Monica García, Université Paris-Saclay
Thick subcategories and g-finiteness in the category of projective presentations


Abstract: The category of projective presentations, or 2-term complexes, behaves in many ways as a mirror of the category of finite-dimensional modules over a finite-dimensional algebra. Both satisfy similar properties, however, unlike the category of modules, the category of projective presentations is not abelian, but rather an extriangulated category. In this talk, we reinforce the relation between these two categories by establishing an analogue of the Ingalls-Thomas correspondences within the category of projective presentations. We will exhibit bijections between the sets of isomorphism classes of 2-term silting objects, complete cotorsion pairs, and thick subcategories with enough injectives of the category of projective presentations. Furthermore, we will show how these correspondences mirror those between the sets of isomorphism classes of support tau-tilting pairs, functorially finite torsion classes, and left wide subcategories of the module category.

Additionally, we introduce new equivalent conditions for an algebra to be g-finite. Namely, we show that an algebra has finitely many isomorphism classes of silting objects if and only if all cotorsion pairs are complete, and if and only if there are finitely many thick subcategories of projective presentations. We will establish that in such cases, all thick subcategories have enough injectives.