Lattice theory of torsion classes: Beyond 𝜏-tilting theory

<p>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</mml:annotation> </mml:semantics> </mml:math> </inline-formula> that are contracted by an algebraic congruence in terms of the brick labelling. In the third part, we study in detail the case of preprojective algebras <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Pi"> <mml:semantics> <mml:mi mathvariant="normal">Π<!-- Π --></mml:mi> <mml:annotation encoding="application/x-tex">\Pi</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, for which <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 normal upper Pi"> <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 mathvariant="normal">Π<!-- Π --></mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathsf {tors} \Pi</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is the Weyl group endowed with the weak order. In particular, we give a new, more representation theoretical proof of the isomorphism between <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 k upper Q"> <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>k</mml:mi> <mml:mi>Q</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathsf {tors} k Q</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and the Cambrian lattice when <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper Q"> <mml:semantics> <mml:mi>Q</mml:mi> <mml:annotation encoding="application/x-tex">Q</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is a Dynkin quiver. We also prove that, in type <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>, the algebraic quotients 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 normal upper Pi"> <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 mathvariant="normal">Π<!-- Π --></mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathsf {tors} \Pi</mml:annotation> </mml:semantics> </mml:math> </inline-formula> are exactly its Hasse-regular lattice quotients.</p>