Lattice theory of torsion classes: Beyond 𝜏-tilting theory

Abstract

The aim of this paper is to establish a lattice theoretical framework to study the partially ordered set <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="sans-serif t sans-serif o sans-serif r sans-serif s upper A"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="sans-serif">t</mml:mi> <mml:mi mathvariant="sans-serif">o</mml:mi> <mml:mi mathvariant="sans-serif">r</mml:mi> <mml:mi mathvariant="sans-serif">s</mml:mi> </mml:mrow> <mml:mi>A</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathsf {tors} A</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of torsion classes over a finite-dimensional algebra <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper A"> <mml:semantics> <mml:mi>A</mml:mi> <mml:annotation encoding="application/x-tex">A</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="sans-serif t sans-serif o sans-serif r sans-serif s upper A"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="sans-serif">t</mml:mi> <mml:mi mathvariant="sans-serif">o</mml:mi> <mml:mi mathvariant="sans-serif">r</mml:mi> <mml:mi mathvariant="sans-serif">s</mml:mi> </mml:mrow> <mml:mi>A</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathsf {tors} A</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is a complete lattice which enjoys very strong properties, as <italic>bialgebraicity</italic> and <italic>complete semidistributivity</italic> . Thus its Hasse quiver carries the important part of its structure, and we introduce the brick labelling of its Hasse quiver and use it to study lattice congruences of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="sans-serif t sans-serif o sans-serif r sans-serif s upper A"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="sans-serif">t</mml:mi> <mml:mi mathvariant="sans-serif">o</mml:mi> <mml:mi mathvariant="sans-serif">r</mml:mi> <mml:mi mathvariant="sans-serif">s</mml:mi> </mml:mrow> <mml:mi>A</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathsf {tors} A</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . In particular, we give a representation-theoretical interpretation of the so-called <italic>forcing order</italic> , and we prove that <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="sans-serif t sans-serif o sans-serif r sans-serif s upper A"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="sans-serif">t</mml:mi> <mml:mi mathvariant="sans-serif">o</mml:mi> <mml:mi mathvariant="sans-serif">r</mml:mi> <mml:mi mathvariant="sans-serif">s</mml:mi> </mml:mrow> <mml:mi>A</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathsf {tors} A</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is <italic>completely congruence uniform</italic> . When <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper I"> <mml:semantics> <mml:mi>I</mml:mi> <mml:annotation encoding="application/x-tex">I</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is a two-sided ideal of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper A"> <mml:semantics> <mml:mi>A</mml:mi> <mml:annotation encoding="application/x-tex">A</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="sans-serif t sans-serif o sans-serif r sans-serif s left-parenthesis upper A slash upper I right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="sans-serif">t</mml:mi> <mml:mi mathvariant="sans-serif">o</mml:mi> <mml:mi mathvariant="sans-serif">r</mml:mi> <mml:mi mathvariant="sans-serif">s</mml:mi> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>A</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>/</mml:mo> </mml:mrow> <mml:mi>I</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathsf {tors} (A/I)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is a lattice quotient of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="sans-serif t sans-serif o sans-serif r sans-serif s upper A"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="sans-serif">t</mml:mi> <mml:mi mathvariant="sans-serif">o</mml:mi> <mml:mi mathvariant="sans-serif">r</mml:mi> <mml:mi mathvariant="sans-serif">s</mml:mi> </mml:mrow> <mml:mi>A</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathsf {tors} A</mml:annotation> </mml:semantics> </mml:math> </inline-formula> which is called an <italic>algebraic quotient</italic> , and the corresponding lattice congruence is called an <italic>algebraic congruence</italic> . The second part of this paper consists in studying algebraic congruences. We characterize the arrows of the Hasse quiver of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="sans-serif t sans-serif o sans-serif r sans-serif s upper A"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="sans-serif">t</mml:mi> <mml:mi mathvariant="sans-serif">o</mml:mi> <mml:mi mathvariant="sans-serif">r</mml:mi> <mml:mi mathvariant="sans-serif">s</mml:mi> </mml:mrow> <mml:mi>A</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathsf {tors} A<

Topics & Concepts