τ-Tilting Modules Over One-Point Extensions by a Projective Module

Abstract

Let A be the one point extension of an algebra B by a projective B-module. We prove that the extension of a given support τ -tilting B-module is a support τ -tilting Amodule; and, conversely, the restriction of a given support τ -tilting A-module is a support τ -tilting B-module. Moreover, we prove that there exists a full embedding of quivers between the corresponding poset of support τ -tilting modules.

Silting Modules over Triangular Matrix Rings

“…For example, Assem, Happel and Trepode [5] studied how to extend and restrict tilting modules by given tilting modules for the one-point extension of an algebra by a projective module. Suarez [19] generalized this result to the case for support τ -tilting modules.…”

Support $τ$-tilting modules over one-point extensions

“…In [4], Assem, Happel and Trepode studied how to extend and restrict tilting modules for onepoint extension algebras by a projective module. In [8], Suarez generalized this result for the context of support support τ -tilting modules. More precisely, let B = A[P ] be the one-point extension of an algebra A by a projective A-module P and e the identity of A.…”

“…If M is a support τ -tilting A-module, then Hom B (eB, M) ⊕ S a is a support τ -tilting Bmodule, where S a is the simple module corresponding to the new point a (see [8,Theorem A]). An example shown that Hom B (eB, M) ⊕S a may not be a support τ -tilting B-module if P is not projective (see [8,Example 4.7]).…”

Extending (τ-)tilting subcategories and (co)silting modules

“…Using previous result, we are able to provide another proof for the main result, i.e. Proposition 3.2, of [18]. (i) Let T be a support τ -tilting module of mod-A.…”

“…It turned out that these two functors, being an adjoint pair, have nice homological properties. P. Suarez [18] followed this approach and studied the restriction and extension of (support) τ -tilting modules. The notion of τ -tilting theory was introduced by Adachi, Iyama and Reiten in [1] as a new approach for studying two classical branches of the representation theory of finite dimensional algebras, namely tilting theory and Auslander-Reiten theory.…”