Evans-Griffith Syzygy TheoremComparison of Order Ideals of Consecutive Syzygies

The Strong Syzygy Theorem

Syzygy Theorems viaComparison of Order Ideals

Alexandra Seceleanu

University of Illinois at Urbana-Champaign

AMS National Meeting in San FranciscoJanuary 2010

Alexandra Seceleanu Syzygy Theorems

Evans-Griffith Syzygy TheoremComparison of Order Ideals of Consecutive Syzygies

The Strong Syzygy Theorem

Evans-Griffith Syzygy Theorem

The Evans-Griffith Syzygy Theorem (1981) asserts:

Theorem (Evans-Griffith)

A non-free, finitely generated and finite projective dimension kth

module of syzygies over a Cohen-Macaulay local ring R containinga field, has rank at least k.

It is known

for rings R containing a field

for graded resolutions over R = V [[x1, . . . , xn]], V a DVR

for R of any (including mixed) characteristic and dim R ≤ 5.

Alexandra Seceleanu Syzygy Theorems

Evans-Griffith Syzygy TheoremComparison of Order Ideals of Consecutive Syzygies

The Strong Syzygy Theorem

Order Ideals

Definition

If E is an R-module and e ∈ E , the order ideal of e is defined by

OE (e) = {f (e)|f ∈ HomR(E ,R)}.

Then (a1,1, . . . , a1,n) ⊆ OE (e1):

E // Rm

26664a1,1 . . . a1,m

......

...an,1 . . . an,m

37775// Rn

π

**UUUUUUUUUUUUUUUUUU

e1� // e1

� // (a1,1, . . . , a1,n)

**UUUUUUUUUUUUUU R

a1,j

Alexandra Seceleanu Syzygy Theorems

Evans-Griffith Syzygy TheoremComparison of Order Ideals of Consecutive Syzygies

The Strong Syzygy Theorem

Order Ideals

Definition

If E is an R-module and e ∈ E , the order ideal of e is defined by

OE (e) = {f (e)|f ∈ HomR(E ,R)}.

If E is the kernel of

E // Rm

26664a1,1 . . . a1,m

......

...an,1 . . . an,m

37775// Rn

and if e1 ∈ E

Alexandra Seceleanu Syzygy Theorems

Evans-Griffith Syzygy TheoremComparison of Order Ideals of Consecutive Syzygies

The Strong Syzygy Theorem

Order Ideals

Definition

If E is an R-module and e ∈ E , the order ideal of e is defined by

OE (e) = {f (e)|f ∈ HomR(E ,R)}.

Then (a1,1, . . . , a1,n) ⊆ OE (e1):

E // Rm

26664a1,1 . . . a1,m

......

...an,1 . . . an,m

37775// Rn

π

**UUUUUUUUUUUUUUUUUU

e1� // e1

� // (a1,1, . . . , a1,n)

**UUUUUUUUUUUUUU R

a1,j

Alexandra Seceleanu Syzygy Theorems

Evans-Griffith Syzygy TheoremComparison of Order Ideals of Consecutive Syzygies

The Strong Syzygy Theorem

Connection between order ideals and ranks of syzygies

characteristic p

Order Ideal Theorem:heightOE (e) ≥ k

Syzygy Theorem holds.

Order Ideal Theo-rem in mixed char

unramified mixed char p

Syzygy Theorem holdsfor E such that

pExt1(E , ·) ≡ 0.

Comparison

Alexandra Seceleanu Syzygy Theorems

Evans-Griffith Syzygy TheoremComparison of Order Ideals of Consecutive Syzygies

The Strong Syzygy Theorem

Connection between order ideals and ranks of syzygies

characteristic p

Order Ideal Theorem:heightOE (e) ≥ k

Syzygy Theorem holds.

Order Ideal Theo-rem in mixed char

unramified mixed char p

Syzygy Theorem holdsfor E such that

pExt1(E , ·) ≡ 0.

Comparison

Alexandra Seceleanu Syzygy Theorems

Evans-Griffith Syzygy TheoremComparison of Order Ideals of Consecutive Syzygies

The Strong Syzygy Theorem

Connection between order ideals and ranks of syzygies

characteristic p

Order Ideal Theorem:heightOE (e) ≥ k

Syzygy Theorem holds.

Order Ideal Theo-rem in mixed char

unramified mixed char p

Syzygy Theorem holdsfor E such that

pExt1(E , ·) ≡ 0.

Comparison

Alexandra Seceleanu Syzygy Theorems

Evans-Griffith Syzygy TheoremComparison of Order Ideals of Consecutive Syzygies

The Strong Syzygy Theorem

Connection between order ideals and ranks of syzygies

characteristic p

Order Ideal Theorem:heightOE (e) ≥ k

Syzygy Theorem holds.

Order Ideal Theo-rem in mixed char

unramified mixed char p

Syzygy Theorem holdsfor E such that

pExt1(E , ·) ≡ 0.

Comparison

Alexandra Seceleanu Syzygy Theorems

Evans-Griffith Syzygy TheoremComparison of Order Ideals of Consecutive Syzygies

The Strong Syzygy Theorem

Superficial elements

Definition (Samuel)

Let R be a ring , I an ideal, M an R-module. We say a non-zerodivisor x ∈ I is a superficial element of I with respect to M if

(I n+1M :M x) = I nM

.

(For the purposes of this talk one may just think x ∈ m −m2.)

Alexandra Seceleanu Syzygy Theorems

Evans-Griffith Syzygy TheoremComparison of Order Ideals of Consecutive Syzygies

The Strong Syzygy Theorem

Superficial elements

Definition (Samuel)

Let R be a ring , I an ideal, M an R-module. We say a non-zerodivisor x ∈ I is a superficial element of I with respect to M if

(I n+1M :M x) = I nM

.

(For the purposes of this talk one may just think x ∈ m −m2.)

Alexandra Seceleanu Syzygy Theorems

Evans-Griffith Syzygy TheoremComparison of Order Ideals of Consecutive Syzygies

The Strong Syzygy Theorem

Comparison Theorem for heights of order ideals

Theorem (-, Comparison Theorem)

Let E have minimal presentation 0 −→ Zι−→ F → E → 0 and

suppose we have

x ∈ m superficial with respect to Z

xExt1R(M, ·) ≡ 0

Then one can build a diagram

0 // Zι //

·x��

F //

f����������E // 0

Z

Alexandra Seceleanu Syzygy Theorems

Evans-Griffith Syzygy TheoremComparison of Order Ideals of Consecutive Syzygies

The Strong Syzygy Theorem

Comparison Theorem for heights of order ideals

Theorem (-, Comparison Theorem)

Let E have minimal presentation 0 −→ Zι−→ F → E → 0 and

suppose we have

x ∈ m superficial with respect to Z

xExt1R(M, ·) ≡ 0

Then one can build a diagram

0 // Zι //

·x��

F //

f����������E // 0

Z

Alexandra Seceleanu Syzygy Theorems

Evans-Griffith Syzygy TheoremComparison of Order Ideals of Consecutive Syzygies

The Strong Syzygy Theorem

Comparison Theorem for heights of order ideals

Theorem (-, Comparison Theorem)

Let E have minimal presentation 0 −→ Zι−→ F → E → 0 and

suppose we have

x ∈ m superficial with respect to Z

xExt1R(M, ·) ≡ 0

Then one can build a diagram

0 // Zι //

·x��

F //

f����������E // 0

Z

Alexandra Seceleanu Syzygy Theorems

Evans-Griffith Syzygy TheoremComparison of Order Ideals of Consecutive Syzygies

The Strong Syzygy Theorem

Comparison Theorem for heights of order ideals

Theorem (-, Comparison Theorem)

Let E have minimal presentation 0 −→ Zι−→ F → E → 0 and

suppose we have

x ∈ m superficial with respect to Z

xExt1R(M, ·) ≡ 0

Then one can build a diagram

0 // Zι //

·x��

F //

f����������E // 0

Z

Alexandra Seceleanu Syzygy Theorems

Evans-Griffith Syzygy TheoremComparison of Order Ideals of Consecutive Syzygies

The Strong Syzygy Theorem

Comparison Theorem for heights of order ideals

Theorem (-, Comparison Theorem)

Let E have minimal presentation 0 −→ Zι−→ F → E → 0 and

suppose we have

x ∈ m superficial with respect to Z

xExt1R(E , ·) ≡ 0

Then one can build a diagram

0 // Zι //

·x��

F //

fvvmmmmmmmmmm E //

��0

Zπ��

E = E/xE

fssgggggggggggggg

Z = Z/xZ

such that Im(f ) 6⊆ mZ

Alexandra Seceleanu Syzygy Theorems

Evans-Griffith Syzygy TheoremComparison of Order Ideals of Consecutive Syzygies

The Strong Syzygy Theorem

Comparison Theorem for heights of order ideals

Theorem (-, Comparison Theorem)

Let E have minimal presentation 0 −→ Zι−→ F → E → 0 and

suppose we have

x ∈ m superficial with respect to Z

xExt1R(E , ·) ≡ 0

Then one can build a diagram

0 // Zι //

·x��

F //

fvvmmmmmmmmmm E //

��0

Zπ��

E = E/xE

fssgggggggggggggg

Z = Z/xZ

such that Im(f ) 6⊆ mZ

Alexandra Seceleanu Syzygy Theorems

Evans-Griffith Syzygy TheoremComparison of Order Ideals of Consecutive Syzygies

The Strong Syzygy Theorem

Comparison Theorem for heights of order ideals

Theorem (-, Comparison Theorem)

Let E have minimal presentation 0 −→ Zι−→ F → E → 0 and

suppose we have

x ∈ m superficial with respect to Z

xExt1R(M, ·) ≡ 0

Then one can build a diagram

0 // Zι //

·x��

F //

fvvmmmmmmmmmm E //

��0

Zπ��

E = E/xE

fssgggggggggggggg

Z = Z/xZ

such that Im(f ) 6⊆ mZ and consequently htOE (e) ≥ htOZ (f (e)).

Alexandra Seceleanu Syzygy Theorems

Evans-Griffith Syzygy TheoremComparison of Order Ideals of Consecutive Syzygies

The Strong Syzygy Theorem

Main Consequences on the Comparison Theorem

To apply the Comparison Theorem with x = p we need that p beunramified i.e. p ∈ m −m2.

Theorem (Griffith, -)

Let (R,m) be an unramified Cohen-Macaulay local ring of mixedcharacteristic. Assume that M is a finite projective dimensionR-module such that for a fixed k, pExtk+1

R (M, ·) ≡ 0. Then theSyzygy theorem holds for every j th syzygy of M with j ≥ k.

Alexandra Seceleanu Syzygy Theorems

Evans-Griffith Syzygy TheoremComparison of Order Ideals of Consecutive Syzygies

The Strong Syzygy Theorem

Main Consequences on the Comparison Theorem

Theorem (Griffith, -)

Let (R,m) be an unramified Cohen-Macaulay local ring of mixedcharacteristic. Then the Syzygy Theorem holds over R for syzygiesof modules of the type R/Q with Q ∈ Spec(R).

Alexandra Seceleanu Syzygy Theorems

Evans-Griffith Syzygy TheoremComparison of Order Ideals of Consecutive Syzygies

The Strong Syzygy Theorem

The Strong Syzygy Theorem

Theorem (Strong Syzygy Theorems)

Let (R,m) be a local ring of unmixed ramified characteristic p andM an R-module that satisfies any of the hypotheses

M is annihilated by p (Shamash)

M viewed as R/(p)-module is weakly liftable to R (ADS)

M = M/xM ' (0 :M x) i.e. there is a four-term exact

sequence 0→ Mδ→ F → E → M → 0

. Then

1 there is a short exact sequence0→ Syzk−1(M)→ Syzk(M)→ Syzk(M)→ 0

2 rank Syzk(M) ≥ 2k − 1 for 1 ≤ k ≤ pd(M)− 3

3 βRk (M) = βR

k−1(M) + βRk (M) for 2 ≤ k ≤ pdM − 1,

Alexandra Seceleanu Syzygy Theorems

Evans-Griffith Syzygy TheoremComparison of Order Ideals of Consecutive Syzygies

The Strong Syzygy Theorem

The Strong Syzygy Theorem

Theorem (Strong Syzygy Theorems)

Let (R,m) be a local ring of unmixed ramified characteristic p andM an R-module that satisfies any of the hypotheses

M is annihilated by p (Shamash)

M viewed as R/(p)-module is weakly liftable to R (ADS)

M = M/xM ' (0 :M x)

. Then

1 there is a short exact sequence0→ Syzk−1(M)→ Syzk(M)→ Syzk(M)→ 0

2 rank Syzk(M) ≥ 2k − 1 for 1 ≤ k ≤ pd(M)− 3

3 βRk (M) = βR

k−1(M) + βRk (M) for 2 ≤ k ≤ pdM − 1,

Alexandra Seceleanu Syzygy Theorems

Evans-Griffith Syzygy TheoremComparison of Order Ideals of Consecutive Syzygies

The Strong Syzygy Theorem

Applications to Weak Lifting

Definition

An R-module N is weakly liftable to R if it is a direct summand ofa module M ⊗ R such that TorR

i (M,R/(p)) = 0 for i ≥ 1.

rankRSyzk(N) < 2k − 1 =⇒ N is not liftable

Find interesting classes of examples !

Alexandra Seceleanu Syzygy Theorems

Evans-Griffith Syzygy TheoremComparison of Order Ideals of Consecutive Syzygies

The Strong Syzygy Theorem

Applications to Weak Lifting

Definition

An R-module N is weakly liftable to R if it is a direct summand ofa module M ⊗ R such that TorR

i (M,R/(p)) = 0 for i ≥ 1.

rankRSyzk(N) < 2k − 1 =⇒ N is not liftable

Find interesting classes of examples !

Alexandra Seceleanu Syzygy Theorems

Evans-Griffith Syzygy TheoremComparison of Order Ideals of Consecutive Syzygies

The Strong Syzygy Theorem

Applications to Weak Lifting

Definition

An R-module N is weakly liftable to R if it is a direct summand ofa module M ⊗ R such that TorR

i (M,R/(p)) = 0 for i ≥ 1.

rankRSyzk(N) < 2k − 1 =⇒ N is not liftable

Find interesting classes of examples !

Alexandra Seceleanu Syzygy Theorems

Evans-Griffith Syzygy TheoremComparison of Order Ideals of Consecutive Syzygies

The Strong Syzygy Theorem

Thank You

Thank You !

Alexandra Seceleanu Syzygy Theorems