A depth formula
for Tate Tor independent modules
over Gorenstein rings
Joint work with David A. Jorgensen
arXiv:1107.3102[math.AC]
Lars Winther Christensen
Texas Tech University
Lincoln NE, 16 October 2011
Auslander’s depth formula
Setup
R a commutative noetherian local ring
Auslander’s depth formula
Let M and N be finitely generated R-modules; assume M has
finite projective dimension. If TorR
>0 (M, N) = 0, then
depthR (M ⊗R N) = depthR M + depthR N − depth R
Corollary
Assume R is regular; let M and N be finitely generated R-modules.
If TorR
>0 (M, N) = 0, then
depthR (M ⊗R N) = depthR M + depthR N − depth R
Lars Winther Christensen
Depth formula for Tate Tor independent modules
Improvements of Auslander’s depth formula
Theorem (Huneke and Wiegand; Iyengar)
Let M and N be R-modules; assume M has finite CI-dimension.
If TorR
>0 (M, N) = 0, then
depthR (M ⊗R N) = depthR M + depthR N − depth R
Corollary
Assume R is complete intersection; let M and N be R-modules.
If TorR
>0 (M, N) = 0, then
depthR (M ⊗R N) = depthR M + depthR N − depth R
Auslander’s depth formula—derived version (Foxby)
Let M and N be R-modules. If M has finite projective dimension,
then TorR
0 (M, N) = 0 and
depthR (M ⊗LR N) = depthR M + depthR N − depth R
Lars Winther Christensen
Depth formula for Tate Tor independent modules
Improvements of the derived depth formula
Theorem A
Let M and N be R-modules; assume M has finite CI-dimension.
If TorR
0 (M, N) = 0, then
depthR (M ⊗LR N) = depthR M + depthR N − depth R
Theorem B
Let M and N be R-modules; assume M has finite Gorenstein
projective dimension.
c R (M, N) = 0, then TorR (M, N) = 0 and
If Tor
∗
0
depthR (M ⊗LR N) = depthR M + depthR N − depth R
Lars Winther Christensen
Depth formula for Tate Tor independent modules
Complete resolutions
Definition
A complex of projective R-modules
T :
T
∂n+1
∂T
· · · −→ Tn+1 −−−→ Tn −−n→ Tn−1 −→
is totally acyclic if
T is acyclic, i.e. H(T ) = 0
HomR (T , P) is acyclic for every projective R-module P
A diagram
τ
π
T −→ P −→ M,
where
π is a projective resolution of M
τ is an isomorphism in high degrees
is called a complete projective resolution of M
Lars Winther Christensen
Depth formula for Tate Tor independent modules
Tate homology
Definition
A module M has finite Gorenstein projective dimension if it has a
complete resolution
T → P → M,
and then
c R (M, N) = Hi (T ⊗R N)
Tor
i
Theorem B
Let M and N be R-modules; assume M has finite Gorenstein
projective dimension.
c R (M, N) = 0, then TorR (M, N) = 0 and
If Tor
0
∗
depthR (M ⊗LR N) = depthR M + depthR N − depth R
Lars Winther Christensen
Depth formula for Tate Tor independent modules
Gorenstein rings
Corollary
Assume R is Gorenstein; let M and N be R-modules.
c R (M, N) = 0, then
If Tor
∗
depthR (M ⊗LR N) = depthR M + depthR N − depth R
Corollary
Assume R is AB; let M and N be finitely generated R-modules.
c R (M, N) = 0, then
If Tor
0
depthR (M ⊗LR N) = depthR M + depthR N − depth R
Lars Winther Christensen
Depth formula for Tate Tor independent modules
Vanishing of cohomology
Theorem C
Assume R is AB; let M and N be finitely generated R-modules.
If ExtiR (M, N) = 0 for i 0, then
sup{ i ∈ Z | ExtiR (M, N) 6= 0 } = depth R − depthR M
Theorem C’
Let M and N be finitely generated R-modules, such that M has
finite Gorenstein projective dimension or N has finite Gorenstein
c ∗ (M, N) = 0, then
injective dimension. If Ext
R
sup{ i ∈ Z | ExtiR (M, N) 6= 0 } = depth R − depthR M
Lars Winther Christensen
Depth formula for Tate Tor independent modules