Rigidity of Ext and Tor via flat-cotorsion theory

2021/11/30 by Christensen, Lars Winther, Ferraro, Luigi, Thompson, Peder
#13D05 #13D07 #Commutative Algebra (math.AC) #FOS: Mathematics

paper · doi:10.48550/arxiv.2112.00103

Abstract

Let p be a prime ideal in a commutative noetherian ring R and denote by k(p) the residue field of the local ring R_p. We prove that if an R-module M satisfies Ext_Rⁿ(k(p),M) = 0 for some n >= dim R, then Ext_Rⁱ(k(p),M) = 0 holds for all i >= n. This improves a result of Christensen, Iyengar, and Marley by lowering the bound on n. We also improve existing results on Tor-rigidity. This progress is driven by the existence of minimal semi-flat-cotorsion replacements in the derived category as recently proved by Nakamura and Thompson.

Related

• Rigidity of Ext and Tor with coefficients in residue fields of a\n commutative noetherian ring
• Two theorems on the vanishing of Ext
• Rigidity properties of the cotangent complex
• A Rigidity Theorem for Ext
• Serre's condition for tensor products and n-Tor-rigidity of modules