Graded self-injective algebras "are" trivial extensions

Abstract

For a positively graded artin algebra $A={\oplus }_{n\ge 0}{A}_n$ 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.

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References in corpus (2)

• Derived Categories of Coherent Sheaves and Triangulated Categories of Singularities
• Rings graded equivalent to the Weyl algebra