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The notions of quasi-embedding, quasi-equality, and quasi-isomorphism are generalized and their properties are studied.

$\newcommand{\poly}{_{\operatorname{poly}}^{\bullet}}\newcommand{\td}{(\operatorname{td}_{L/A}^{\nabla})^{\frac{1}{2}}}\newcommand{\cx}[1]{\operatorname{tot}\big(\Gamma(\Lambda^\bullet A^\vee)\otimes_R\mathcal{#1}\poly\big)}\newcommand{\cy}[1]{\mathbb{H}^\bullet_{\operatorname{CE}}(A,\mathcal{#1}\poly)}$Kontsevich's formality theorem states that there exists an $L_\infty$ quasi-isomorphism from the dgla $T\poly(M)$ of polyvector fields on a smooth manifold $M$ to the dgla $D\poly(M)$ of polydifferential operators on $M$, which extends the classical Hochschild--Kostant--Rosenberg map. In this paper, we extend Kontsevich's formality theorem to Lie pairs, a framework which includes a range of diverse geometric contexts such as complex manifolds, foliations, and $\mathfrak{g}$-manifolds. The spaces $\cx{T}$ and $\cx{D}$ associated with a Lie pair $(L,A)$ each carry an $L_\infty$ algebra structure canonical up to $L_\infty$ isomorphism. These two spaces serve as replacements for the spaces of polyvector fields and polydifferential operators, respectively. Their corresponding cohomology groups $\cy{T}$ and $\cy{D}$ admit canonical Gerstenhaber algebra structures. We establish the following formality theorem for Lie pairs: there exists an $L_\infty$ quasi isomorphism from $\cx{T}$ to $\cx{D}$ whose first Taylor coefficient is equal to $\operatorname{hkr}\circ\td$. Here $\td$ acts on $\cx{T}$ by contraction. Furthermore, we prove a Kontsevich--Duflo type theorem for Lie pairs: the Hochschild--Kostant--Rosenberg map twisted by the square root of the Todd class of the Lie pair $(L,A)$ is an isomorphism of Gerstenhaber algebras from $\cy{T}$ to $\cy{D}$. As applications, we establish formality theorems and Kontsevich--Duflo type theorems for complex manifolds, foliations, and $\mathfrak{g}$-manifolds. In the case of complex manifolds, we recover the Kontsevich--Duflo theorem of complex geometry., Comment: 55 pages, several typos corrected, some references added, some minor cosmetic changes in the presentation

Let O be a differential graded (possibly colored) operad defined over rationals. Let us assume that there exists a zig-zag of quasi-isomorphisms connecting O ⊗ K to its cohomology , where K is any field extension of Q . We show that for a large class of such dg operads, a formality quasi-isomorphism for O exists and can be constructed recursively. Every step of our recursive procedure involves a solution of a finite dimensional linear system and it requires no explicit knowledge about the zig-zag of quasi-isomorphisms connecting O ⊗ K to its cohomology.

It is believed arXiv:0808.2762, arXiv:math/9904055 that, among the coefficients entering Kontsevich's formality quasi-isomorphism arXiv:q-alg/9709040, there are irrational (possibly even transcendental) numbers. In this paper, we prove that a formality quasi-isomorphism for Hochschild cochains of a polynomial algebra over rationals can be constructed recursively. The proof that the proposed recursive algorithm works, is based on the existence of formality quasi-isomorphism over reals. However, the algorithm requires no explicit knowledge of the coefficients entering Kontsevich's construction. Although this algorithm completely bypasses Tamarkin's approach arXiv:math/0003052, arXiv:math/9803025, the construction is inspired by Proposition 5.8 from the classical paper (Algebra i Analiz, 1990) by V. Drinfeld.

This chapter proves the Quasi-Isomorphism Theorem modulo two technical lemmas, which will be dealt with in the next two chapters. Section 18.2 introduces the affine transformation TA from the Quasi-Isomorphism Theorem. Section 18.3 defines the graph grid GA = TA(Z2) and states the Grid Geometry Lemma, a result about the basic geometric properties of GA. Section 18.4 introduces the set Z* that appears in the Renormalization Theorem and states the main result about it, the Intertwining Lemma. Section 18.5 explains how the Orbit Equivalence Theorem sets up a canonical bijection between the nontrivial orbits of the plaid PET and the orbits of the graph PET. Section 18.6 reinterprets the orbit correspondence in terms of the plaid polygons and the arithmetic graph polygons. Everything is then put together to complete the proof of the Quasi-Isomorphism Theorem. Section 18.7 deduces the Projection Theorem (Theorem 0.2) from the Quasi-Isomorphism Theorem.

This monograph is devoted to the study of homological and homotopic properties of various algebraic structures. The problems considered and line of investigation taken fall under the general headings of non-Abelian homological algebra and simplicial methods in category theory, with applications to K-theory and cyclic homology.

In this chapter, we introduce cohomology groups by means of injective resolutions and show the duality theorem and spectral sequences. Readers who already know these concepts can skip this chapter.

We study the collection of homological equivalence relations on a fixed curve. We construct a moduli space for pairs consisting of a curve of genus g and a homological equivalence relation of degree n on the curve, and a classifying set for homological equivalence relations of degree n on a fixed curve, modulo automorphisms of the curve. We identify a special type of homological equivalence relations, and we characterize the special homological equivalence relations in terms of the existence of elliptic curves in the Jacobian of the curve.

[EN] In the paper one defines topological invariants of type degree for morphisms in the category Top₍₂₎ of topological pairs of spaces and continuous single valued maps, which admit homological n-spheres as target and arbitrary topological pairs of spaces as source. The different described degrees are acquired by means homological methods, and are a powerful tool in the root theory. Several existence theorems are obtained for equations with multivalued transformations., Project supported by CNEPRU No B00920110092 and PNR / ATRST No 8/u250/713. Laboratory M.M.E.R.E. U.M.1 Constantine and Russian FBR grant 11-01 -00382-a.

There is a well-known equivalence between the homotopy types of connected CW-spaces X with πnX = 0 for n ≠ 1, 2 and the quasi-isomorphism classes of crossed modules ∂: M → P [Mac Lane and Whitehead 50]. When the homotopy groups π1X and π2X are finite, one can represent the homotopy type of X by a crossed module in which M and P are finite groups. We define the order of such a crossed module to be |∂| = |M| × |P|, and the order of a quasi-isomorphism class of crossed modules to be the least order among all crossed modules in the class. We then define the order of a homotopy 2-type X to be the order of the corresponding quasi-isomorphism class of crossed modules. In this paper, we describe a computer implementation that inputs a finite crossed module of reasonably small order and returns a quasi-isomorphic crossed module of least order. Underlying the function is a catalog of all quasi-isomorphism classes of order m ≤ 127, m ≠ 32, 64, 81, 96, and a catalog of all isomorphism classes of crossed modules of or...

We prove that a certain homological epimorphism between two algebras induces a triangle equivalence between their singularity categories. Applying the result to a construction of matrix algebras, we describe the singularity categories of some non-Gorenstein algebras.

The goal of this paper is to generalize some of the existing toolkit of combinatorial algebraic topology in order to study the homology of abstract chain complexes. We define shellability of chain complexes in a similar way as for cell complexes and introduce the notion of regular chain complexes. In the case of chain complexes coming from simplicial complexes we recover the classical notions but, in contrast to the topological case, in the abstract setting shellings turn out to be a weaker homological invariant. In particular, we study special chain complexes, which are cones, and a class of regular chain complexes, which we call totally regular, for which we can obtain complete homological information if an additional condition is fulfilled.

In [Chu12], Church used representation stability to prove that the space of configurations of distinct unordered points in a closed manifold exhibit rational homological stability. A second proof was also given by Randal-Williams in [RW11] using transfer maps. We give a third proof of this fact using localization and rational homotopy theory. This gives new insight into the role that the rationals play in homological stability. Our methods also yield new information about stability for torsion in the homology of configuration spaces of points in a closed manifold., 14 pages

The classes of GC homological modules over commutative ring, where C is a semidualizing module, extend Holm and Jgensen's notions of C-Gorenstein homological modules to the non-Noetherian set- ting and generalize the classical classes of homological modules and the classes of Gorenstein homological modules within this setting. On the other hand, transfer of homological properties along ring homomorphisms is already a classical eld of study. Motivated by the ideas mentioned above, in this article we will investigate the transfer properties of C and GC homological dimension.

The isomorphism and quasi-isomorphism relations on the $p$-local torsion-free abelian groups of rank $n\geq3$ are incomparable with respect to Borel reducibility., This is a part of my doctoral thesis. 23 pages, 1 figure

For QTAG-modules the concept of quasi-isomorphism was defined by Mehdi [5]. Two QTAG-modules are quasi-isomorphic when they contain cobounded submodules. Here we prove that cobounded submodules inherit basic submodules. Then we impose some restrictions on quasi-isomorphism and find that quasi-isomorphic modules with these restrictions are very different. We also study the quasi-isomorphism of direct sum of uniserial modules and some quasi-isomorphic variants. It is natural to ask if for two quasi-isomorphic QTAG-modules M, M′ if M is a direct sum of countably generated uniserial modules, then M′ is also the direct sum of countably generated uniserial modules or not. In the last section we obtain the condition under which this is possible.

Let $\Lambda$ be a finite dimensional algebra over an algebraically closed field $k$. We will investigate homological properties of piecewise hereditary algebras $\Lambda$. In particular we give lower and upper bounds of the strong global dimension, show the behavior of the strong global under one point extensions and tilting. Moreover we show that the pieces of mod $\Lambda$ have Auslander-Reiten sequence., Oberwolfach Preprints;2009,11

In this appendix we recall some basic notions from homological algebra. We start by defining complexes of modules and the notion of homotopy between morphisms of complexes. This is in fact the cohomological shadow of the topological notion of homotopy (see the remark after Proposition 11.12). One obtains the homotopy category of modules, which is a fundamental notion in homological algebra. After a short interlude on diagram chases we introduce a central notion for the definition of cohomology: injective modules and K-injective complexes. Until then all notions were explained for modules over a ring, but in fact they make sense much more generally in arbitrary abelian categories. This is explained in the last section of this appendix.

Relative homological algebra was conceived by Auslander and Bridger [14, 15] and was formed by Enochs, Jenda, and Torrecillas (for example, [54, 55, 57]).

International audience; By nature, tree structures frequently present similarities between their sub-parts. Making use of this redundancy, different types of tree compression techniques have been designed in the literature to reduce the complexity of tree structures. A popular and efficient way to compress a tree consists of merging its isomorphic subtrees, which produces a directed acyclic graph (DAG) equivalent to the original tree. An important property of this method is that the compressed structure (i.e. the DAG), has the same height as the original tree, thus limiting partially the possibility of compression. In this paper we address the problem of further compressing this DAG in height. The difficulty is that compression must be carried out on substructures that are not exactly isomorphic as they are strictly nested within each-other. We thus introduced a notion of quasi-isomorphism between subtrees, that makes it possible to define similar patterns along any given path in a tree. We then proposed an algorithm to detect these patterns and to merge them, thus leading to compressed structures corresponding to DAGs augmented with return edges. In this way, redundant information is removed from the original tree in both width and height, thus achieving minimal structural compression. The complete compression algorithm is then illustrated on the compression of various plant-like structures.

Let be a commutative unital Banach algebra with infinite spectrum. Then by Helemskiĭ's global dimension theorem the global homological dimension of is strictly greater than one. This estimate has no analogue for abstract algebras or non-normable topological algebras. It is proved in the present paper that for every unital Banach algebra the global homological dimensions and the homological bidimensions of the Banach algebras and (assuming certain restrictions on ) are related by and . Thus, a partial extension of Helemskiĭ's theorem to tensor products is obtained. Bibliography: 28 titles.

We develop the homological algebra of coefficient systems on a group, in particular from the point of view of calculating higher limits. We show how various sequences of modules associated to a class of subgroups of a given group can be analysed by methods from homological algebra. We are particularly interested in when these sequences are exact, or, if not, when their homology is equal to the higher limits of the coefficient system.

A formality on a manifold M is a quasi isomorphism between the space of polyvector fields (Tpoly(M)) and the space of multidifferential operators (Dpoly(M)). In the case M=R d , such a mapping was explicitly built by Kontsevich, using graphs drawn in configuration spaces. Looking for such a construction step by step, we have to consider several cohomologies (Hochschild, Chevalley, and Harrison and Chevalley) for mappings defined on Tpoly. Restricting ourselves to the case of mappings defined with graphs, we determine the corresponding coboundary operators directly on the spaces of graphs. The last cohomology vanishes.

We prove that the chain operad of little disks is formal in characteristic zero, and discuss briefly the relation with Kontsevich formality in deformation quantization.

Our aim in this paper is to introduce a generalization of some notions in homological algebra. We define the concepts of chain U-complex, U-homology, chain (U, U')-map, chain (U, U')-homotopy and -functor. We also obtain some interesting results. We use these results to find a generalization of Lambek Lemma, Snake Lemma, Connecting Homomorphism and Exact Triangle.

Homological algebra is one of the most important tools if not the most important of all for the theory of representations of algebras. It gathers information of different points of views about its main subject, which is that of short exact sequences. First an equivalence relation is established on short exact sequences with fixed end terms. Then a left and a right multiplication with certain morphisms are defined and it is shown that this two multiplications are compatible. This enables to define a structure of vector space on the equivalence classes of short exact sequences with fixed end terms. Finally, for each module and each short exact sequence two so-called “long exact” sequences are associated. This makes it possible to compare the extension spaces for different end terms. Homological algebra prepares the basics for the second step for understanding the structure of module categories, which is called Auslander-Reiten theory and developed in the next chapter.

Let A be a C*-algebra. For each Banach A-bimodule X, the second continuous Hochschild cohomology group H2(A, X) of A with coefficients in X is defined (see [6]); there is a natural correspondence between the elements of this group and equivalence classes of singular, admissible extensions of A by X. Specifically this means that H2(A, X) ≠ {0} for some X if and only if there exists a Banach algebra B with Jacobson radical R such that R2 = {0}, R is complemented as a Banach space, and B/R ≅ A, but B has no strong Wedderburn decomposition; i.e., there is no closed subalgebra C of B such that B ≅ C © R. In turn this is equivalent to db A ≥ 2, where db A is the homological bidimension of A; i.e., the homological dimension of A#, the unitization of A, as an,A-bimodule [6, III. 5.15]. This paper is concerned with the following basic question, which was posed in [7].

This paper represents a general study of the (Yoneda) Ext-algebra A* of a finite dimensional K-algebra A. Our motivation lies in the problem of establishing conditions under which (i) the species of A* coincides with the dual species of A and (ii) the quasi-heredity of A (or A*) yields the quasi-heredity of A* (or A, respectively). These questions are closely related to the Kazhdan—Lusztig Theory as presented by [CPS2]. The main results include introducing the concept of a solid algebra and the relevant Theorem 4.5 as well as a rather complete description of the situation in the case of monomial algebras in Section 5.

Using a generalization of the definition of the projective cover of a module, a special type of surjective free resolution, known as the projective cover of a complex, may be defined. The projective cover is shown to be a direct summand of every surjective free resolution and to be the direct sum of the minimal free resolution and an exact complex. Necessary and sufficient conditions for the projective cover and minimal free resolution to be identical are discussed.

Introduction Semiexact categories Homological Categories Subquotients, Homology and Exact Couples Satellites Universal Constructions Applications to Algebraic Topology Homological Theories and Biuniversal Models Appendix A. Some Points of Category Theory.