The Q-shaped derived category of a ring — compact and perfect objects

Publikation: Bidrag til tidsskriftTidsskriftartikelForskningfagfællebedømt


  • Fulltext

    Indsendt manuskript, 396 KB, PDF-dokument

A chain complex can be viewed as a representation of a certain self-injective quiver with relations, Q. To define Q, include a vertex qn and an arrow qn ∂ → qn-1 for each integer n. The relations are ∂2 = 0. Replacing Q by a general self-injective quiver with relations, it turns out that some of the key properties of chain complexes generalise. Indeed, consider the representations of such a Q with values in AMod where A is a ring. We showed in earlier work that these representations form the objects of the Qshaped derived category, DQ(A), which is triangulated and generalises the classic derived category D(A). This follows ideas of Iyama and Minamoto. While DQ(A) has many good properties, it can also diverge dramatically from D(A). For instance, let Q be the quiver with one vertex q, one loop ∂, and the relation ∂2 = 0. By analogy with perfect complexes in the classic derived category, one may expect that a representation with a finitely generated free module placed at q is a compact object of DQ(A), but we will show that this is, in general, false. The purpose of this paper, then, is to compare and contrast DQ(A) and D(A) by investigating several key classes of objects: Perfect and strictly perfect, compact, fibrant, and cofibrant.

TidsskriftTransactions of the American Mathematical Society
Udgave nummer5
Sider (fra-til)3095-3128
Antal sider34
StatusUdgivet - maj 2024

Bibliografisk note

Funding Information:
Received by the editors September 28, 2022, and, in revised form, April 21, 2023. 2020 Mathematics Subject Classification. Primary 16E35, 18G80, 18N40. Key words and phrases. (Co)fibrant objects, compact objects, derived categories, differential modules, Frobenius categories, perfect objects, projective and injective model structures, quivers with relations, stable categories, Zeckendorf expansions. This work was supported by a DNRF Chair from the Danish National Research Foundation (grant DNRF156), by a Research Project 2 from the Independent Research Fund Denmark (grant 1026-00050B), and by Aarhus University Research Foundation (grant AUFF-F-2020-7-16).

Publisher Copyright:
© 2024 American Mathematical Society.

ID: 392564708