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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