Representation Theory of Geigle-Lenzing Complete Intersections

Abstract

Weighted projective lines, introduced by Geigle and Lenzing in 1987, are important objects in representation theory. They have tilting bundles, whose endomorphism algebras are the canonical algebras introduced by Ringel. The aim of this paper is to study their higher dimensional analogs. First, we introduce a certain class of commutative Gorenstein rings <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper R"> <mml:semantics> <mml:mi>R</mml:mi> <mml:annotation encoding="application/x-tex">R</mml:annotation> </mml:semantics> </mml:math> </inline-formula> graded by abelian groups <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper L"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">L</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathbb {L}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of rank <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="1"> <mml:semantics> <mml:mn>1</mml:mn> <mml:annotation encoding="application/x-tex">1</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, which we call Geigle-Lenzing complete intersections. We study the stable category <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="ModifyingBelow sans-serif upper C sans-serif upper M With bar Superscript double-struck upper L Baseline upper R"> <mml:semantics> <mml:mrow> <mml:msup> <mml:munder> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="sans-serif">C</mml:mi> <mml:mi mathvariant="sans-serif">M</mml:mi> </mml:mrow> <mml:mo>_</mml:mo> </mml:munder> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">L</mml:mi> </mml:mrow> </mml:mrow> </mml:msup> <mml:mi>R</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\underline {\mathsf {CM}}^{\mathbb {L}}R</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of Cohen-Macaulay representations, which coincides with the singularity category <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="sans-serif upper D Subscript normal s normal g Superscript double-struck upper L Baseline left-parenthesis upper R right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msubsup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="sans-serif">D</mml:mi> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="normal">s</mml:mi> <mml:mi mathvariant="normal">g</mml:mi> </mml:mrow> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">L</mml:mi> </mml:mrow> </mml:mrow> </mml:msubsup> <mml:mo stretchy="false">(</mml:mo> <mml:mi>R</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathsf {D}^{\mathbb {L}}_{\mathrm {sg}}(R)</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. We show that <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="ModifyingBelow sans-serif upper C sans-serif upper M With bar Superscript double-struck upper L Baseline upper R"> <mml:semantics> <mml:mrow> <mml:msup> <mml:munder> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="sans-serif">C</mml:mi> <mml:mi mathvariant="sans-serif">M</mml:mi> </mml:mrow> <mml:mo>_</mml:mo> </mml:munder> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">L</mml:mi> </mml:mrow> </mml:mrow> </mml:msup> <mml:mi>R</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\underline {\mathsf {CM}}^{\mathbb {L}}R</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is triangle equivalent to <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="sans-serif upper D Superscript normal b Baseline left-parenthesis sans-serif m sans-serif o sans-serif d upper A Superscript normal upper C normal upper M Baseline right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="sans-serif">D</mml:mi> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="normal">b</mml:mi> </mml:mrow> </mml:mrow> </mml:msup> <mml:mo stretchy="false">(</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="sans-serif">m</mml:mi> <mml:mi mathvariant="sans-serif">o</mml:mi> <mml:mi mathvariant="sans-serif">d</mml:mi> </mml:mrow> <mml:msup> <mml:mi>A</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="normal">C</mml:mi> <mml:mi mathvariant="normal">M</mml:mi> </mml:mrow> </mml:mrow> </mml:msup> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathsf {D}^{\mathrm {b}}(\mathsf {mod} A^{\mathrm {CM}})</mml:annotation> </mml:semantics> </mml:math> </inline-formula> for a finite dimensional algebra <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper A Superscript normal upper C normal upper M"> <mml:semantics> <mml:msup> <mml:mi>A</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="normal">C</mml:mi> <mml:mi mathvariant="normal">M</mml:mi> </mml:mrow> </mml:mrow> </mml:msup> <mml:annotation encoding="application/x-tex">A^{\mathrm {CM}}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, which we call the CM-canonical algebra. As an application, we classify the <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis upper R comma double-struck upper L right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>R</mml:mi> <mml:mo>,</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">L</mml:mi> </mml:mrow> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">(R,\mathbb {L})</mml:annotation> </mml:semantics> </mml:math> </inline-formula> that are Cohen-Macaulay finite. We also give sufficient conditions for <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis upper R comma double-struck upper L right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>R</mml:mi> <mml:mo>,</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">L</mml:mi> </mml:mrow> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">(R,\mathbb {L})</mml:annotation> </mml:semantics> </mml:math> </inline-formula> to be <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="d"> <mml:semantics> <mml:mi>d</mml:mi> <mml:annotation encoding="application/x-tex">d</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-Cohen-Macaulay finite in the sense of higher Auslander-Reiten theory. Secondly, we study a new class of non-commutative projective schemes in the sense of Artin-Zhang, i.e. the category <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="sans-serif c sans-serif o sans-serif h double-struck upper X equals sans-serif m sans-serif o sans-serif d Superscript double-struck upper L Baseline upper R slash sans-serif m sans-serif o sans-serif d Subscript 0 Superscript double-struck upper L Baseline upper R"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="sans-serif">c</mml:mi> <mml:mi mathvariant="sans-serif">o</mml:mi> <mml:mi mathvariant="sans-serif">h</mml:mi> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">X</mml:mi> </mml:mrow> <mml:mo>=</mml:mo> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="sans-serif">m</mml:mi> <mml:mi mathvariant="sans-serif">o</mml:mi> <mml:mi mathvariant="sans-serif">d</mml:mi> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">L</mml:mi> </mml:mrow> </mml:mrow> </mml:msup> <mml:mi>R</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>/</mml:mo> </mml:mrow> <mml:msubsup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="sans-serif">m</mml:mi> <mml:mi mathvariant="sans-serif">o</mml:mi> <mml:mi mathvariant="sans-serif">d</mml:mi> </mm