Tor-persistence and the graded Tachikawa Conjecture
Tracing its roots in the representation theory of Artin algebras, the Tachikawa conjecture (for commutative rings) states that if R is a Cohen-Macaulay ring with canonical module w, then the vanishing of Ext^i_R(w,R) for all i>0 forces R to be Gorenstein. We relate this conjecture to an open question of Avramov-Iyengar-Nasseh-Sather-Wagstaff which asks whether the vanishing of Tor^R_i(M,M) for all i>0 forces M to be projective. We prove the Tachikawa conjecture in the graded case when R has characteristic different from 2, and we answer the question of Avramov-Iyengar-Nasseh-Sather-Wagstaff in the affirmative for local rings whose maximal ideal has cube 0. Our approach to both of these problems is to exploit properties of exterior and symmetric powers, for both modules and complexes. This is based on joint work with Jonathan Montaño and Sean Sather-Wagstaff.
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