Graded self-injective algebras "are" trivial extensions

TL;DR

This paper shows that well-graded self-injective algebras can be characterized as trivial extensions of their Beilinson algebras, establishing a deep connection between their module categories and derived categories.

Contribution

It introduces the Beilinson algebra for positively graded artin algebras and proves an equivalence between module categories of such algebras and trivial extensions, clarifying their structure.

Findings

Category of graded modules over A is equivalent to modules over T(b(A))

Full embedding of derived category of b(A) into stable category of A

Equivalence holds if A_0 has finite global dimension

Abstract

For a positively graded artin algebra $A=\oplus_{n\geq 0}A_n$ we introduce its Beilinson algebra $\mathrm{b}(A)$. We prove that if $A$ is well-graded self-injective, then the category of graded $A$-modules is equivalent to the category of graded modules over the trivial extension algebra $T(\mathrm{b}(A))$. Consequently, there is a full exact embedding from the bounded derived category of $\mathrm{b}(A)$ into the stable category of graded modules over $A$; it is an equivalence if and only if the 0-th component algebra $A_0$ has finite global dimension.