Gorenstein categories, singular equivalences and finite generation of cohomology rings in recollements (Q2826798)

Given any Artin algebra \(\Lambda\) over a commutative ring \(k\) and any idempotent \(a \in \Lambda\), the article under review compares several homological properties of the rings \(\Lambda\) and \(a \Lambda a\), such as the Gorenstein property and the \textbf{Fg} property. They also study the relation between the corresponding singularity categories. Given an algebra \(A\) over a ring \(k\) such that \(A\) is a flat \(k\)-module, we recall that \(A\) satisfies the \textit{ \textbf{Fg} property} if the Hochschild cohomology \(HH^{\bullet}(A)\) is a noetherian ring and the module \(\text{Ext}^{\bullet}_{A}(A/\text{rad}(A), A/\text{rad}(A))\) over \(HH^{\bullet}(A)\) is of finite type.NEWLINENEWLINEMore precisely, on the one hand they give necessary and sufficient conditions for the canonical functor \(\text{mod}\Lambda \rightarrow \text{mod}a\Lambda a\) given by mutiplication by \(a\) to induce an equivalence NEWLINE\[NEWLINE \text{D}^{b}(\text{mod}\Lambda)/\text{perf}(\Lambda) \rightarrow \text{D}^{b}(\text{mod}a\Lambda a)/\text{perf}(a\Lambda a) NEWLINE\]NEWLINE between the associated singularity categories (see Cor. 5.4). On the other hand, they show that under certain assumptions on the idempotent, \(\Lambda\) is Gorenstein if and only if \(a \Lambda a\) is so (see Cor. 4.7). They have also extended the previous results to the case of abelian categories. Furthermore, they also show that, under the same assumptions on the idempotent \(a\) as before and if \(k\) is a field such that \(\Lambda/\text{rad}(\Lambda) \otimes_{k} \Lambda/\text{rad}(\Lambda)\) is a semisimple \(\Lambda\)-bimodule, \(\Lambda\) satisfies the \textbf{Fg} property if and only if \(a \Lambda a\) does so (see Thm. 7.10). The last section contains several nice examples (e.g. triangular matrix algebras) and applications.