r + · · · + pnℓ b(ℓ−1)(rℓ +1)nℓ (4) ℓ +1)0 ℓ +1)1 pnℓ −i ¯ℓ−1 )∗ for i = 0, 1, . . . , nℓ . with ¯b(ℓ−1)(r ∈ (J¯ℓ−1 B ℓ +1)i

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The Case when ℓ ≥ 2 and rℓ ≥ 2

qℓ and Φℓ

qℓ and Φℓ Let

Mℓ−1

q

p cℓ := uℓ−1 γℓ−1 x1 ℓ−1

(1)

Assume that Kℓ ∈ N satisfies ¯ c¯ℓ ̸∈ (x1Kℓ , x3Kℓ , . . . , xdKℓ )R

(2)

By 9.6 (1), we may assume Kℓ−1 Nℓ−1 pnℓ−1 + Kℓ−2 Nℓ−2 pnℓ−2 +eℓ−1 + · · · · · · + K1 N1 pn1 +e2 +···+eℓ−1 < qℓ−1 (3)

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The Case when ℓ ≥ 2 and rℓ ≥ 2

qℓ and Φℓ

Take eBℓ−1 and Nℓ in 7.3. Then we can find eℓ ∈ N (eℓ > eBℓ−1 ) enough large such that Kℓ Nℓ pnℓ + Kℓ−1 Nℓ−1 pnℓ−1 +eℓ + · · · · · · + K1 N1 pn1 +e2 +···+eℓ−1 +eℓ < qℓ−1 peℓ (4) Let qℓ := qℓ−1 peℓ

(5)

Then we have a type rℓ relation on Deℓ with respect to the system of parameters x1qℓ , p, x3qℓ , . . . , xdqℓ : x1qℓ φeℓ (b∗(ℓ−1)1 ) + pφeℓ (b(ℓ−1)2 ) + · · · · · · + xrqℓℓ φeℓ (b(ℓ−1)rℓ ) + xrqℓℓ+1 φeℓ (b(ℓ−1)(rℓ +1) ) = 0 (6) Jun-ichi NISHIMURA (OECU)

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The Case when ℓ ≥ 2 and rℓ ≥ 2

qℓ and Φℓ

¯ e with respect to the system And we have a type rℓ − 1 relation on D ℓ qℓ qℓ qℓ of parameters x1 , x3 , . . . , xd : x1qℓ φ¯eℓ (b∗(ℓ−1)1 ) + x3qℓ φ¯eℓ (b(ℓ−1)3 ) + · · · · · · + xrqℓℓ φ¯eℓ (b(ℓ−1)rℓ ) + xrqℓℓ+1 φ¯eℓ (b(ℓ−1)(rℓ +1) ) = 0 (7) We remark that nℓ

p φ¯eℓ (b(ℓ−1)(rℓ +1) ) = φ¯eℓ (b(ℓ−1)(r ) = φ¯eℓ (b(ℓ−1)(rℓ +1)0 ) p ℓ +1)0

nℓ

By Theorem 2.8, Remark after Theorem 2.6and 2.12 (3), we may further assume that ℓ ¯e ¯ℓ is a test element for D c¯N ℓ ℓ γ

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The Case when ℓ ≥ 2 and rℓ ≥ 2

qℓ and Φℓ

Then, n

nℓ

p ℓ ℓ (cN φeℓ (b(ℓ−1)(rℓ +1)0 ) p ℓ γℓ ) nℓ

nℓ

nℓ

≡ x1qℓ ωℓ1p + x3qℓ ωℓ3p + · · · + xrqℓℓ ωℓrp ℓ nℓ −1

nℓ −2

+ pηℓ1p + p2 ηℓ2p (mod pnℓ +1 Deℓ )

+ · · · + pnℓ ηℓnℓ

with nℓ

nℓ

nℓ

nℓ −1

ω ¯ ℓ1p , ω ¯ ℓ3p , . . . , ω ¯ ℓrp ℓ , η¯ℓ1p

nℓ −2

, η¯ℓ2p

, . . . , η¯ℓnℓ nℓ ¯ e )∗ ∈ (¯ γℓp φ¯eℓ (J¯ℓ−1 )D ℓ

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The Case when ℓ ≥ 2 and rℓ ≥ 2

qℓ and Φℓ

Hence n

p ℓ ℓ (cN φeℓ (b(ℓ−1)(rℓ +1) ) ℓ γℓ ) nℓ

nℓ

nℓ

= x1qℓ ωℓ1p + x3qℓ ωℓ3p + · · · + xrqℓℓ ωℓrp ℓ { n −1 pnℓ ℓ + (cN γ ) pφeℓ (b(ℓ−1)(rℓ +1)1 ) p ℓ + · · · ℓ ℓ

} · · · + pnℓ φeℓ (b(ℓ−1)(rℓ +1)nℓ )

nℓ −1

+ pηℓ1p

Jun-ichi NISHIMURA (OECU)

nℓ −2

+ p2 ηℓ2p

+ · · · + pnℓ ηℓnℓ + pnℓ +1 ηℓ(nℓ +1)

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The Case when ℓ ≥ 2 and rℓ ≥ 2

qℓ and Φℓ

Let Φℓ : Deℓ ⊕ Derℓℓ → Deℓ be a Deℓ -homomorphism given by ( ℓ pnℓ nℓ nℓ ) nℓ Φℓ := (cN − ωℓrp ℓ . . . − ωℓ3p − η˜ℓ − ωℓ1p ℓ γℓ ) where η˜ℓ :=

pnℓ ℓ (cN ℓ γℓ )

nℓ −1

+ ηℓ1p

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{ n −1 φeℓ (b(ℓ−1)(rℓ +1)1 ) p ℓ + · · ·

} · · · + pnℓ −1 φeℓ (b(ℓ−1)(rℓ +1)nℓ )

nℓ −2

+ pηℓ2p

+ · · · + pnℓ −1 ηℓnℓ + pnℓ ηℓ(nℓ +1) .

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The Case when ℓ ≥ 2 and rℓ ≥ 2

Then Φℓ kills

    1 ⊗ v˜ℓ =   

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qℓ and Φℓ

φeℓ (b(ℓ−1)(rℓ +1) ) xrqℓℓ .. . p x1qℓ

Big Cohen-Macaulay Modules

      

(10)

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The Case when ℓ ≥ 2 and rℓ ≥ 2

ϕℓ and Jℓ

ϕℓ and Jℓ Because Deℓ ⊗Bℓ−1 N(ℓ−1)1 = Deℓ ⊕ Derℓℓ /Deℓ (1 ⊗ v˜ℓ ) Φℓ induces a Deℓ -homomorphism ϕℓ : Deℓ ⊗B N(ℓ−1)1 → Deℓ Thus, we have the following diagram De ℓ



ι

Deℓ ⊕ Derℓℓ −−−→ Deℓ ⊗Bℓ−1 N(ℓ−1)1   Φ ϕ y ℓ y ℓ

−−−→

Deℓ −−−−−−n→ N

(cℓ ℓ γℓ ) p

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ℓ

De ℓ Big Cohen-Macaulay Modules

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The Case when ℓ ≥ 2 and rℓ ≥ 2

ϕℓ and Jℓ

Then ( ℓ pnℓ nℓ nℓ nℓ ) Im ϕℓ = (cN Im ϕℓ−1 , ωℓrp ℓ , . . . , ωℓ3p , η˜ℓ , ωℓ1p Deℓ ℓ γℓ )

(1)

Because aℓ := ϕℓ (1 ⊗ β(ℓ−1)1 )   β (ℓ−1)0 ( )   0   .. = ϕℓ 1 ⊗   . 0   ( φeℓ (bℓ−1 ) )   0  = (cNℓ γℓ ) pnℓ φe (bℓ−1 ) . = ϕℓ  ℓ ℓ .   . 0 Jun-ichi NISHIMURA (OECU)

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135 / 168

The Case when ℓ ≥ 2 and rℓ ≥ 2

ϕℓ and Jℓ

we have a ¯ℓ n nℓ ℓp ℓ = (¯ cN γ¯ℓp )φ¯eℓ (¯bℓ−1 ) ℓ

Nℓ pnℓ

= (¯ cℓ

nℓ

pnℓ

γ¯ℓ

N

ℓ−1 )(¯ cℓ−1

N

ℓ−1 ℓp = (¯ cN c¯ℓ−1 ℓ

(3)

nℓ−1

p

pnℓ−1 +eℓ

nℓ−1

p γ¯ℓ−1 )

1p · · · c¯N 1

× (¯ γℓ K Nℓ pnℓ +Kℓ−1 Nℓ−1

N1 pn1

· · · (¯ c1

n1 +e2 +···+eℓ−1 +eℓ

pnℓ

̸∈ (x1 ℓ

peℓ

pnℓ−1 +eℓ

γ¯ℓ−1

pn1

pe2 +···+eℓ−1 +eℓ

γ¯1 )

) n1 +e2 +···+eℓ−1 +eℓ

· · · γ¯1p

pnℓ−1 +eℓ +···+K1 N1 pn1 +e2 +···+eℓ−1 +eℓ

,

Kℓ Nℓ pnℓ +Kℓ−1 Nℓ−1 pnℓ−1 +eℓ +···+K1 N1 pn1 +e2 +···+eℓ−1 +eℓ

x3

,...

Kℓ Nℓ pnℓ +Kℓ−1 Nℓ−1 pnℓ−1 +eℓ +···+K1 N1 pn1 +e2 +···+eℓ−1 +eℓ

. . . , xd

nℓ

pnℓ−1 +eℓ

× (¯ γℓp γ¯ℓ−1 Jun-ichi NISHIMURA (OECU)

Big Cohen-Macaulay Modules

)

)

pn1 +e2 +···+eℓ−1 +eℓ

· · · γ¯1

2014.10.12

¯ )R

136 / 168

The Case when ℓ ≥ 2 and rℓ ≥ 2

Inductive Notation

Inductive Notation Applying ϕℓ to the top r(ℓ−1)0 + r(ℓ−1)1 = 1 + rℓ components of each vectors, we get the following diagram: Deℓ ⊗Bℓ−1 N(ℓ−1)1 −−−→ Deℓ ⊗Bℓ−1 N(ℓ−1)2 −−−→ · · ·    ϕℓ ⊕1 r(ℓ−1)2 ϕℓ y y De ℓ −−−→

De ℓ

−−−→ · · ·

Mℓ1

· · · −−−→

Deℓ ⊗Bℓ−1 N(ℓ−1)(t−ℓ+1)  ϕℓ−1 ⊕1 r(ℓ−1)2 +···+r(ℓ−1)(t−ℓ+1)  y De ℓ · · · −−−→

Jun-ichi NISHIMURA (OECU)

Mℓ(t−ℓ)

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The Case when ℓ ≥ 2 and rℓ ≥ 2

Inductive Notation

Let (Deℓ , ϕℓ (1 ⊗ β(ℓ−1)1 )) =: (Aℓ , aℓ ) =: (Mℓ0 , αℓ0 )

(1)

(cf. 9.4 (2)), and / Mℓm := (Mℓ(m−1) ⊕ Derℓℓ+m )

(2)

Deℓ (ϕℓ ⊕ 1Drℓ+1 +···+rℓ+m (1 ⊗ v˜ℓ+m )) eℓ

for 1 ≤ m ≤ t − ℓ.

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The Case when ℓ ≥ 2 and rℓ ≥ 2

Inductive Notation

Then we get a new degenerate sequence of modifications of type rℓ := (rℓ1 , . . . , rℓ(t−ℓ) ) = (rℓ+1 , . . . , rt ): Mℓ : (Mℓ0 , αℓ0 ) → (Mℓ1 , αℓ1 ) → . . . . . . → (Mℓ(t−ℓ) , αℓ(t−ℓ) )

(3)

in which (Mℓm , αℓm ) is a modification of (Mℓ(m−1) , αℓ(m−1) ) for a relation ϕℓ ⊕ 1Drℓ+1 +···+rℓ+m (1 ⊗ ρ∗ℓ+m ) eℓ

on Mℓ(m−1) of type rℓm = rℓ+m with respect to the system of parameters x1qℓ , p, x3qℓ , . . . , xdqℓ of Aℓ .

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The Case when ℓ ≥ 2 and rℓ ≥ 2

Inductive Notation

Let nℓ

Jℓ :=γℓp φeℓ (Jℓ−1 )Deℓ

(4)

Then ( Nℓ pnℓ nℓ nℓ ) nℓ ¯ e (5) ¯ ℓ3p , η¯˜ℓ , ω ¯ ℓ1p D cℓ γ¯ℓ ) φ¯eℓ (Im ϕ¯ℓ−1 ), ω ¯ ℓrp ℓ , . . . , ω Im ϕ¯ℓ = (¯ ℓ ( Nℓ pnℓ ) pnℓ pnℓ ¯ pnℓ ¯ ∗ ¯ ⊂ (¯ cℓ γ¯ℓ ) φ¯eℓ (Jℓ−1 ) , ω ¯ ℓrℓ , . . . , ω ¯ ℓ3 , η˜ℓ , ω ¯ ℓ1 Deℓ pnℓ ¯ e )∗ ⊂ (¯ γℓ φ¯eℓ (J¯ℓ−1 )D ℓ = J¯ℓ∗ Hence, by 10.2 (4), 10.3 (3) and 10.4 (4) ( )∗ a ¯ℓ ̸∈ (x1qℓ , x3qℓ , . . . , xdqℓ )J¯ℓ Jun-ichi NISHIMURA (OECU)

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The Case when ℓ ≥ 2 and rℓ = 1

Assumption and Notation

Assumption and Notation Assume that rℓ = 1. We have a type 1 relation on Aℓ−1 : q

x1 ℓ−1 a(ℓ−1)1 + pa(ℓ−1)2 = 0.

(1)

With notation in 5.1, we get a type 1 relation on Sℓ−1 := W [[ξ(ℓ−1)1 , ξ(ℓ−1)3 , . . . , ξ(ℓ−1)d , ξ(ℓ−1)(d+1) , . . . , ξ(ℓ−1)(d+hℓ−1 ) ]] q x1 ℓ−1 s(ℓ−1)1

+ ps(ℓ−1)2 =

rℓ−1 ∑

w(ℓ−1)j f(ℓ−1)j

(2)

j=1

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The Case when ℓ ≥ 2 and rℓ = 1

Assumption and Notation

Then, with notation in Theorem 5.3, a type 1 relation on Eℓ−1 means: ∗ x1 ℓ−1 τ(ℓ−1)1 + pτ(ℓ−1)2 = τ(ℓ−1)0 g(ℓ−1)0 q

(3)

nℓ−1

p where we may assume that τ(ℓ−1)0 = σ ˜(ℓ−1)0 .

Let n

e∗

p ℓ−1 ¯ p ℓ−1 δ¯ℓ−1 := γ¯ℓ−1 δℓ−2

and take a large number Lℓ−1 that satisfies L L L ¯ ℓ−1 δ¯ℓ−1 ̸∈ (x1 ℓ−1 , x3 ℓ−1 , . . . , xd ℓ−1 )R

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The Case when ℓ ≥ 2 and rℓ = 1

Assumption and Notation

Let λℓ−1 be a large number that satisfies 2/pλℓ−1 < qℓ−1 /4ℓ−1 Lℓ−1 . Then λℓ−1

2/p δ¯ℓ−1

( q /4ℓ−1 qℓ−1 /4ℓ−1 )∗ q /4ℓ−1 ¯ ̸∈ (x1 ℓ−1 , x3 , . . . , xd ℓ−1 )Rℓ−1 .

By assumption 9.6 (6), we have ( 2q /4ℓ−1 2qℓ−1 /4ℓ−1 2q /4ℓ−1 ¯ 1−2/pλℓ−1 )∗ a ¯ℓ−1 ̸∈ (x1 ℓ−1 , x3 , . . . , xd ℓ−1 )Jℓ−1

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The Case when ℓ ≥ 2 and rℓ = 1

ε ˇ p ℓ−1 and E

pεℓ−1 and Eˇ For a(ny) given large number εℓ−1 , by taking sufficiently large ϵˇℓ−1 , we have in 11.1 (3) εℓ−1

p ∗ τ¯(ℓ−1)1 = υ¯ℓ−1

¯ ∈ Eˇℓ−1

(1)

(cf. Remark 5.5). By inductive assumption, ∗ ¯ℓ−1 )∗ = (δ¯ℓ−1 B ¯ℓ−1 )∗ τ¯(ℓ−1)1 ∈ (J¯ℓ−1 B

(2)

By Theorem 5.3, 5.5 (1) and 5.5 (2), ¯ℓ−1 = E¯ℓ−1 /¯ B g(ℓ−1)0 E¯ℓ−1 ( ) ¯ ⊂ E¯ˇℓ−1 /¯ g(ℓ−1)0 E¯ˇℓ−1 ↠ E¯ˇℓ−1 / g¯(ℓ−1)0 Eˇℓ−1 ∩ E¯ˇℓ−1 ¯ ¯ ⊂ Eˇ /¯ g Eˇ ℓ−1

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(ℓ−1)0

ℓ−1

2014.10.12

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ε ˇ p ℓ−1 and E

The Case when ℓ ≥ 2 and rℓ = 1

Then

( 1/pεℓ−1 ¯ )∗ ¯ υ¯ℓ−1 ∈ δ¯ℓ−1 (Eˇℓ−1 /¯ g(ℓ−1)0 Eˇℓ−1 )

(3)

By Remark 5.7 (1), for a fixed number Gℓ−1 that is independent on ϵˇℓ−1 , we have G

G

ℓ−1 ℓ−1 χ¯ℓ−1 := υ¯ℓ−1 + ··· + σ ¯i υ¯ℓ−1

−i

¯ + ··· + σ ¯Gℓ−1 ∈ g¯(ℓ−1)0 Eˇℓ−1

i/pεℓ−1 ˇ ¯ . with σ ¯i ∈ δ¯ℓ−1 R ℓ−1

Then εℓ−1

p χ¯ℓ−1

εℓ−1

∗ )Gℓ−1 + · · · + σ ¯ip = (¯ τ(ℓ−1)1 εℓ−1 ¯ ∈ g¯ p Eˇ ∩ E¯ˇ (ℓ−1)0

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ℓ−1

εℓ−1

∗ )Gℓ−1 −i + · · · + σ ¯Gp ℓ−1 (¯ τ(ℓ−1)1

ℓ−1

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The Case when ℓ ≥ 2 and rℓ = 1

ε ˇ p ℓ−1 and E

By Remark 5.7 (2), for a fixed number Hℓ−1 that is independent on ϵˇℓ−1 , we have εℓ−1

εℓ−1

p p (χ¯ℓ−1 )Hℓ−1 + · · · + κ ¯ j (χ¯ℓ−1 )Hℓ−1 −j + · · · + κ ¯ Hℓ−1 = 0 j pεℓ−1 ¯ Eˇℓ−1 . with κ ¯ j ∈ g¯(ℓ−1)0

Then ∗ ∗ (¯ τ(ℓ−1)1 )Gℓ−1 Hℓ−1 + · · · + θ¯k (¯ τ(ℓ−1)1 )Gℓ−1 Hℓ−1 −k + · · · · · · + θ¯G

ℓ−1 Hℓ−1

= 0 (4)

) ( k m pεℓ−1 ¯k−mGℓ−1 pεℓ−1 ¯k−Gℓ−1 δℓ−1 , . . . E¯ˇℓ−1 . , g¯(ℓ−1)0 δℓ−1 , . . . , g¯(ℓ−1)0 with θ¯k ∈ δ¯ℓ−1

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The Case when ℓ ≥ 2 and rℓ = 1

ε ˇ p ℓ−1 and E

For a given large number pλℓ−1 , take sufficiently large ϵˇℓ−1 (cf. Remark 5.7). Then pεℓ−1 in 11.2 (1) satisfies the condition pεℓ−1 ≥ pλℓ−1 Gℓ−1 and m pεℓ−1 ¯k−mGℓ−1 k (δ¯ℓ−1 , . . . , g¯(ℓ−1)0 δℓ−1 , . . . )Eˇ¯ℓ−1 mpλℓ−1 G k−mG k ⊂ (δ¯ℓ−1 , . . . , g¯(ℓ−1)0 ℓ−1 δ¯ℓ−1 ℓ−1 , . . . )Eˇ¯ℓ−1 p ℓ−1 k ¯ ) Eˇℓ−1 ⊂ (δ¯ℓ−1 , g¯(ℓ−1)0 λ

λℓ−1 1/pλℓ−1 ⊂ (δ¯ℓ−1 , g¯(ℓ−1)0 ) p k E¯ˇℓ−1

This means that λ 1/pλℓ−1 ∗ τ¯(ℓ−1)1 ∈ (δ¯ℓ−1 , g¯(ℓ−1)0 ) p ℓ−1 Eˇ¯ℓ−1 Jun-ichi NISHIMURA (OECU)

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The Case when ℓ ≥ 2 and rℓ = 1

ε ˇ p ℓ−1 and E

Thanks to Theorem 2.13 1/pλℓ−1

(δ¯ℓ−1

Thus

λℓ−1

, g¯(ℓ−1)0 ) p

( 1/pλℓ−1 )∗ λℓ−1 E¯ˇℓ−1 ⊂ (δ¯ℓ−1 , g¯(ℓ−1)0 ) p −1 E¯ˇℓ−1 ( 1/pλℓ−1 (pλℓ−1 −2) 2 )∗ ⊂ (δ¯ℓ−1 , g¯(ℓ−1)0 )E¯ˇℓ−1 ( 1−2/pλℓ−1 2 )∗ = (δ¯ℓ−1 , g¯(ℓ−1)0 )Eˇ¯ℓ−1

( 1−2/pλℓ−1 2 )∗ ∗ τ¯(ℓ−1)1 ∈ (δ¯ℓ−1 , g¯(ℓ−1)0 )E¯ˇℓ−1

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The Case when ℓ ≥ 2 and rℓ = 1

ε ˇ p ℓ−1 and E

By Theorem 2.10, we have (

q 1−2/p x1 ℓ−1 (δ¯ℓ−1

λℓ−1

)∗ 2 , g¯(ℓ−1)0 )E¯ˇℓ−1 : g¯(ℓ−1)0 E¯ˇℓ−1 (( )∗ ) q 1−2/pλℓ−1 2 = x1 ℓ−1 (δ¯ℓ−1 , g¯(ℓ−1)0 ) : g¯(ℓ−1)0 E¯ˇℓ−1 ( q )∗ 1−2/pλℓ−1 = x1 ℓ−1 (δ¯ , g¯(ℓ−1)0 )E¯ˇℓ−1 ℓ−1

Then, by 11.1 (3) and 11.2 (5), we have ( q )∗ 1−2/pλℓ−1 pnℓ−1 τ¯(ℓ−1)0 = τ¯˜(ℓ−1)0 ∈ x1 ℓ−1 (δ¯ℓ−1 , g¯(ℓ−1)0 )E¯ˇℓ−1

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The Case when ℓ ≥ 2 and rℓ = 1

Assumption and Notation (continued)

Assumption and Notation (continued) Remember that rℓ = 1. We have a type 1 relation on Eℓ−1 (cf. 11.1 (3)): ∗ x1 ℓ−1 τ(ℓ−1)1 + pτ(ℓ−1)2 = τ(ℓ−1)0 g(ℓ−1)0 q

(1)

With notation above, let ϵ˜ℓ−1 be a sufficiently large number that satisfies ϵ˜ℓ−1 > ϵˇℓ−1 By taking pϵ˜ℓ−1 th root ξ˜(ℓ−1)i of ξ(ℓ−1)i and pϵ˜ℓ−1 th root π ˜ℓ−1 of πℓ−1 , let ˜ ℓ−1 := W [[ξ˜(ℓ−1)1 , ξ˜(ℓ−1)3 , . . . , ξ˜(ℓ−1)d ]] Rℓ−1 ⊂ R Sℓ−1 ⊂ S˜ℓ−1 := W [[ξ˜(ℓ−1)1 , ξ˜(ℓ−1)3 , . . . , ξ˜(ℓ−1)d , π ˜ℓ−1 , ξ˜(ℓ−1)(d+1) , . . . , ξ˜ ˜

(2) (3)

(ℓ−1)(d+hℓ−1 ) ]]

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The Case when ℓ ≥ 2 and rℓ = 1

Assumption and Notation (continued)

With notation in 6.1, we get a type 1 relation on E˜ℓ−1 : ∗ x1 ℓ−1 τ˜(ℓ−1)1 + p˜ τ(ℓ−1)2 = τ˜(ℓ−1)0 g˜(ℓ−1)0 q

(4)

where, by taking a new variable ϖℓ−1 on S˜ℓ−1 , ˜ Q

ν˜

(ℓ−1)0 ℓ−1 g˜(ℓ−1)0 := ϖℓ−1 − c˜ℓ−1 ϖℓ−1 − p

ν˜

(5) ν˜

(ℓ−1)0 (ℓ−1) ∗ ∗ x1 ℓ−1 τ˜(ℓ−1)1 := x1 ℓ−1 τ(ℓ−1)1 − τ(ℓ−1)0 c˜ℓ−1 ϖℓ−1 + c˜ℓ−1 υ˜ℓ−1

q

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q

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The Case when ℓ ≥ 2 and rℓ = 1

Assumption and Notation (continued)

Because ˜ℓ−1 := E˜ℓ−1 /˜ B g(ℓ−1)0 E˜ℓ−1 ˜ℓ−1 : we get a type 1 relation on B q x1 ℓ−1 ˜b∗(ℓ−1)1 + p˜b(ℓ−1)2 = 0

and we put

( v˜ℓ =

Jun-ichi NISHIMURA (OECU)

˜b(ℓ−1)2 q x1 ℓ−1

(7)

) ˜ℓ−1 ⊕ B ˜ℓ−1 ∈B

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The Case when ℓ ≥ 2 and rℓ = 1

Assumption and Notation (continued)

Because we may assume that n ˜ ℓ−1

p ∗ τ(ℓ−1)1 =σ ˜(ℓ−1)1

(9)

n ˜ ( ¯˜ )∗ and that ¯˜(ℓ−1)1 ∈ J¯1/p ℓ−1 B with σ ℓ−1 ℓ−1 nℓ−1

p τ(ℓ−1)0 = τ˜(ℓ−1)0

(10)

( q ) 1−2/pλℓ−1 ¯ pnℓ−1 ˜ℓ−1 ∗ (cf. 11.2 (6)) with τ¯˜(ℓ−1)0 ∈ x1 ℓ−1 J¯ℓ−1 B and because ν˜

˜ Q

(ℓ−1)0 ℓ−1 −p c˜ℓ−1 ϖℓ−1 ≡ ϖℓ−1

(mod g˜(ℓ−1)0 )

(11)

if ϵ˜ℓ−1 is chosen large enough (cf. 5.2 (1)), we have Jun-ichi NISHIMURA (OECU)

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The Case when ℓ ≥ 2 and rℓ = 1

Assumption and Notation (continued)

x1 ℓ−1 ˜b∗(ℓ−1)1 q

n ˜ ℓ−1

q

˜ Q

nℓ−1

ν˜

p p (ℓ−1) ℓ−1 = x1 ℓ−1 σ ˜(ℓ−1)1 − τ˜(ℓ−1)0 (ϖℓ−1 − p) + c˜ℓ−1 υ˜ℓ−1 n ˜

−1

−2

n ˜

p ℓ−1 p ℓ−1 + p2˜b(ℓ−1)12 + · · · + pn˜ ℓ−1 +1˜b(ℓ−1)1˜nℓ−1 +1 (12) = p˜b(ℓ−1)11

with ¯˜b pn˜ ℓ−1 −i ∈ (x qℓ−1 J¯1−2/pλℓ−1 B ¯˜ )∗ for i = 1, . . . , n ˜ ℓ−1 + 1. ℓ−1 1 ℓ−1 (ℓ−1)1i Hence n ˜

n ˜

−1

˜b(ℓ−1)2 = ˜b p ℓ−1 + p˜b p ℓ−1 + · · · + pn˜ ℓ−1 ˜b(ℓ−1)2˜n ℓ−1 (ℓ−1)20 (ℓ−1)21

(13)

with ¯˜b pn˜ ℓ−1 −i ∈ (x qℓ−1 J¯1−2/pλℓ−1 B ¯˜ )∗ for i = 0, . . . , n ˜ ℓ−1 . ℓ−1 1 ℓ−1 (ℓ−1)2i Jun-ichi NISHIMURA (OECU)

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The Case when ℓ ≥ 2 and rℓ = 1

qℓ and Φℓ

qℓ and Φℓ Let

Mℓ−1

q

p cℓ := u˜ℓ−1 γ˜ℓ−1 x1 ℓ−1

(1)

Assume that Kℓ ∈ N satisfies ¯ c¯ℓ ̸∈ (x1Kℓ , x3Kℓ , . . . , xdKℓ )R

(2)

By 9.3 (3), we may assume ∗

Kℓ−1 Nℓ−1 pnℓ−1 + Kℓ−2 Nℓ−2 pnℓ−2 +eℓ−1 + · · · ∗

∗

· · · + K1 N1 pn1 +e2 +···+eℓ−1 < qℓ−1 (3)

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The Case when ℓ ≥ 2 and rℓ = 1

qℓ and Φℓ

Take eB˜ℓ−1 and Nℓ in 7.3. Then we can find eℓ ∈ N (eℓ > eB˜ℓ−1 ) enough large such that Kℓ Nℓ pnℓ + Kℓ−1 Nℓ−1 pnℓ−1 +eℓ + · · · ∗

∗

· · · + K1 N1 pn1 +e2 +···+eℓ−1 +eℓ < qℓ−1 peℓ (4) Let qℓ := qℓ−1 peℓ

(5)

Then we have a type 1 relation on Deℓ with respect to the system of parameters x1qℓ , p, x3qℓ , . . . , xdqℓ : x1qℓ φeℓ (˜b∗(ℓ−1)1 ) + pφeℓ (˜b(ℓ−1)2 ) = 0

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The Case when ℓ ≥ 2 and rℓ = 1

qℓ and Φℓ

Here n ˜

n ˜

−1

p ℓ−1 p ℓ−1 φeℓ (˜b(ℓ−1)2 ) = φeℓ (˜b(ℓ−1)20 ) + pφeℓ (˜b(ℓ−1)21 ) + ···

· · · + pn˜ ℓ−1 φeℓ (˜b(ℓ−1)2˜nℓ−1 ) (7) with ( ˜ ℓ−1 −i 1−2/pλℓ−1 ¯ )∗ pn φeℓ (¯˜b(ℓ−1)2i ) ∈ x1qℓ φeℓ (J¯ℓ−1 )Deℓ for i = 0, . . . , n ˜ ℓ−1 .

By Theorem 2.8, Remark after Theorem 2.6 and 2.12 (3), we may assume that ℓ ¯e c¯N ¯ℓ is a test element for D (8) ℓ ℓ γ

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The Case when ℓ ≥ 2 and rℓ = 1

qℓ and Φℓ

Then, p ℓ (cN ℓ γℓ )

n ˜ ℓ−1 −i

˜ ℓ−1 −i pn φeℓ (¯˜b(ℓ−1)2i ) n ˜ ℓ−1 −i

p ℓ = (cN ℓ γℓ )

n ˜ ℓ−1 −i

≡ x1qℓ ωℓip with

n ˜ ℓ−1 −i

ω ¯ ℓip

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n ˜ ℓ−1 −i

∈ (¯ γℓp

n ˜ ℓ−1 −i φeℓ (¯˜b(ℓ−1)2i ) p

(mod pn˜ ℓ−1 −i+1 Deℓ )

1−2/p φ¯eℓ (J¯ℓ−1

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¯ e )∗ )D ℓ

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The Case when ℓ ≥ 2 and rℓ = 1

qℓ and Φℓ

Letting pnℓ := pn˜ ℓ−1 and

nℓ −i

ηℓip

nℓ −i

i p ℓ := (cN ℓ γℓ ) ωℓi

we have pnℓ ℓ (cN φeℓ (˜b(ℓ−1)2 ) ℓ γℓ ) ( nℓ ) nℓ −1 ≡ x1qℓ ηℓ0p + pηℓ1p + · · · + pnℓ ηℓnℓ (mod pnℓ +1 Deℓ )

with nℓ −i

η¯ℓip

nℓ −i

ℓ = (¯ cN ¯ℓ )i ω ¯ ℓip ℓ γ

Jun-ichi NISHIMURA (OECU)

nℓ

1−2/pλℓ−1

∈ (¯ γℓp φ¯eℓ (J¯ℓ−1

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The Case when ℓ ≥ 2 and rℓ = 1

qℓ and Φℓ

Hence pnℓ ℓ (cN φeℓ (˜b(ℓ−1)2 ) ℓ γℓ ) nℓ

nℓ −1

= x1qℓ (ηℓ0p + pηℓ1p

+ · · · + pnℓ ηℓnℓ ) + pnℓ +1 ηℓ(nℓ +1)

Remark that x1qℓ | ηℓ(nℓ +1) by 11.4 (6) That is,

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′ ηℓ(nℓ +1) = x1qℓ ηℓ(n ℓ +1)

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The Case when ℓ ≥ 2 and rℓ = 1

qℓ and Φℓ

Let Φℓ : Deℓ ⊕ Deℓ → Deℓ be a Deℓ -homomorphism given by ( ℓ pnℓ Φℓ := (cN ℓ γℓ ) nℓ

nℓ −1

− (ηℓ0p + pηℓ1p

Then Φℓ kills

( 1 ⊗ v˜ℓ =

Jun-ichi NISHIMURA (OECU)

) ′ + · · · + pnℓ ηℓnℓ + pnℓ +1 ηℓ(n ) (9) +1) ℓ

φeℓ (˜b(ℓ−1)2 ) x1qℓ

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The Case when ℓ ≥ 2 and rℓ = 1

ϕℓ and Jℓ

ϕℓ and Jℓ Because Deℓ ⊗Bℓ−1 N(ℓ−1)1 = Deℓ ⊕ Deℓ /Deℓ (1 ⊗ v˜ℓ ) Φℓ induces a Deℓ -homomorphism ϕℓ : Deℓ ⊗B N(ℓ−1)1 → Deℓ Thus, we have the following diagram De ℓ



ι

−−−→

Deℓ ⊕ Deℓ −−−→ Deℓ ⊗Bℓ−1 N(ℓ−1)1   Φ ϕ y ℓ y ℓ

Deℓ −−−−−−n→ N

(cℓ ℓ γℓ ) p

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ℓ

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The Case when ℓ ≥ 2 and rℓ = 1

ϕℓ and Jℓ

Then ( ℓ pnℓ nℓ nℓ −1 Im ϕℓ−1 , ηℓ0p + pηℓ1p Im ϕℓ = (cN + ··· ℓ γℓ )

) ′ · · · + pnℓ ηℓnℓ + pnℓ +1 ηℓ(n Deℓ (1) ℓ +1)

Because aℓ := ϕℓ (1 ⊗ β(ℓ−1)1 ) ( ( )) β(ℓ−1)0 = ϕℓ 1 ⊗ 0 (( )) φeℓ (bℓ−1 ) pnℓ ℓ = ϕℓ = (cN φeℓ (bℓ−1 ) ℓ γℓ ) 0

(2)

we have Jun-ichi NISHIMURA (OECU)

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The Case when ℓ ≥ 2 and rℓ = 1

ϕℓ and Jℓ

n nℓ ℓp ℓ a ¯ℓ = (¯ cN γ¯ℓp )φ¯eℓ (¯bℓ−1 ) ℓ

Nℓ pnℓ

= (¯ cℓ

nℓ

pnℓ

γ¯ℓ

N

ℓ−1 )(¯ cℓ−1

N

ℓ−1 ℓp = (¯ cN c¯ℓ−1 ℓ

(3)

pnℓ−1

pnℓ−1 +eℓ

pnℓ−1

peℓ

γ¯ℓ−1 )

N1 pn1

· · · (¯ c1

n1 +e2 +···+eℓ−1 +eℓ

1p · · · c¯N 1

pnℓ

× (¯ γℓ

pnℓ−1 +eℓ

γ¯ℓ−1

pn1

pe2 +···+eℓ−1 +eℓ

γ¯1 )

) n1 +e2 +···+eℓ−1 +eℓ

· · · γ¯1p

Kℓ Nℓ pnℓ +Kℓ−1 Nℓ−1 pnℓ−1 +eℓ +···+K1 N1 pn1 +e2 +···+eℓ−1 +eℓ

̸∈ (x1

,

Kℓ Nℓ pnℓ +Kℓ−1 Nℓ−1 pnℓ−1 +eℓ +···+K1 N1 pn1 +e2 +···+eℓ−1 +eℓ

x3

,...

Kℓ Nℓ pnℓ +Kℓ−1 Nℓ−1 pnℓ−1 +eℓ +···+K1 N1 pn1 +e2 +···+eℓ−1 +eℓ

. . . , xd

nℓ

pnℓ−1 +eℓ

× (¯ γℓp γ¯ℓ−1

Jun-ichi NISHIMURA (OECU)

Big Cohen-Macaulay Modules

)

)

n1 +e2 +···+eℓ−1 +eℓ

· · · γ¯1p

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The Case when ℓ ≥ 2 and rℓ = 1

Inductive Notation

Inductive Notation Applying ϕℓ to the top r(ℓ−1)0 + r(ℓ−1)1 = 1 + rℓ = 2 components of each vectors, we get the following diagram: Deℓ ⊗Bℓ−1 N(ℓ−1)1 −−−→ Deℓ ⊗Bℓ−1 N(ℓ−1)2 −−−→ · · ·    ϕℓ ⊕1 r(ℓ−1)2 ϕℓ y y De ℓ −−−→

De ℓ

−−−→ · · ·

Mℓ1

· · · −−−→

Deℓ ⊗Bℓ−1 N(ℓ−1)(t−ℓ+1)  ϕℓ−1 ⊕1 r(ℓ−1)2 +···+r(ℓ−1)(t−ℓ+1)  y De ℓ · · · −−−→

Jun-ichi NISHIMURA (OECU)

Mℓ(t−ℓ)

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The Case when ℓ ≥ 2 and rℓ = 1

Inductive Notation

Let (Deℓ , ϕℓ (1 ⊗ β(ℓ−1)1 )) =: (Aℓ , aℓ ) = (Mℓ0 , αℓ0 )

(1)

(cf. 9.4 (2)), and / Mℓm := (Mℓ(m−1) ⊕Derℓℓ+m )

(2)

Deℓ (ϕℓ ⊕ 1Drℓ+1 +···+rℓ+m (1 ⊗ v˜ℓ+m )) eℓ

for 1 ≤ m ≤ t − ℓ.

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The Case when ℓ ≥ 2 and rℓ = 1

Inductive Notation

Then we get a new degenerate sequence of modifications of type rℓ := (rℓ1 , . . . , rℓ(t−ℓ) ) = (rℓ+1 , . . . , rt ): Mℓ : (Mℓ0 , αℓ0 ) → (Mℓ1 , αℓ1 ) → · · · → (Mℓ(t−ℓ) , αℓ(t−ℓ) )

(3)

in which (Mℓm , αℓm ) is a modification of (Mℓ(m−1) , αℓ(m−1) ) for a relation ϕℓ ⊕ 1Drℓ+1 +···+rℓ+m (1 ⊗ ρ∗ℓ+m ) eℓ

on Mℓ(m−1) of type rℓm = rℓ+m with respect to the system of parameters x1qℓ , p, x3qℓ , . . . , xdqℓ of Aℓ .

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The Case when ℓ ≥ 2 and rℓ = 1

Inductive Notation

Let nℓ

1−2/pλℓ−1

Jℓ :=γℓp φeℓ (Jℓ−1

)Deℓ

(4)

Then ( Nℓ pnℓ nℓ ) ¯e Im ϕ¯ℓ = (¯ cℓ γ¯ℓ ) φ¯eℓ (Im ϕ¯ℓ−1 ), η¯ℓ0p D ℓ ( Nℓ pnℓ ) n ℓ p ∗ ¯e ⊂ (¯ cℓ γ¯ℓ ) φ¯eℓ (J¯ℓ−1 ) , η¯ℓ0 D ℓ nℓ 1−2/p ⊂ (¯ γℓp φ¯eℓ (J¯ℓ−1 = J¯ℓ∗

λℓ−1

(5)

¯ e )∗ )D ℓ

Hence, by 11.4 (4), 11.5 (3) and 11.6 (4) ( )∗ a ¯ℓ ̸∈ (x1qℓ , x3qℓ , . . . , xdqℓ )J¯ℓ Jun-ichi NISHIMURA (OECU)

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References

References [1] Bruns, W.–Herzog, J., Cohen–Macaulay rings, Cambridge University Press 1993. [2] EGA, chapitre IV, Publ. Math. I.H.E.S. 20 (1964), 24 (1965). [3] Flenner, H., Die S¨atze von Bertini f¨ur lokale Ringe, Math. Ann. 229 (1977), 97–111. [4] Heitmann, R. C.,The direct summand conjecture in dimension three, Ann. of Math., 156 (2002), 695–712. [5] Hochster, M.,Topics in the homological theory of modules over commutative rings, Regional Conference Ser. Math., 24, Amer. Math. Soc., 1975. Jun-ichi NISHIMURA (OECU)

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References

[6] Hochster, M., Big Cohen–Macaulay algebras in dimension three via Heitmann’s theorem, J. Algebra 254 (2002), 395–408. [7] Huneke, C., Tight closure and its applications, Regional Conference Ser. Math., 88, Amer. Math. Soc., 1996. [8] Matsumura, H., Commutative Algebra, Benjamin 1970 (second ed. 1980). [9] Matsumura, H., Commutative ring theory, Cambridge University Press 1986. [10] Mumford, D., Lectures on Curves on an Algebraic Surface, Princeton University Press 1966. [11] Nagata, M., Local Rings, John Wiley 1962 (reprt. ed. Krieger 1975). Jun-ichi NISHIMURA (OECU)

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References

[12] Peskine, C.–Szpiro, L., Dimension projective finie et cohomologie locale, Publ. Math. I.H.E.S. 42(1973), 323–395. [13] Raynaud, M., Anneaux Locaux Hens´eliens, Lect. Notes Math. 169, Springer–Verlag, 1970. [14] Roberts, P., Two applications of dualizing complexes over local rings, Ann. ENS (4) 9 (1976), 103–106. [15] Roberts, P., Le th´eor`eme d’intersection, C. R. Acad Paris 304 (1987), 177–180. [16] Swanson, I.–Huneke, C., Integral Closure of Ideals, Rings, and Modules, London Math. Soc. Lect. Note Ser. 336, Cambridge Univ. Press 2006. Jun-ichi NISHIMURA (OECU)

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