Sarah Witherspoon

Given an exact module category over a finite tensor category with finitely generated cohomology, we show that if there exists an object of complexity at least three, then the category is of wild representation type. In particular, if the Krull dimension of the cohomology ring of the finite tensor category is at least three, then this category is wild.

Given an exact module category over a finite tensor category with finitely generated cohomology, we show that if there exists an object of complexity at least three, then the category is of wild representation type. In particular, if the Krull dimension of the cohomology ring of the finite tensor category is at least three, then this category is wild.

Alexander-Whitney and Eilenberg-Zilber maps traditionally convert between the tensor product of standard resolutions and the standard resolution of a tensor product of algebras. We examine Alexander-Whitney and Eilenberg-Zilber maps for twisted tensor products, which include skew group algebras, smash products of Hopf algebras, Ore extensions, and universal enveloping algebras. These maps convert between the twist of standard resolutions and the standard resolution of a twist. We extend these to chain maps to and from twists of other resolutions. This allows one to transfer homological information between various resolutions of algebras and to expedite results on the deformation theory of twisted tensor product algebras.

International Press of Boston - publishers of scholarly mathematical and scientific journals and books

For some exact monoidal categories, we describe explicitly a connection between topological and algebraic definitions of the Lie bracket on the extension algebra of the unit object. The topological definition, due to Schwede and Hermann, involves loops in extension categories. The algebraic definition, due to the first author, involves homotopy liftings of maps. As a consequence of our description, we prove that the topological definition indeed yields a Gerstenhaber algebra structure in this monoidal category setting. This answers a question of Hermann for those exact monoidal categories in which the unit object has a particular type of resolution that is called power flat. For use in proofs, we generalize $A_{\infty}$-coderivation and homotopy lifting techniques from bimodule categories to these exact monoidal categories.

The Hochschild cohomology of a tensor product of algebras is isomorphic to a graded tensor product of Hochschild cohomology algebras, as a Gerstenhaber algebra. A similar result holds when the tensor product is twisted by a bicharacter. We present new proofs of these isomorphisms, using Volkov's homotopy liftings that were introduced for handling Gerstenhaber brackets expressed on arbitrary bimodule resolutions. Our results illustrate the utility of homotopy liftings for theoretical purposes.