A depth formula for Tate Tor independent modules over Gorenstein - - PowerPoint PPT Presentation

A depth formula for Tate Tor independent modules

• ver 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 ⊗L

R 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 ⊗L

R N) = depthR M + depthR N − depth R

Theorem B Let M and N be R-modules; assume M has finite Gorenstein projective dimension. If TorR

∗ (M, N) = 0, then TorR ≫0(M, N) = 0 and

depthR(M ⊗L

R 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 : · · · − → Tn+1

∂T

n+1

− − − → Tn

∂T

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 T

τ

− → P

π

− → 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

• TorR

i (M, N) = Hi(T ⊗R N)

Theorem B Let M and N be R-modules; assume M has finite Gorenstein projective dimension. If TorR

∗ (M, N) = 0, then TorR ≫0(M, N) = 0 and

depthR(M ⊗L

R 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. If TorR

∗ (M, N) = 0, then

depthR(M ⊗L

R N) = depthR M + depthR N − depth R

Corollary Assume R is AB; let M and N be finitely generated R-modules. If TorR

≫0(M, N) = 0, then

depthR(M ⊗L

R 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 Exti

R(M, N) = 0 for i ≫ 0, then

sup{ i ∈ Z | Exti

R(M, N) = 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 injective dimension. If Ext∗

R(M, N) = 0, then

sup{ i ∈ Z | Exti

R(M, N) = 0 } = depth R − depthR M

Lars Winther Christensen Depth formula for Tate Tor independent modules