TY - JOUR

T1 - The Q-shaped derived category of a ring — compact and perfect objects

AU - Holm, Henrik

AU - Jørgensen, Peter

N1 - Publisher Copyright: © 2024 American Mathematical Society.

PY - 2024/5

Y1 - 2024/5

N2 - 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.

AB - 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.

KW - (Co)fibrant objects

KW - compact objects

KW - derived categories

KW - differential modules

KW - Frobenius categories

KW - perfect objects

KW - projective and injective model structures

KW - quivers with relations

KW - stable categories

KW - Zeckendorf expansions

U2 - 10.1090/tran/8979

DO - 10.1090/tran/8979

M3 - Journal article

AN - SCOPUS:85192689971

VL - 377

SP - 3095

EP - 3128

JO - Transactions of the American Mathematical Society

JF - Transactions of the American Mathematical Society

SN - 0002-9947

IS - 5

ER -