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
SN - 0002-9947
VL - 377
SP - 3095
EP - 3128
JO - Transactions of the American Mathematical Society
JF - Transactions of the American Mathematical Society
IS - 5
ER -