Extension category algebras and LHS--spectral sequences

TL;DR

This paper introduces new algebraic constructions called extension category algebras and Grothendieck constructions, establishing their properties and spectral sequences, thus advancing the understanding of module categories over these algebras.

Contribution

It defines the extension category algebra and Grothendieck construction, proving their equivalence under certain conditions and deriving spectral sequences for their module categories.

Findings

The category of modules over the Grothendieck construction is equivalent to modules over the extension algebra.

Two LHS-spectral sequences are established for the Grothendieck construction.

Extension category algebra generalizes trivial and skew category algebras.

Abstract

Let $\mathcal{C}$ be a small category, $\mathfrak{A}$ be a precosheaf of unital $k$-algebras on $\mathcal{C}$ and $\mathfrak{M}$ be an $\mathfrak{A}$-bimodule. We introduce two new notions, namely, the Grothendieck construction $Gr_{\mathcal{C}}(\mathfrak{A}, \mathfrak{M})$ of $\mathfrak{A}$ and $\mathfrak{M}$, as well as the extension category algebra $\mathfrak{A} \ltimes \mathfrak{M}$. The extension category algebra contains the trivial extension algebra and the skew category algebra as special cases. If $\mathcal{C}$ is object-finite, we prove that the category of modules of $Gr_{\mathcal{C}}(\mathfrak{A}, \mathfrak{M})$ is equivalent to the category of modules over $\mathfrak{A} \ltimes \mathfrak{M}$. Finally, we obtain two LHS-spectral sequences about $Gr_{\mathcal{C}}(\mathfrak{A}, \mathfrak{N})$ for a right $\mathfrak{A}$-module $\mathfrak{N}$.