On gg-finiteness in the category of projective presentations

We provide new equivalent conditions for an algebra $\Lambda$ to be $g$-finite, analogous to those established by L. Demonet, O. Iyama, and G. Jasso, but within the category of projective presentations $\mathcal{K}^{[-1,0]}(\text{proj} \Lambda)$. We show that an algebra has finitely many isomorphism classes of basic $2$-term silting objects if and only if all cotorsion pairs in $\mathcal{K}^{[-1,0]}(\text{proj} \Lambda)$ are complete. Furthermore, we establish that this criterion is also equivalent to all thick subcategories in $\mathcal{K}^{[-1,0]}(\text{proj} \Lambda)$ having enough injective and projective objects.Comment: 27 pages, 1 figure. V2 : Added Proposition 2.1

Stratifying systems and Jordan-H\"{o}lder extriangulated categories

Stratifying systems, which have been defined for module, triangulated and exact categories previously, were developed to produce examples of standardly stratified algebras. A stratifying system $\Phi$ is a finite set of objects satisfying some orthogonality conditions. One very interesting property is that the subcategory $\mathcal{F}(\Phi)$ of objects admitting a composition series-like filtration with factors in $\Phi$ has the Jordan-H{\"{o}}lder property on these filtrations. This article has two main aims. First, we introduce notions of subobjects, simple objects and composition series for an extriangulated category, in order to define a Jordan-H{\"{o}}lder extriangulated category. Moreover, we characterise Jordan-H{\"{o}}lder, length, weakly idempotent complete extriangulated categories in terms of the associated Grothendieck monoid and Grothendieck group. Second, we develop a theory of stratifying systems in extriangulated categories. We define projective stratifying systems and show that every stratifying system $\Phi$ in an extriangulated category is part of a minimal projective one $(\Phi,Q)$. We prove that $\mathcal{F}(\Phi)$ is a Jordan-H{\"{o}}lder extriangulated category when $(\Phi,Q)$ satisfies a left exactness condition.Comment: v3: 32 pages; Proposition 3.6 and Corollary 4.5 added; this has simplified several results in Section 3 and allowed us to show every stratifying system is part of a minimal projective one in Section 4; a missing assumption added to Theorem 3.19(=Theorem C); other minor changes. v2: 32 pages; Remark 5.3 added; typos corrected; other minor changes. v1: 31 pages. Comments very welcome

On thick subcategories of the category of projective presentations

We study thick subcategories of the category of 2-term complexes of projective modules over an associative algebra. We show that those thick subcategories that have enough injectives are in explicit bijection with 2-term silting complexes and complete cotorsion pairs. We also provide a bijection with left finite wide subcategories of the module category and prove that all these maps are compatible with previously known correspondences. We discuss possible applications to stability conditions.Comment: 35 pages V2: typos corrected V3: new introduction for section 4, typos correcte

Relative left Bongartz completions and their compatibility with mutations

In this paper, we introduce relative left Bongartz completions for a given basic $\tau$-rigid pair $(U,Q)$ in the module category of a finite dimensional algebra $A$. They give a family of basic $\tau$-tilting pairs containing $(U,Q)$ as a direct summand. We prove that relative left Bongartz completions have nice compatibility with mutations. Using this compatibility we are able to study the existence of maximal green sequences under $\tau$-tilting reduction. We also explain our construction and some of the results in the setting of silting theory.Comment: 21 page

Aspects of representation theory: τ-exceptional sequences, modular Fuss-Catalan numbers and idempotent completion of extriangulated categories

Abstract This thesis is concerned with various aspects of the representation the- ory of finite dimensional algebras, with a focus on combinatorial and homological aspects. We explore the aspects of representation theory relating to tilting modules, cluster algebras, τ-exceptional sequences, and extriangulated categories. The notion of a τ-exceptional sequence was introduced by Buan and Marsh in Buan & Marsh (2021) as a generalisation of an exceptional sequence for finite-dimensional algebras. We calculate the number of complete τ-exceptional sequences of certain classes of Nakayama al- gebras. In some cases, we obtain closed formulas which also count other well-known combinatorial sets and exceptional sequences of path algebras of Dynkin quivers. The modular Catalan numbers C(k,n), introduced in Hein & Huang (2017) count equivalence classes of parenthesizations of x0 ∗ · · · ∗ xn, where ∗ is a binary k-associative operation and k is a positive inte- ger. The classical notion of associativity coincides with 1-associativity, in which case C(1,n) = 1, and the single 1-equivalence class has size given by the Catalan number Cn. We introduce modular Fuss-Catalan numbers Cm which count k-equivalence classes of parenthesizations of k,n x0 ∗ · · · ∗ xn where ∗ is an m-ary k-associative operation for m ≥ 2. Our main results are an explicit formula for Cm , and a characterisation of k-associativity. Extriangulated categories were introduced by Nakaoka and Palu in Nakaoka & Palu (2019a) as a simultaneous generalisation of exact cat- egories and triangulated categories. We show that the idempotent completion of an extriangulated category is also extriangulated. A possible consequence of this is a methodology for constructing Krull- Remak-Schmidt extriangulated categories, since an additive category A has the Krull-Remak-Schmidt property if and only if A is idempotent complete and the endomorphism ring of every object is semi-perfect; see (Krause, 2015, Corollary 4.4)