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I'm not sure this question is research level question. Sorry in advance. Hypothesis $k$ is a commutative ring. $A$ is an augmented $k$-algebra. $A^e$ is defined as the $k$-algebra $A\otimes_{k}A^{op}$...

Let $R$ be a local Gorenstein ring of Krull dimension $d$ with an isolated singularity. Defined $D_{sing}(R)$ as the Verdier quotient $D^b(R)/Perf(R)$). Then, a famous result of Auslander says that ...

An Azumaya variety over a field is by definition a pair $(X,\mathcal A_X)$, where $X$ is an algebraic variety of finite type over that field and $\mathcal A_X$ is a sheaf of Azumaya algebras, namely ...