Evans-Griffith Syzygy Theorem
Comparison of Order Ideals of Consecutive Syzygies
The Strong Syzygy Theorem
Syzygy Theorems via
Comparison of Order Ideals
Alexandra Seceleanu
University of Illinois at Urbana-Champaign
AMS National Meeting in San Francisco
January 2010
Alexandra Seceleanu
Syzygy Theorems
Evans-Griffith Syzygy Theorem
Comparison 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 k th
module of syzygies over a Cohen-Macaulay local ring R containing
a 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 Theorem
Comparison 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)}.
Alexandra Seceleanu
Syzygy Theorems
Evans-Griffith Syzygy Theorem
Comparison 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
2
E
3
a1,1 . . . a1,m
6
..
.. 7
7
6 ..
6 .
.
. 7
4
5
an,1 . . . an,m
/ Rn
/ Rm
and if e1 ∈ E
Alexandra Seceleanu
Syzygy Theorems
Evans-Griffith Syzygy Theorem
Comparison 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 ):
2
3
a1,1 . . . a1,m
..
..
.. 7
7
.
.
. 7
5
an,1 . . . an,m
6
6
6
4
E
/ Rm
e1 / e1 / R n UU
UUUU
UUUπU
UUUU
UUUU
*
/ (a1,1 , . . . , a1,n )
R
UUUU
UUUU
UUUU
UU*
a
1,j
Alexandra Seceleanu
Syzygy Theorems
Evans-Griffith Syzygy Theorem
Comparison 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.
Evans-Griffith Syzygy Theorem
Comparison 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.
unramified mixed char p
Comparison
Evans-Griffith Syzygy Theorem
Comparison 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.
unramified mixed char p
Comparison
Order Ideal Theorem in mixed char
Evans-Griffith Syzygy Theorem
Comparison 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
unramified mixed char p
Comparison
Syzygy Theorem holds.
Alexandra Seceleanu
Order Ideal Theorem in mixed char
Syzygy Theorem holds
for E such that
pExt 1 (E , ·) ≡ 0.
Syzygy Theorems
Evans-Griffith Syzygy Theorem
Comparison 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 nonzerodivisor x ∈ I is a superficial element of I with respect to M if
(I n+1 M :M x) = I n M
.
Alexandra Seceleanu
Syzygy Theorems
Evans-Griffith Syzygy Theorem
Comparison 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 nonzerodivisor x ∈ I is a superficial element of I with respect to M if
(I n+1 M :M x) = I n M
.
(For the purposes of this talk one may just think x ∈ m − m2 .)
Alexandra Seceleanu
Syzygy Theorems
Evans-Griffith Syzygy Theorem
Comparison of Order Ideals of Consecutive Syzygies
The Strong Syzygy Theorem
Comparison Theorem for heights of order ideals
Theorem (-, Comparison Theorem)
ι
Let E have minimal presentation 0 −→ Z −→ F → E → 0 and
suppose we have
Alexandra Seceleanu
Syzygy Theorems
Evans-Griffith Syzygy Theorem
Comparison of Order Ideals of Consecutive Syzygies
The Strong Syzygy Theorem
Comparison Theorem for heights of order ideals
Theorem (-, Comparison Theorem)
ι
Let E have minimal presentation 0 −→ Z −→ F → E → 0 and
suppose we have
x ∈ m superficial with respect to Z
Alexandra Seceleanu
Syzygy Theorems
Evans-Griffith Syzygy Theorem
Comparison of Order Ideals of Consecutive Syzygies
The Strong Syzygy Theorem
Comparison Theorem for heights of order ideals
Theorem (-, Comparison Theorem)
ι
Let E have minimal presentation 0 −→ Z −→ F → E → 0 and
suppose we have
x ∈ m superficial with respect to Z
xExtR1 (M, ·) ≡ 0
Alexandra Seceleanu
Syzygy Theorems
Evans-Griffith Syzygy Theorem
Comparison of Order Ideals of Consecutive Syzygies
The Strong Syzygy Theorem
Comparison Theorem for heights of order ideals
Theorem (-, Comparison Theorem)
ι
Let E have minimal presentation 0 −→ Z −→ F → E → 0 and
suppose we have
x ∈ m superficial with respect to Z
xExtR1 (M, ·) ≡ 0
Then one can build a diagram
0
/Z
/F


·x 
 f


ι
/E
Z
Alexandra Seceleanu
Syzygy Theorems
/0
Evans-Griffith Syzygy Theorem
Comparison of Order Ideals of Consecutive Syzygies
The Strong Syzygy Theorem
Comparison Theorem for heights of order ideals
Theorem (-, Comparison Theorem)
ι
Let E have minimal presentation 0 −→ Z −→ F → E → 0 and
suppose we have
x ∈ m superficial with respect to Z
xExtR1 (E , ·) ≡ 0
Then one can build a diagram
0
/Z
/F
mmm
m
m
·xvmmmmmf
ι
Z
π
Z̄ = Z /xZ
/E
Ē = E /xE
gggg
g
g
g
g
g
s gggg f¯
g
Alexandra Seceleanu
Syzygy Theorems
/0
Evans-Griffith Syzygy Theorem
Comparison of Order Ideals of Consecutive Syzygies
The Strong Syzygy Theorem
Comparison Theorem for heights of order ideals
Theorem (-, Comparison Theorem)
ι
Let E have minimal presentation 0 −→ Z −→ F → E → 0 and
suppose we have
x ∈ m superficial with respect to Z
xExtR1 (E , ·) ≡ 0
Then one can build a diagram
0
/Z
/F
mmm
m
m
·xvmmmmmf
ι
Z
π
Z̄ = Z /xZ
/E
Ē = E /xE
gggg
g
g
g
g
g
s gggg f¯
g
such that Im(f¯) 6⊆ mZ̄
Alexandra Seceleanu
Syzygy Theorems
/0
Evans-Griffith Syzygy Theorem
Comparison of Order Ideals of Consecutive Syzygies
The Strong Syzygy Theorem
Comparison Theorem for heights of order ideals
Theorem (-, Comparison Theorem)
ι
Let E have minimal presentation 0 −→ Z −→ F → E → 0 and
suppose we have
x ∈ m superficial with respect to Z
xExtR1 (M, ·) ≡ 0
Then one can build a diagram
0
/Z
/F
mmm
m
m
·xvmmmmmf
ι
Z
π
Z̄ = Z /xZ
/E
/0
Ē = E /xE
gggg
g
g
g
g
g
s gggg f¯
g
such that Im(f¯) 6⊆ mZ̄ and consequently htOE (e) ≥ htOZ̄ (f¯(ē)).
Alexandra Seceleanu
Syzygy Theorems
Evans-Griffith Syzygy Theorem
Comparison 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 be
unramified i.e. p ∈ m − m2 .
Theorem (Griffith, -)
Let (R, m) be an unramified Cohen-Macaulay local ring of mixed
characteristic. Assume that M is a finite projective dimension
R-module such that for a fixed k, pExtRk+1 (M, ·) ≡ 0. Then the
Syzygy theorem holds for every j th syzygy of M with j ≥ k.
Alexandra Seceleanu
Syzygy Theorems
Evans-Griffith Syzygy Theorem
Comparison 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 mixed
characteristic. Then the Syzygy Theorem holds over R for syzygies
of modules of the type R/Q with Q ∈ Spec(R).
Alexandra Seceleanu
Syzygy Theorems
Evans-Griffith Syzygy Theorem
Comparison 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 and
M 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)
M̄ = M/xM ' (0 :M x) i.e. there is a four-term exact
δ
sequence 0 → M̄ → F̄ → Ē → M̄ → 0
Alexandra Seceleanu
Syzygy Theorems
Evans-Griffith Syzygy Theorem
Comparison 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 and
M 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)
M̄ = M/xM ' (0 :M x)
. Then
1
there is a short exact sequence
0 → Syzk−1 (M̄) → Syzk (M) → Syzk (M̄) → 0
2
rank Syzk (M) ≥ 2k − 1 for 1 ≤ k ≤ pd(M) − 3
3
R̄ (M) + β R̄ (M) for 2 ≤ k ≤ pdM − 1,
βkR (M) = βk−1
k
Alexandra Seceleanu
Syzygy Theorems
Evans-Griffith Syzygy Theorem
Comparison of Order Ideals of Consecutive Syzygies
The Strong Syzygy Theorem
Applications to Weak Lifting
Definition
An R̄-module N is weakly liftable to R if it is a direct summand of
a module M ⊗ R̄ such that ToriR (M, R/(p)) = 0 for i ≥ 1.
Alexandra Seceleanu
Syzygy Theorems
Evans-Griffith Syzygy Theorem
Comparison of Order Ideals of Consecutive Syzygies
The Strong Syzygy Theorem
Applications to Weak Lifting
Definition
An R̄-module N is weakly liftable to R if it is a direct summand of
a module M ⊗ R̄ such that ToriR (M, R/(p)) = 0 for i ≥ 1.
rankR Syzk (N) < 2k − 1 =⇒ N is not liftable
Alexandra Seceleanu
Syzygy Theorems
Evans-Griffith Syzygy Theorem
Comparison of Order Ideals of Consecutive Syzygies
The Strong Syzygy Theorem
Applications to Weak Lifting
Definition
An R̄-module N is weakly liftable to R if it is a direct summand of
a module M ⊗ R̄ such that ToriR (M, R/(p)) = 0 for i ≥ 1.
rankR Syzk (N) < 2k − 1 =⇒ N is not liftable
Find interesting classes of examples !
Alexandra Seceleanu
Syzygy Theorems
Evans-Griffith Syzygy Theorem
Comparison of Order Ideals of Consecutive Syzygies
The Strong Syzygy Theorem
Thank You
Thank You !
Alexandra Seceleanu
Syzygy Theorems