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

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 Tor R > 0 ( M , N ) = 0 , then depth R ( M ⊗ R N ) = depth R M + depth R N − depth R Corollary Assume R is regular; let M and N be finitely generated R-modules. If Tor R > 0 ( M , N ) = 0 , then depth R ( M ⊗ R N ) = depth R M + depth R 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 Tor R > 0 ( M , N ) = 0 , then depth R ( M ⊗ R N ) = depth R M + depth R N − depth R Corollary Assume R is complete intersection; let M and N be R-modules. If Tor R > 0 ( M , N ) = 0 , then depth R ( M ⊗ R N ) = depth R M + depth R N − depth R Auslander’s depth formula—derived version (Foxby) Let M and N be R-modules. If M has finite projective dimension, then Tor R ≫ 0 ( M , N ) = 0 and depth R ( M ⊗ L R N ) = depth R M + depth R 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 Tor R ≫ 0 ( M , N ) = 0 , then depth R ( M ⊗ L R N ) = depth R M + depth R N − depth R Theorem B Let M and N be R-modules; assume M has finite Gorenstein projective dimension. If � Tor R ∗ ( M , N ) = 0 , then Tor R ≫ 0 ( M , N ) = 0 and depth R ( M ⊗ L R N ) = depth R M + depth R 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 n T : · · · − → T n +1 − − − → T n − − → T n − 1 − → is totally acyclic if T is acyclic, i.e. H( T ) = 0 Hom R ( T , P ) is acyclic for every projective R-module P A diagram τ π − → P − → M , T 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 � Tor R i ( M , N ) = H i ( T ⊗ R N ) Theorem B Let M and N be R-modules; assume M has finite Gorenstein projective dimension. If � Tor R ∗ ( M , N ) = 0 , then Tor R ≫ 0 ( M , N ) = 0 and depth R ( M ⊗ L R N ) = depth R M + depth R 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 � Tor R ∗ ( M , N ) = 0 , then depth R ( M ⊗ L R N ) = depth R M + depth R N − depth R Corollary Assume R is AB; let M and N be finitely generated R-modules. If � Tor R ≫ 0 ( M , N ) = 0 , then depth R ( M ⊗ L R N ) = depth R M + depth R 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 Ext i R ( M , N ) = 0 for i ≫ 0 , then sup { i ∈ Z | Ext i R ( M , N ) � = 0 } = depth R − depth R 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 | Ext i R ( M , N ) � = 0 } = depth R − depth R M Lars Winther Christensen Depth formula for Tate Tor independent modules