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

Abstract

Given an artin algebra <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Lamda"> <mml:semantics> <mml:mi mathvariant="normal">Λ<!-- Λ --></mml:mi> <mml:annotation encoding="application/x-tex">\Lambda</mml:annotation> </mml:semantics> </mml:math> </inline-formula> with an idempotent element <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="a"> <mml:semantics> <mml:mi>a</mml:mi> <mml:annotation encoding="application/x-tex">a</mml:annotation> </mml:semantics> </mml:math> </inline-formula> we compare the algebras <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Lamda"> <mml:semantics> <mml:mi mathvariant="normal">Λ<!-- Λ --></mml:mi> <mml:annotation encoding="application/x-tex">\Lambda</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="a normal upper Lamda a"> <mml:semantics> <mml:mrow> <mml:mi>a</mml:mi> <mml:mi mathvariant="normal">Λ<!-- Λ --></mml:mi> <mml:mi>a</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">a\Lambda a</mml:annotation> </mml:semantics> </mml:math> </inline-formula> with respect to Gorensteinness, singularity categories and the finite generation condition <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="sans-serif Fg"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mtext mathvariant="sans-serif">Fg</mml:mtext> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathrm {\textsf {Fg}}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> for the Hochschild cohomology. In particular, we identify assumptions on the idempotent element <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="a"> <mml:semantics> <mml:mi>a</mml:mi> <mml:annotation encoding="application/x-tex">a</mml:annotation> </mml:semantics> </mml:math> </inline-formula> which ensure that <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Lamda"> <mml:semantics> <mml:mi mathvariant="normal">Λ<!-- Λ --></mml:mi> <mml:annotation encoding="application/x-tex">\Lambda</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is Gorenstein if and only if <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="a normal upper Lamda a"> <mml:semantics> <mml:mrow> <mml:mi>a</mml:mi> <mml:mi mathvariant="normal">Λ<!-- Λ --></mml:mi> <mml:mi>a</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">a\Lambda a</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is Gorenstein, that the singularity categories of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Lamda"> <mml:semantics> <mml:mi mathvariant="normal">Λ<!-- Λ --></mml:mi> <mml:annotation encoding="application/x-tex">\Lambda</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="a normal upper Lamda a"> <mml:semantics> <mml:mrow> <mml:mi>a</mml:mi> <mml:mi mathvariant="normal">Λ<!-- Λ --></mml:mi> <mml:mi>a</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">a\Lambda a</mml:annotation> </mml:semantics> </mml:math> </inline-formula> are equivalent and that <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="sans-serif Fg"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mtext mathvariant="sans-serif">Fg</mml:mtext> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathrm {\textsf {Fg}}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> holds for <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Lamda"> <mml:semantics> <mml:mi mathvariant="normal">Λ<!-- Λ --></mml:mi> <mml:annotation encoding="application/x-tex">\Lambda</mml:annotation> </mml:semantics> </mml:math> </inline-formula> if and only if <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="sans-serif Fg"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mtext mathvariant="sans-serif">Fg</mml:mtext> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathrm {\textsf {Fg}}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> holds for <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="a normal upper Lamda a"> <mml:semantics> <mml:mrow> <mml:mi>a</mml:mi> <mml:mi mathvariant="normal">Λ<!-- Λ --></mml:mi> <mml:mi>a</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">a\Lambda a</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. We approach the problem by using recollements of abelian categories and we prove the results concerning Gorensteinness and singularity categories in this general setting. The results are applied to stable categories of Cohen–Macaulay modules and classes of triangular matrix algebras and quotients of path algebras.

Locations

• Transactions of the American Mathematical Society Series B - View - PDF
• arXiv (Cornell University) - View - PDF