arXiv subjects

Anastasios Slaftsos

Publications and source records attributed to Anastasios Slaftsos.

2 recordsLinked to original sources

arXiv2026

Silting t-structures in $Q$-shaped derived categories

Torsion pairs, and in particular t-structures, play a central role in the study of triangulated categories. Specifically, t-structures induced by silting (or tilting) objects often admit desirable properties with strong connections to derived equivalences. In this paper, using the correspondence of Saor\'in-\v{S}\v{t}ov\'i\v{c}ek between cohereditary cotorsion pairs in Frobenius exact categories and t-structures in their stable categories, we construct a family of t-structures in the $Q$-shaped derived category of Holm and Jorgensen, arising from admissible partitions of $Q$. We give an explicit description of the associated cotorsion pairs inside the Frobenius exact category of the bifibrant objects, and we identify the corresponding co-aisles by certain homological vanishing conditions. Such t-structures are proved to be induced by a silting object, that can be completely determined by the combinatorics of $Q$. Finally, we illustrate our results by recovering well-known equivalences in the $Q$-shaped setting, while also providing examples where the combinatorial conditions fail (e.g. cyclic quivers), showing that such categories may admit no non-trivial t-structures, revealing phenomena analogous to those observed by Linckelmann in stable module categories.

math.RT↗

arXiv2026

Relative $Q$-shaped homological algebra

Exact categories are a natural generalisation of abelian categories and provide a fertile ground to develop relative homological algebra. In this paper, starting from a class of relative Gorenstein projective objects in an exact category $(\mathcal{A},\mathscr{E})$, we define exact model structures on $\mathcal{A}$ and cohomology functors that detect trivial objects and weak equivalences. Moreover, we show that varying the exact structure on $\mathcal{A}$ induces Bousfield (co)localisation sequences between the corresponding homotopy categories. We use these techniques to study the category ${}_{Q,A}\operatorname{Mod}$ of ${}_{A}\operatorname{Mod}$-valued representations, for a ring $A$, of a suitable $\Bbbk$-linear small category $Q$, where we apply our results to a range of objectwise exact structures, ranging from the split exact structure to the abelian one. In particular, we recover the $Q$-shaped derived category of Holm and Jorgensen and construct an intermediate $Q$-shaped homotopy category, analogous to the homotopy category of complexes. Finally, we show that the $Q$-shaped derived category is a Verdier quotient of the $Q$-shaped homotopy category, and that this quotient functor is part of recollement - generalising results of Verdier, Krause, and Iyama-Kato-Miyachi for complexes and $N$-complexes, respectively.

math.RT↗