… , for commutative rings in terms of the vanishing of Tor. This has led to the notion of Tor-persistent
rings. Our main result shows that the class of Tor-persistent local rings is closed under …

… Our approach to both of our main results provides an explanation as to why the vanishing
of Tor i R ( M , N ) is special when M = N ; namely, the vanishing of Tor i R ( M , M ) has …

… rings over which all modules satisfy the n-Tor rigid property for some n. We first consider the
Tor rigid test property of Ulrich modules. We show that faithful Ulrich modules over a Cohen–…

… modular forms with non-vanishing central critical values and linear independence of Fourier
coefficients of modular … a theorem due to VanderKam to modular symbols of higher weights. …

… If dim R = 1 the claim is in [8, 1.4] that we do not need the tor-rigidity assumption. Recall
that (ϕnR)p = ϕ(Rp) and that ϕ(Rp) is e-rigid. Apply this for any height one prime ideal, we …

… vanishing of Tor, while condition (2) is the strongest that one can impose on the asymptotic
vanishing … assumptions, eg, if R has a canonical module (Theorem 3.2). Following Jorgensen …

… coinduced from R-modules from C. Assuming that the class F … -modules cofiltered by A-modules
coinduced from R-modules … to expect these Ext groups to vanish for all modules F 2 F . …

… of general conjectures on the non-vanishing in self-dual families. … ) tor . Based on the results
for Galois image of a GL 2 -type abelian variety, we show that the torsion ⋃ K A ( H K , c ) tor …

… vanishing and Ext-Tor dualities, and derive a number of freeness criteria for finite modules
… such as Cohen-Macaulayness of tensor products and Tor-independence, and inspired by a …

… Recently, Kimura, Otake, and Takahashi proved a theorem about the vanishing of Ext of
finitely generated modules over Cohen–Macaulay rings. The aim of this paper is to obtain …