OAI.Tachikawa.main_theorem

The theorem states that the defined proposition Counterexample holds. Here k is the fraction field of the polynomial ring in three variables over the field with two elements, F₂(x₀,x₁,x₂). Counterexample asserts that there exist a ring A with a k-algebra structure, finite-dimensional over k, and symmetric over k, together with an A-module M that is also a k-module, with the two structures compatible (k acts through A, a scalar tower), and M finite-dimensional over k, such that M is not projective as an A-module, yet Ext^n_A(M,M) is a subsingleton (trivial) for every positive integer n. Symmetric over k means there is a k-linear isomorphism e from A to its k-dual such that, for all a, b, c in A, e(ab)(c) = e(b)(ca) and e(ab)(c) = e(a)(bc). Thus the statement is existential: it asserts a finite-dimensional symmetric k-algebra with a finitely generated non-projective module having vanishing self-extensions in all positive degrees. The theorem is admitted with sorry in the source, so no proof is claimed.