Calabi–Yau categories and Poincaré duality

Abstract

The singular cochain complex of a topological space is a classical object. It is a Differential Graded algebra which has been studied intensively with a range of methods, not least within rational homotopy theory.More recently, the tools of Auslander-Reiten theory have also been applied to the singular cochain complex. One of the highlights is that by these methods, each Poincaré duality space gives rise to a Calabi-Yau category. This paper is a review of the theory.has the full subcategory D c (C * (X; k)) consisting of compact DG modules; these play the role of finitely generated representations. Theorem 6.4 now says that if k has characteristic 0, then D c (C * (X; k)) is an n-Calabi-Yau category (1.1) ⇔ X has n-dimensional Poincaré duality over k.Let me briefly explain the terminology. A triangulated category T, such as for instance D c (C * (X; k)), is called n-Calabi-Yau if n is the smallest non-negative integer for which Σ n , the nth power of the suspension functor, is a Serre functor, that is, permits natural isomorphismsThe topological space X is said to have n-dimensional Poincaré duality over k if there is an isomorphismExamples of n-Calabi-Yau categories are higher cluster categories, see [15, sec. 4], and examples of spaces with n-dimensional Poincaré duality are compact ndimensional manifolds. Equation (1.1) provides a link between the currently popular theory of Calabi-Yau categories and algebraic topology. It also gives a new class of examples of Calabi-Yau categories which, so far, typically have been exemplified by higher cluster categories. The new categories appear to behave very differently from higher cluster categories, cf. Section 7, Problem 7.8.A number of other results are also obtained, not least on the structure of the AR quiver of D c (C * (X; k)) which, for a space with Poincaré duality, consists of copies of the repetitive quiver ZA ∞ , see Theorem 6.5.In a speculative vein, the theory presented here ties in with the version of noncommutative geometry in which a DG algebra, or more generally a DG category, is viewed as a non-commutative scheme. The idea is to think of the derived category of the DG algebra or DG category as being the derived category of quasi-coherent sheaves on a non-commutative scheme (which does not actually exist). There appear so far to be no published references for this viewpoint which has been brought forward by Drinfeld and Kontsevich, but it does seem to call for a detailed study of the derived categories of DG algebras and DG categories. Auslander-Reiten theory is an obvious tool to try, and [11], [12], and [20] along with this paper can, perhaps, be viewed as a first, modest step.As indicated, the paper is a review. The results were known previously, the main references being [11], [12], and [20]; more details of the origin of individual results are given in the introductions to the sections. There is no claim to originality, except that some of the proofs are new. It is also the first time this material has appeared together.Most of the paper is phras...

Almost Split Sequences and Approximations

“…The result says in particular that if the ambient category has almost split sequences, then the almost split sequences in a Hom-finite Krull-Schmidt exact subcategory are precisely the minimal projectively or injectively stable approximations of the almost split sequences in the ambient category; see (3.5) and (3.7). This is a strengthened analogous version, by means of a very different approach, of Jørgensen's result stated in [15]. Since our categories do not necessarily have projective or injective objects, one can not simply apply Jørgensen's result in our situation as is done in [22].…”

“…The Auslander-Reiten theory of almost split sequences has been playing a fundamental role in the representation theory of artin algebras with a great impact in other areas such as algebraic geometry and algebraic topology; see [6,3,15]. It is a long standing problem to determine which categories have almost split sequences.…”

Representation theory of strongly locally finite quivers

“…On the other hand, the Auslander-Reiten theory of irreducible morphisms and almost split sequences provides an indispensable powerful tool for the representation theory and it appears in many other areas such as algebraic geometry and algebraic topology; see [6,3,25]. The impact of these two theories to other branches of mathematics is best illustrated by their recent interaction with the theory of cluster algebras via the cluster category; see, for example, [12,17].…”

Auslander-Reiten theory in a Krull-Schmidt category

“…It has also an important impact to many other areas of mathematics such as algebraic geometry and algebraic topology; see, for example, [1,14,15]. Indeed, it appears naturally in abelian categories such as the module category of an artin algebra; see [4], additive categories with an exact structure such as the representation category of a bocs; see [7], and triangulated categories such as the derived category of bounded complexes in an abelian category; see [10,14] and cluster categories; see [9].…”