Let T be a set of elements in a ring A. The set T is right permutable if for any a ∈ A and t ∈ T, there exist b ∈ A, u ∈ T such that au = tb. A multiplicative set in a ring A is any subset T of A such that 1 ∈ T,0 ∉ T and T is closed under multiplication. A completely prime ideal in a ring A is any proper ideal B such that A\B is a multiplicative set (i.e. A/Bis a domain). A minimal prime ideal (resp. minimal completely prime ideal) in a ring A is any prime (resp. completely prime) ideal P such that P contains no properly any other prime ideal (resp. completely prime ideal) of A. Let I be any proper ideal of a ring A. The set of all elements a ∈ A such that a + I is a regular element of A/I is denoted by c(I). In particular, c(0) is the set of all regular elements of A.

Rings of quotients

The class of support $\\tau$-tilting modules was introduced to provide a completion of the class of tilting modules from the point of view of mutations. In this article we study $\\tau$-tilting finite algebras, i.e. finite dimensional algebras $A$ with finitely many isomorphism classes of indecomposable $\\tau$-rigid modules. We show that $A$ is $\\tau$-tilting finite if and only if very torsion class in $\\mod A$ is functorially finite. We observe that cones generated by $g$-vectors of indecomposable direct summands of each support $\\tau$-tilting module form a simplicial complex $\\Delta(A)$. We show that if $A$ is $\\tau$-tilting finite, then $\\Delta(A)$ is homeomorphic to an $(n-1)$-dimensional sphere, and moreover the partial order on support $\\tau$-tilting modules can be recovered from the geometry of $\\Delta(A)$. Finally we give a bijection between indecomposable $\\tau$-rigid $A$-modules and bricks of $A$ satisfying a certain finiteness condition, which is automatic for $\\tau$-tilting finite algebras.

$\boldsymbol{\tau}$-Tilting Finite Algebras, Bricks, and $\boldsymbol{g}$-Vectors

The class of support $\tau$-tilting modules was introduced to provide a completion of the class of tilting modules from the point of view of mutations. In this article we study $\tau$-tilting finite algebras, i.e. finite dimensional algebras $A$ with finitely many isomorphism classes of indecomposable $\tau$-rigid modules. We show that $A$ is $\tau$-tilting finite if and only if very torsion class in $\mod A$ is functorially finite. We observe that cones generated by $g$-vectors of indecomposable direct summands of each support $\tau$-tilting module form a simplicial complex $\Delta(A)$. We show that if $A$ is $\tau$-tilting finite, then $\Delta(A)$ is homeomorphic to an $(n-1)$-dimensional sphere, and moreover the partial order on support $\tau$-tilting modules can be recovered from the geometry of $\Delta(A)$. Finally we give a bijection between indecomposable $\tau$-rigid $A$-modules and bricks of $A$ satisfying a certain finiteness condition, which is automatic for $\tau$-tilting finite algebras.

$\tau$-tilting finite algebras, bricks and $g$-vectors

It is shown that the silting reduction T /thickP of a triangulated category T with respect to a presilting subcategory P can be realised as a certain subfactor category of T , and that there is a one-to-one correspondence between the set of (pre)silting subcategories of T containing P and the set of (pre)silting subcategories of T /thickP . This is analogous to a result for Calabi–Yau reduction. This result is applied to show that the Amiot–Guo–Keller construction of d-Calabi–Yau triangulated categories with d-cluster-tilting objects takes silting reduction to Calabi–Yau reduction, and conversely, Calabi–Yau reduction lifts to silting reduction.

Silting reduction and Calabi–Yau reduction of triangulated categories

Silting theory in triangulated categories with coproducts

We introduce the new concept of silting modules These modules generalise tilting modules over an arbitrary ring, as well as support $\tau$-tilting modules over a finite dimensional algebra recently introduced by Adachi, Iyama and Reiten We show that silting modules generate torsion classes that provide left approximations, and that every partial silting module admits an analogue of the Bongartz complement Furthermore, we prove that silting modules are in bijection with 2-term silting complexes and with certain t-structures and co-t-structures in the derived module category We also see how some of these bijections hold for silting complexes of arbitrary finite length

Silting modules

For a finite-dimensional algebra A, we establish correspondences between torsion classes and wide subcategories in mod(A). In case A is representation finite, we obtain an explicit bijection between these two classes of subcategories. Moreover, we translate our results to the language of ring epimorphisms and universal localisations. It turns out that universal localisations over representation finite algebras are classified by torsion classes and support τ-tilting modules.

Torsion classes, wide subcategories and localisations

Silting modules and ring epimorphisms

For a finite dimensional algebra $A$, we establish correspondences between torsion classes and wide subcategories in $mod(A)$. In case $A$ is representation finite, we obtain an explicit bijection between these two classes of subcategories. Moreover, we translate our results to the language of ring epimorphisms and universal localisations. It turns out that universal localisations over representation finite algebras are classified by torsion classes and support $\tau$-tilting modules.

In the setting of compactly generated triangulated categories, we show that the heart of a (co)silting t-structure is a Grothendieck category if and only if the (co)silting object satisfies a purity assumption. Moreover, in the cosilting case the previous conditions are related to the coaisle of the t-structure being a definable subcategory. If we further assume our triangulated category to be algebraic, it follows that the heart of any nondegenerate compactly generated t-structure is a Grothendieck category.

Frederik Marks

Papers

Silting modules

Torsion classes, wide subcategories and localisations

Silting modules and ring epimorphisms

Torsion classes, wide subcategories and localisations

Torsion pairs in silting theory