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# On diagram-chasing in double complexes

AbstractWe 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.

Topics: Snake lemma (74%), Five lemma (72%), Abelian category (66%), Exact sequence (65%)

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Journal Article
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
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.

6 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
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.

4 citations

Posted Content
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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Journal ArticleDOI
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.

4 citations

##### References
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Book
01 Jan 1971

9,164 citations

Book
01 Jan 1968

2,028 citations

Book
31 Dec 1971
Abstract: I. Modules.- 1. Modules.- 2. The Group of Homomorphisms.- 3. Sums and Products.- 4. Free and Projective Modules.- 5. Projective Modules over a Principal Ideal Domain.- 6. Dualization, Injective Modules.- 7 Injective Modules over a Principal Ideal Domain.- 8. Cofree Modules.- 9. Essential Extensions.- II. Categories and Functors.- 1. Categories.- 2. Functors.- 3. Duality.- 4. Natural Transformations.- 5. Products and Coproducts Universal Constructions.- 6. Universal Constructions (Continued) Pull-backs and Push-outs.- 7. Adjoint Functors.- 8. Adjoint Functors and Universal Constructions.- 9. Abelian Categories.- 10. Projective, Injective, and Free Objects.- III. Extensions of Modules.- 1. Extensions.- 2. The Functor Ext.- 3. Ext Using Injectives.- 4. Computation of some Ext-Groups.- 5. Two Exact Sequences.- 6. A Theorem of Stein-Serre for Abelian Groups.- 7. The Tensor Product.- 8. The Functor Tor.- IV. Derived Functors.- 1. Complexes.- 2. The Long Exact (Co) Homology Sequence.- 3. Homotopy.- 4. Resolutions.- 5. Derived Functors.- 6. The Two Long Exact Sequences of Derived Functors.- 7. The Functors Extn? Using Projectives.- 8. The Functors % MathType!MTEF!2!1!+- % feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbWexLMBb50ujbqegm0B % 1jxALjharqqr1ngBPrgifHhDYfgasaacH8srps0lbbf9q8WrFfeuY- % Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0-yr0RYxir-Jbba9q8aq % 0-yq-He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaae % aaeaaakeaadaqdaaqaaGqaaiaa-veacaWF4bGaa8hDaaaadaqhaaWc % baacciGae43MdWeabaGaamOBaaaaaaa!40A3! $$\overline {Ext} _\Lambda ^n$$ Using Injectives.- 9. Extn and n-Extensions.- 10. Another Characterization of Derived Functors.- 11. The Functor Torn?.- 12. Change of Rings.- V. The Kiinneth Formula.- 1. Double Complexes.- 2. The Kunneth Theorem.- 3. The Dual Kunneth Theorem.- 4. Applications of the Kunneth Formulas.- VI. Cohomology of Groups.- 1. The Group Ring.- 2. Definition of (Co) Homology.- 3. H0, H0.- 4. H1, H1 with Trivial Coefficient Modules.- 5. The Augmentation Ideal, Derivations, and the Semi-Direct Product.- 6. A Short Exact Sequence.- 7. The (Co) Homology of Finite Cyclic Groups.- 8. The 5-Term Exact Sequences.- 9. H2, Hopf's Formula, and the Lower Central Series.- 10. H2 and Extensions.- 11. Relative Projectives and Relative Injectives.- 12. Reduction Theorems.- 13. Resolutions.- 14. The (Co) Homology of a Coproduct.- 15. The Universal Coefficient Theorem and the (Co)Homology of a Product.- 16. Groups and Subgroups.- VII. Cohomology of Lie Algebras.- 1. Lie Algebras and their Universal Enveloping Algebra.- 2. Definition of Cohomology H0, H1.- 3. H2 and Extensions.- 4. A Resolution of the Ground Field K.- 5. Semi-simple Lie Algebras.- 6. The two Whitehead Lemmas.- 7. Appendix : Hubert's Chain-of-Syzygies Theorem.- VIII. Exact Couples and Spectral Sequences.- 1. Exact Couples and Spectral Sequences.- 2. Filtered Differential Objects.- 3. Finite Convergence Conditions for Filtered Chain Complexes.- 4. The Ladder of an Exact Couple.- 5. Limits.- 6. Rees Systems and Filtered Complexes.- 7. The Limit of a Rees System.- 8. Completions of Filtrations.- 9. The Grothendieck Spectral Sequence.- IX. Satellites and Homology.- 1. Projective Classes of Epimorphisms.- 2. ?-Derived Functors.- 3. ?-Satellites.- 4. The Adjoint Theorem and Examples.- 5. Kan Extensions and Homology.- 6. Applications: Homology of Small Categories, Spectral Sequences.- X. Some Applications and Recent Developments.- 1. Homological Algebra and Algebraic Topology.- 2. Nilpotent Groups.- 3. Finiteness Conditions on Groups.- 4. Modular Representation Theory.- 5. Stable and Derived Categories.

964 citations

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

• ...3, the vertical exactness assumption would simply take the form Im(δ1) = Ker(δ1), and the diagram obtained, (65), would reduce to what is called in [6] an exact couple: A - A ] A· ; (78)...

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Book
15 Sep 2011
Abstract: Preface.- Introduction.- Glossary of Notation.- I First Concepts.- 1 Two Definitions of Lattices.- 2 How to Describe Lattices.- 3 Some Basic Concepts.- 4 Terms, Identities, and Inequalities.- 5 Free Lattices.- 6 Special Elements.- II Distributive Lattices.- 1 Characterization and Representation Theorems.- 2 Terms and Freeness.- 3 Congruence Relations.- 4 Boolean Algebras.- 5 Topological Representation.- 6 Pseudocomplementation.- III Congruences.- 1 Congruence Spreading.- 2 Distributive, Standard, and Neutral Elements.- 3 Distributive, Standard, and Neutral Ideals.- 4 Structure Theorems.- IV Lattice Constructions.- 1 Adding an Element.- 2 Gluing.- 3 Chopped Lattices.- 4 Constructing Lattices with Given Congruence Lattices.- 5 Boolean Triples.- V Modular and Semimodular Lattices.- 1 Modular Lattices.- 2 Semimodular Lattices.- 3 Geometric Lattices.- 4 Partition Lattices.- 5 Complemented Modular Lattices.- VI Varieties of Lattices.- 1 Characterizations of Varieties 397.- 2 The Lattice of Varieties of Lattices.- 3 Finding Equational Bases.- 4 The Amalgamation Property.- VII Free Products.- 1 Free Products of Lattices.- 2 The Structure of Free Lattices.- 3 Reduced Free Products.- 4 Hopfian Lattices.- Afterword.- Bibliography.

506 citations

Journal ArticleDOI

18 citations

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

• ...n the above proof fulﬁll the promise that “long” connections would be reduced to composites of “short” ones. The proofs of the next three lemmas continue this theme. Lemma 11 (Snake Lemma, [1, p.23], [4], [9, p.158], [11, p.50]). If in the commuting diagram at left below, both rows are exact, and we append a row of kernels and a row of cokernels to the vertical maps, as in the diagram at right, (16) ...

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