Peter Jorgensen’s research while affiliated with Aarhus University and other places

The main result of this paper is that there is sometimes a triangulated equivalence between $D_Q( A )$, the Q-shaped derived category of an algebra A, and $D( B )$, the classic derived category of a different algebra B. By construction, $D_Q( A )$ consists of Q-shaped diagrams of A-modules for a suitable small category Q. Our result concerns the case where Q consists of shifts of indecomposable projective modules over a self-injective $\mathbb{Z}$-graded algebra $\Lambda$. A notable special case is the result by Iyama, Kato, and Miyachi that $D_N( A )$, the N-derived category of A, is triangulated equivalent to $D( T_{ N-1 }A )$, the classic derived category of $T_{ N-1 }( A )$, which denotes upper diagonal $( N-1 ) \times ( N-1 )$-matrices over A. Several other special cases will also be discussed.

This paper studies a non-commutative generalisation of Coxeter friezes due to Berenstein and Retakh. It generalises several earlier results to this situation: A formula for frieze determinants, a T-path formula expressing the Laurent phenomenon, and results on gluing friezes together. One of our tools is a non-commutative version of the weak friezes introduced by Canakci and Jorgensen.

Starting from its original definition in module categories with respect to projective modules, the index has played an important role in various aspects of homological algebra, categorification of cluster algebras and K-theory. In the last few years, the notion of index has been generalised to several different contexts in (higher) homological algebra, typically with respect to a (higher) cluster-tilting subcategory $\mathcal{X}$ of the relevant ambient category $\mathcal{C}$. The recent tools of extriangulated and higher-exangulated categories have permitted some conditions on the subcategory $\mathcal{X}$ to be relaxed. In this paper, we introduce the index with respect to a generating, contravariantly finite subcategory of a d-exact category that has d-kernels. We show that our index has the important property of being additive on d-exact sequences up to an error term.

Palu defined the index with respect to a cluster tilting object in a suitable triangulated category, in order to better understand the Caldero–Chapoton map that exhibits the connection between cluster algebras and representation theory. We push this further by proposing an index with respect to a contravariantly finite, rigid subcategory, and we show this index behaves similarly to the classical index. Let ${\mathcal{C}}$ be a skeletally small triangulated category with split idempotents, which is thus an extriangulated category $({\mathcal{C}},{\mathbb{E}},{\mathfrak{s}})$. Suppose ${\mathcal{X}}$ is a contravariantly finite, rigid subcategory of ${\mathcal{C}}$. We define the index ${\operatorname{\textrm{ind}}}{_{{\mathcal{X}}}}(C)$ of an object $C\in{\mathcal{C}}$ with respect to ${\mathcal{X}}$ as the ${K}{_{0}}$-class $[C {]}{_{{\mathcal{X}}}}$ in Grothendieck group ${K}{_{0}}({\mathcal{C}}, {{\mathbb{E}}}{_{{\mathcal{X}}}}, {{\mathfrak{s}}}{_{{\mathcal{X}}}})$ of the relative extriangulated category $({\mathcal{C}}, {{\mathbb{E}}}{_{{\mathcal{X}}}}, {{\mathfrak{s}}}{_{{\mathcal{X}}}})$. By analogy to the classical case, we give an additivity formula with error term for ${\operatorname{\textrm{ind}}}{_{{\mathcal{X}}}}$ on triangles in ${\mathcal{C}}$. In case ${\mathcal{X}}$ is contained in another suitable subcategory ${\mathcal{T}}$ of ${\mathcal{C}}$, there is a surjection $Q\colon{K}{_{0}}({\mathcal{C}}, {{\mathbb{E}}}{_{{\mathcal{T}}}}, {{\mathfrak{s}}}{_{{\mathcal{T}}}}) \twoheadrightarrow{K}{_{0}}({\mathcal{C}}, {{\mathbb{E}}}{_{{\mathcal{X}}}}, {{\mathfrak{s}}}{_{{\mathcal{X}}}})$. Thus, in order to describe ${K}{_{0}}({\mathcal{C}}, {{\mathbb{E}}}{_{{\mathcal{X}}}}, {{\mathfrak{s}}}{_{{\mathcal{X}}}})$, it suffices to determine ${K}{_{0}}({\mathcal{C}}, {{\mathbb{E}}}{_{{\mathcal{T}}}}, {{\mathfrak{s}}}{_{{\mathcal{T}}}})$ and $\operatorname{Ker}\nolimits Q$. We do this under certain assumptions.

A chain complex can be viewed as a representation of a certain quiver with relations, $Q^{\operatorname{cpx}}$. The vertices are the integers, there is an arrow $q \xrightarrow{} q-1$ for each integer q, and the relations are that consecutive arrows compose to 0. Hence the classic derived category $\mathscr{D}$ can be viewed as a category of representations of $Q^{\operatorname{cpx}}$. It is an insight of Iyama and Minamoto that the reason $\mathscr{D}$ is well behaved is that, viewed as a small category, $Q^{\operatorname{cpx}}$ has a Serre functor. Generalising the construction of $\mathscr{D}$ to other quivers with relations which have a Serre functor results in the Q-shaped derived category ${\mathscr{D}}_Q$. Drawing on methods of Hovey and Gillespie, we developed the theory of ${\mathscr{D}}_Q$ in three recent papers. This paper offers a brief introduction to ${\mathscr{D}}_Q$, aimed at the reader already familiar with the classic derived category.

A chain complex can be viewed as a representation of a certain self-injective quiver with relations, Q Q . To define Q Q , include a vertex q n q_n and an arrow q n → ∂ q n − 1 q_n \xrightarrow {\partial } q_{n-1} for each integer n n . The relations are ∂ 2 = 0 \partial ^2 = 0 . Replacing Q 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 Q with values in A Mod {}_{A}\mspace {-1mu}\operatorname {Mod} where A A is a ring. We showed in earlier work that these representations form the objects of the Q Q -shaped derived category , D Q ( A ) \mathcal {D}_{Q}(A) , which is triangulated and generalises the classic derived category D ( A ) \mathcal {D}_{}(A) . This follows ideas of Iyama and Minamoto. While D Q ( A ) \mathcal {D}_{Q}(A) has many good properties, it can also diverge dramatically from D ( A ) \mathcal {D}_{}(A) . For instance, let Q Q be the quiver with one vertex q q , one loop ∂ \partial , and the relation ∂ 2 = 0 \partial ^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 q is a compact object of D Q ( A ) \mathcal {D}_{Q}(A) , but we will show that this is, in general, false. The purpose of this paper, then, is to compare and contrast D Q ( A ) \mathcal {D}_{Q}(A) and D ( A ) \mathcal {D}_{}(A) by investigating several key classes of objects: Perfect and strictly perfect, compact, fibrant, and cofibrant.