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On diagram-chasing in double complexes

TL;DR: In this paper, for any double complex in an abelian category, certain short-distance maps and an exact sequence involving these, instances of which can be pieced together to give the "long-distance" maps and exact sequences of results such as the Snake Lemma.
Abstract: We construct, for any double complex in an abelian category, certain "short-distance" maps, and an exact sequence involving these, instances of which can be pieced together to give the "long-distance" maps and exact sequences of results such as the Snake Lemma. Further applications are given. We also note what the building blocks of an analogous study of triple complexes would be.

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TL;DR: In this article, Iyama's higher Auslander-Reiten theory for Artin algebras from the viewpoint of higher homological algebra is introduced, and the existence of morphisms determined by objects in $d$-cluster-tilting subcategories is established.
Abstract: This article consists of an introduction to Iyama's higher Auslander-Reiten theory for Artin algebras from the viewpoint of higher homological algebra. We provide alternative proofs of the basic results in higher Auslander-Reiten theory, including the existence of $d$-almost-split sequences in $d$-cluster-tilting subcategories, following the approach to classical Auslander-Reiten theory due to Auslander, Reiten, and Smalo. We show that Krause's proof of Auslander's defect formula can be adapted to give a new proof of the defect formula for $d$-exact sequences. We use the defect formula to establish the existence of morphisms determined by objects in $d$-cluster-tilting subcategories.

17 citations

Posted Content
TL;DR: In this paper, the authors used relative symplectic cohomology, recently studied by the second author, to understand rigidity phenomena for compact subsets of symplectic manifolds.
Abstract: This article uses relative symplectic cohomology, recently studied by the second author, to understand rigidity phenomena for compact subsets of symplectic manifolds. As an application, we consider a symplectic crossings divisor in a Calabi-Yau symplectic manifold $M$ whose complement is a Liouville manifold. We show that, for a carefully chosen Liouville structure, the skeleton as a subset of $M$ exhibits strong rigidity properties akin to super-heavy subsets of Entov-Polterovich. Along the way, we expand the toolkit of relative symplectic cohomology by introducing products and units. We also develop what we call the contact Fukaya trick, concerning the behaviour of relative symplectic cohomology of subsets with contact type boundary under adding a Liouville collar.

9 citations

Journal Article
TL;DR: In this paper, it was shown that a double complex with exact rows and columns yields an acyclic cochain complex under totalisation using right (resp., left) truncated products.
Abstract: We consider a non-standard totalisation functor to produce a cochain complex from a given double complex D∗,∗: instead of sums or products, totalisation is defined via truncated products of modules. We give an elementary proof of the fact that a double complex with exact rows (resp., columns) yields an acyclic cochain complex under totalisation using right (resp., left) truncated products. As an application we consider the algebraic mapping torus T (h) of a self map h of a cochain complex C. We show that if C consists of finitely presented modules then T (h) has trivial negative Novikov cohomology; if in addition h is a quasi-isomorphism, then T (h) has trivial positive Novikov cohomology as well. As a consequence we obtain a new proof that a finitely dominated cochain complex over a Laurent polynomial ring has trivial Novikov cohomology. Finiteness conditions for chain complexes of modules play an important role in both algebra and topology. For example, given a group G one might ask whether the trivial G-module Z admits a resolution by finitely generated projective Z[G]-modules; existence of such resolutions is relevant for the study of group cohomology of G, and has applications in the theory of duality groups [B75]. For topologists, finite domination of chain complexes is related, among other things, to questions about finiteness of CW complexes, the topology of ends of manifolds, and obstructions for the existence of nonsingular closed 1-forms [Ran95, S06]. A cochain complex C of R[z, z−1]-modules is called finitely dominated if it is homotopy equivalent, as a complex of R-modules, to a bounded complex of finitely generated projective R-modules. Finite domination of C can be characterised in various ways; Brown considered compatibility of the functors M 7→ H∗(C;M) and M 7→ H∗(C;M) with products and direct limits, respectively [B75, Theorem 1], while Ranicki showed that C is finitely dominated if and only if the Novikov cohomology of C is trivial [Ran95, Theorem 2] (see also Definition 2.3 and Corollary 2.7 below). Our approach to Novikov cohomology involves a non-standard totalisation functor. The key fact is that certain double complexes are converted into acyclic cochain complexes (Proposition 1.2) which is proved by an elementary calculation. As an application we obtain a new result for vanishing of Novikov cohomology of algebraic mapping tori (Theorem 2.5), and recover the “only-if” part of Ranicki’s criterion for finite domination over Date: 08.09.2011. 2000 Mathematics Subject Classification. Primary 18G35; Secondary 55U15. This work was supported by the Engineering and Physical Sciences Research Council [grant number EP/H018743/1]. 1

9 citations

Dissertation
01 Aug 2017
TL;DR: In this article, the authors introduce a class of semiperfect rings which generalise the class of finite-dimensional gentle algebras and consider complexes of modules over these rings, which have finitely generated projective homogeneous components.
Abstract: In this thesis we introduce a class of semiperfect rings which generalise the class of finite-dimensional gentle algebras. We consider complexes of modules over these rings which have finitely generated projective homogeneous components. We then classify them up to homotopy equivalence. The method we use to solve this classification problem is called the functorial filtrations method. The said method was previously only used to classify modules.

8 citations


Cites background from "On diagram-chasing in double comple..."

  • ...We follow: books by Aluffi [1] and Freyd [26] and papers by Bergman [10] and Krause [45] for ideas about abelian categories; and the books by Gelfand and Manin [33], Neeman [52], Weibel [64] and Zimmerman [67] for ideas about triangulated categories....

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  • ...In [10] Bergman presented an idea which captures the notion of diagram chasing in an abelian category....

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Posted Content
TL;DR: In this paper, two cohomology theories for structured spaces are developed, one arises from the presheaves and the other arises from vector bundles, and they can be applied also in many other situations.
Abstract: In [1] we defined a new kind of space called 'structured space' which locally resembles, near each of its points, some algebraic structure. We noted in the conclusion of the cited paper that the maps $f_s$ and $h$, which are of great importance in the theory of structured spaces, have some connections with the notions of presheaves (and hence also sheaves) and vector bundles. There are well known cohomology theories involving such objects; this suggests the possibility of the existence of (co)homology theories for structured spaces which are somehow related to $f_s$ and $h$. In this paper we indeed develop two cohomology theories for structured spaces: one of them arises from $f_s$, while the other one arises from $h$. In order to do this, we first develop a more general cohomology theory (called rectangular cohomology in the finite case, and square cohomology in the infinite case), which can actually be applied also in many other situations, and then we obtain the cohomology theories for structured spaces as simple consequences of this theory.

4 citations


Cites background from "On diagram-chasing in double comple..."

  • ...We conclude noting that, if the square/rectangle is actually a double complex, the salamander lemma (see [39]) uses a similar idea to our ”following the path π”....

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  • ...The construction of the square (or, in the finite case, of the rectangle) is really similar to a double complex (for more details on double complexes, see [36-39])....

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References
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Journal ArticleDOI
TL;DR: In this paper, a homological formulation of Goursat's theorem is presented, and some rudimentary concepts of homological algebra can be introduced with the help of this form of the theorem.
Abstract: This expository note consists of two parts: In the first we present a homological formulation of Goursat's theorem. In the second we indicate how some rudimentary concepts of homological algebra can be introduced with the help of this form of Goursat's theorem. The second part is addressed to those readers with an algebraic background who wish to be initiated into homological algebra as painlessly as possible. The ideas developed here are close to the spirit of [5, Chapter II, § 6], where further references may be found. In fact, these ideas are very much in the air, and any originality in the present note is purely coincidental.

16 citations

Journal ArticleDOI
TL;DR: A category with zero-maps is called "quasi-exact" in the sense of D. Puppe as mentioned in this paper, if it satisfies the following axioms: (Q1) every may f is a product f=μe of an epimorphisrn efollowed by a monomorphism μ, where Ker and Coker are characterized by the familiar universality properties.
Abstract: A category with zero-maps is called "quasi-exact" in the sense of D. Puppe (see [4], page 8, 2. 4), if it satisfies the following axioms: (Q1) Every may f is a product f=μe of an epimorphisrn efollowed by a monomorphism μ (Q2) a) Every epimorphism e has a kernel k = ker e b) Every monomorphism μ has a cokernel γ = Coker e, where Ker and Coker are characterized by the familiar universality properties (see [3], page 252, (1. 10) and (1. 11)).

6 citations


"On diagram-chasing in double comple..." refers background in this paper

  • ...Leicht [11], Kopylov [7], and others have given more general conditions on a category under which Lambek’s result holds....

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Posted Content
TL;DR: In this paper, the invariants Ker and Im for commutative squares in quasi-abelian categories were considered and they were shown to be invariants for groups and then studied by Hilton and Nomura in exact categories.
Abstract: We consider the invariants Ker and Im for commutative squares in quasi-abelian categories. These invariants were introduced by Lambek for groups and then studied by Hilton and Nomura in exact categories.

2 citations