Finite generation of the cohomology rings for the extension algebras of monomial algebras

We describe the cohomology ring of a monomial algebra in the language of dimension tree or minimal resolution graph and in this context we study the finite generation of the cohomology rings of the extension algebras, showing among others that the cohomology ring [Formula: see text] is finitely generated [Formula: see text] is [Formula: see text] is, where [Formula: see text] is the dual extension of a monomial algebra [Formula: see text] and [Formula: see text] is the opposite algebra of [Formula: see text].

On the relative K-group in the ETNC, Part II

In a previous paper we showed that, under some assumptions, the relative K-group in the Burns–Flach formulation of the equivariant Tamagawa number conjecture (ETNC) is canonically isomorphic to a K-group of locally compact equivariant modules. Our approach as well as the standard one both involve presentations: One due to Bass–Swan, applied to categories of finitely generated projective modules; and one due to Nenashev, applied to our topological modules without finite generation assumptions. In this paper we provide an explicit isomorphism.