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In this paper, we first construct the controlling algebras of embedding tensors and Lie-Leibniz triples, which turn out to be a graded Lie algebra and an $L_\infty$-algebra respectively. Then we introduce representations and cohomologies of embedding tensors and Lie-Leibniz triples, and show that there is a long exact sequence connecting various cohomologies. As applications, we classify infinitesimal deformations and central extensions using the second cohomology groups. Finally, we introduce the notion of a homotopy embedding tensor which will induce a Leibniz$_\infty$-algebra. We realize Kotov and Strobl's construction of an $L_\infty$-algebra from an embedding tensor, to a functor from the category of homotopy embedding tensors to that of Leibniz$_\infty$-algebras, and a functor further to that of $L_\infty$-algebras., 32 pages, comments are welcome

In the preceeding paper we constructed an infinite exact sequence a la Villamayor–Zelinsky for a symmetric finite tensor category. It consists of cohomology groups evaluated at three types of coefficients which repeat periodically. In the present paper we interpret the middle cohomology group in the second level of the sequence. We introduce the notion of coring categories and we obtain that the mentioned middle cohomology group is isomorphic to the relative group of Azumaya quasi coring categories. This result is a categorical generalization of the classical Crossed Product Theorem, which relates the relative Brauer group and the second Galois cohomology group with respect to a Galois field extension. We construct the colimit over symmetric finite tensor categories of the relative groups of Azumaya quasi coring categories and the full group of Azumaya quasi coring categories over vec. We prove that the latter two groups are isomorphic.

Let p>3 be a prime number and K a finite extension of Q_p. We consider a proper and smooth surface XK over K, with a semistable model X over the ring of integers OK of K. In this thesis, we give a criterion for the good reduction of XK for the case of K3 surfaces, in terms of the monodromy operator in the second De Rham cohomology group. We don't use trascendental methods nor p-adic Hodge Theory as in other works concerning this problem. Instead, we first get a p-adic version of the Clemens-Schmid exact sequence and use it to study the degree of nilpotency of the monodromy operator N on the log-crystalline cohomology group of the special fiber Xs of the semistable model X. By the work of Nakkajima, we can assume that Xs is a combinatorial K3 surface. Then, we prove that Xs is of type I iff N=0; Xs if of type II iff N is not zero and N^2=0; Xs is of type III iff N^2 is not zero. In particular, this implies that XK has good reduction if and only if the monodromy operator on its second De Rham cohomology is zero. Finally, we also give some ideas on how to address the same problem for the case of Enriques surfaces. In particular, we prove that we are reduced to the case of K3 surfaces.

Let B be the trivial extension of a Poisson algebra A by a left Poisson module M. Then there is a Poisson structure on B. We define a morphism of graded algebras $$\Theta : {\text {HP}}(B)\rightarrow {\text {HP}}(A)$$ from the Poisson cohomology ring of B to that of A. In the case that both A and M are finite dimensional as vector space, there is a long exact sequence computing the Poisson cohomology of B. We study the connecting homomorphism and then obtain a necessary and sufficient condition for each $$\Theta ^n$$ to be surjective. Finally, we study the kernel of $$\Theta ^1$$ explicitly.

Let $$k$$ be an algebraically closed field of characteristic $$0$$ . For a log curve $$X/k^{\times }$$ over the standard log point (Kato in Int J Math 11(2):215–232, 2000), we define (algebraically) a combinatorial monodromy operator on its log-de Rham cohomology group. The invariant part of this action has a cohomological description, it is the Du Bois cohomology of $$X$$ (Du Bois in Bull Soc Math Fr 109(1):41–81, 1981). This can be seen as an analogue of the invariant cycles exact sequence for a semistable family (as in the complex, etale and $$p$$ -adic settings). In the specific case in which $$k={\mathbb {C}}$$ and $$X$$ is the central fiber of a semistable degeneration over the complex disc, our construction recovers the topological monodromy and the classical local invariant cycles theorem. In particular, our description allows an explicit computation of the monodromy operator in this setting.

If K is a number field, arithmetic duality theorems for tori and complexes of tori over K are crucial to understand local-global principles for linear algebraic groups over K. When K is a global field of positive characteristic, we prove similar arithmetic duality theorems, including a Poitou-Tate exact sequence for Galois hypercohomology of complexes of tori. One of the main ingredients is Artin-Mazur-Milne duality theorem for fppf cohomology of finite flat commutative group schemes.

Let $\Gamma$ be a torsion-free arithmetic group acting on its associated global symmetric space $X$. Assume that $X$ is of non-compact type and let $\Gamma$ act on the geodesic boundary $\partial X$ of $X$. Via general constructions in KK-theory, we endow the K-groups of the arithmetic manifold $X/\Gamma$, of the reduced group C*-algebra of $\Gamma$ and of the boundary crossed product algebra associated to the action of $\Gamma$ on $\partial X$, with Hecke operators. The K-theory and K-homology groups of these C*-algebras are related by a Gysin six-term exact sequence. In the case when $\Gamma$ is a group of real hyperbolic isometries, we show that this Gysin sequence is Hecke equivariant. Finally, in the case when $\Gamma$ is a subgroup of a Bianchi group, we construct explicit Hecke-equivariant maps between the integral cohomology of $\Gamma$ and each of these K-groups. Our methods apply to torsion-free finite index subgroups of $PSL(2,\mathbb{Z})$ as well. These results are achieved in the context of unbounded Fredholm modules, shedding light on noncommutative geometric aspects of the purely infinite boundary crossed product algebra., Comment: 50 pages

By using a representation of a Lie algebra on the second Hochschild cohomology group, we construct an obstruction class to extensibility of derivations and a short exact sequence of Wells type for an abelian extension of an associative algebra.

In this paper we study the low dimensional cohomology groups of Hom-Lie algebras and their relation with derivations, abelian extensions and crossed modules. On one hand, we introduce the notion of α -abelian extensions and we obtain a five term exact sequence in cohomology. On the other hand, we study crossed modules of Hom-Lie algebras showing their equivalence with cat 1 -Hom-Lie algebras, and we introduce α -crossed modules to have a better understanding of the third cohomology group.

We adapt and generalize results of Loganathan on the cohomology of inverse semigroups to the cohomology of ordered groupoids. We then derive a five-term exact sequence in cohomology from an extension of ordered groupoids, and show that this sequence leads to a classification of extensions by a second cohomology group. Our methods use structural ideas in cohomology as far as possible, rather than computation with cocycles.

We define a comonad cohomology of track categories and we show it is linked by a long exact sequence to its Dwyer-Kan-Smith cohomology . Under mild hypothesis on the track category, we show that its comonad cohomology coincides, up to dimension shift, with its Dwyer-Kan-Smith cohomology, therefore obtaining an algebraic formulation of the latter. We also specialize our results to the case where the track category is a $2$-groupoid., Comment: The paper has been re-written more concisely to improve readability. To appear in Journal of Homotopy and Related Structures

In earlier work the authors determined the Brauer kernel of extensions of degree p in characteristic p > 2 where the Galois group is a semidirect product of order ps for s | ( p − 1 ) . This result is extended here and tools are developed to compute the cohomological kernels H p m n + 1 ( E m / F ) for all n ≥ 0 where [ E m : F ] = p m and the Galois closure is a semidirect product of cyclic groups order p m and s where s | ( p − 1 ) . A six-term exact sequence describing the K-theory and cohomology of the extension is obtained. As an application it is shown that any F-division p-algebra of index p m split in E m is cyclic; a characteristic p analogue of a result of Vishne. The proofs use the de Rham Witt complex and Izhboldin groups, extending techniques developed earlier for the study of degree 4 extensions in characteristic two. The paper also provides background on the de Rham Witt Complex and Izhboldin groups difficult to track down in the literature.

The main purpose of this note is to construct a divisorial version of the Chase-Harrison- Rosenberg exact sequence, thus reducing the calculation of involutive invariants of a Krull domain to that of certain Amitsur and Galois cohomology groups.

We describe how the Hochschild (co)homology of a bound quiver algebra changes when adding or deleting arrows to the quiver. The main tools are relative Hochschild (co)homology, the Jacobi-Zariski long exact sequence obtained by A. Kaygun and a one step relative projective resolution of a tensor algebra., Comment: Typos corrected To appear in Proc. Amer. Math. Soc

For a particular class of pseudo manifolds, we show that the intersection cohomology groups for any perversity may be naturally represented by extended weighted $L^2$ harmonic forms for a complete metric on the regular stratum with respect to some weight determined by the perversity. Extended weighted $L^2$ harmonic forms are harmonic forms that are almost in the given weighted $L^2$ space for the metric in question, but not quite. This result is akin to the representation of absolute and relative cohomology groups for a manifold with boundary by extended harmonic forms on the associated manifold with cylindrical ends. As in that setting, in the unweighted $L^2$ case, the boundary values of the extended harmonic forms define a Lagrangian splitting of the boundary space in the long exact sequence relating upper and lower middle perversity intersection cohomology groups., 26 pages

In this paper the rigid cohomology of Drinfeld's upper half space over a finite field is computed in two ways. The first method proceeds by computation of the rigid cohomology of the complement of Drinfeld's upper half space in the ambient projective space and then use of the associated long exact sequence for rigid cohomology with proper supports. The second method proceeds by direct computation of rigid cohomology as a direct limit of de Rham cohomologies of a family of strict open neighborhoods of the tube of Drinfeld's upper half space in the ambient rigid-analytic projective space. The resulting cohomology formula has been known since 2007, when Grosse-Kloenne proved that it is the same as the one obtained from l-adic cohomology., 30 pages, v2, final version as appeared in Math. Z (2017)

In this article we study various forms of $\ell$-independence (including the case $\ell =p$) for the cohomology and fundamental groups of varieties over finite fields and equicharacteristic local fields. Our first result is a strong form of $\ell$-independence for the unipotent fundamental group of smooth and projective varieties over finite fields. By then proving a certain ‘spreading out’ result we are able to deduce a much weaker form of $\ell$-independence for unipotent fundamental groups over equicharacteristic local fields, at least in the semistable case. In a similar vein, we can also use this to deduce $\ell$-independence results for the cohomology of smooth and proper varieties over equicharacteristic local fields from the well-known results on $\ell$-independence for smooth and proper varieties over finite fields. As another consequence of this ‘spreading out’ result we are able to deduce the existence of a Clemens–Schmid exact sequence for formal semistable families. Finally, by deforming to characteristic $p$, we show a similar weak version of $\ell$-independence for the unipotent fundamental group of a semistable curve in mixed characteristic.

To a Legendrian knot, one can associate an $\mathcal{A}_{\infty}$ category, the augmentation category. An exact Lagrangian cobordism between two Legendrian knots gives a functor of the augmentation categories of the two knots. We study the functor and establish a long exact sequence relating the corresponding cohomology of morphisms of the two ends. As applications, we prove that the functor between augmentation categories is injective on the level of equivalence classes of objects and find new obstructions to the existence of exact Lagrangian cobordisms in terms of linearized contact homology and ruling polynomials., We added Theorem 5.13 and made the results in Theorem 1.6 and Corollary 1.7 more general than the previous version. 46 pages, 20 figures

For any Moore spectrum M and any homology theory \({{\mathcal {H}}}_*\), we associate a homology theory \({{\mathcal {H}}}_*^M\) which is related to \({{\mathcal {H}}}_*\) by a universal coefficient exact sequence of classical type. On the other hand the category of Moore spectra is not the category of \({\mathbb {Z}}\)-modules, but it can be identified to a full subcategory of an abelian category \({{\mathscr {D}}}\). We prove that \({{\mathcal {H}}}_*\) can be lifted to a homology theory \(\widehat{{\mathcal {H}}}_*\) with values in \({{\mathscr {D}}}\) and we give a new universal coefficient exact sequence relating \({{\mathcal {H}}}_*^M\) and \(\widehat{{\mathcal {H}}}_*\) which is in general more precise than the classical one. We prove also a similar result for cohomology theories and we illustrate its convenience by computing the real K-theory of Moore spaces.

Let $A$ be a Koszul (or more generally, $N$-Koszul) Calabi-Yau algebra. Inspired by the works of Kontsevich, Ginzburg and Van den Bergh, we show that there is a derived non-commutative Poisson structure on $A$, which induces a graded Lie algebra structure on the cyclic homology of $A$; moreover, we show that the Hochschild homology of $A$ is a Lie module over the cyclic homology and the Connes long exact sequence is in fact a sequence of Lie modules. Finally, we show that the Leibniz-Loday bracket associated to the derived non-commutative Poisson structure on $A$ is naturally mapped to the Gerstenhaber bracket on the Hochschild cohomology of its Koszul dual algebra and hence on that of $A$ itself. Relations with some other brackets in literature are also discussed and several examples are given in detail.

Given a fusion system $\mathcal{F}$ defined on a $p$-group $S$, there exist infinite group models, constructed by Leary and Stancu, and Robinson, that realize $\mathcal{F}$. We study these models when $\mathcal{F}$ is a fusion system of a finite group $G$ and prove a theorem which relates the cohomology of an infinite group model $\pi$ to the cohomology of the group $G$. We show that for the groups $GL(n,2)$, where $n\geq 5$, the cohomology of the infinite group obtained using the Robinson model is different than the cohomology of the fusion system. We also discuss the signalizer functors $P\to \Theta(P)$ for infinite group models and obtain a long exact sequence for calculating the cohomology of a centric linking system with twisted coefficients., Comment: 23 pages

We provide a formula for the change of the dimension of the first Hoch\-schild cohomology vector space of bound quiver algebras when adding new arrows. For this purpose we show that there exists a short exact sequence which relates the first cohomology vector spaces of the algebras to the first relative cohomology. Moreover, we show that the first Hochschild homologies are isomorphic when adding new arrows., Comment: 17 pages - small changes made

We compute the mod $2$ cohomology groups of real Lagrangians in Calabi-Yau threefolds using well-behaved torus fibrations constructed by Gross. To do this we study a long exact sequence introduced by Casta\~{n}o-Bernard and Matessi, which relates the cohomology of the Lagrangians to the cohomology of the Calabi-Yau. We show that the connecting homomorphism in this sequence is given by the map squaring divisor classes in the mirror Calabi-Yau., Comment: 31 pages, 8 figures. Improved exposition of Appendix A and corrected Example 4.11. Minor adjustments throughout

The goal of this thesis is to classify all extensions where the kernel has order p^s and the cokernel has order p^t, p is a prime, and 1 ≤ s,t ≤ 2. We determine (up to weak congruence) the different combinations of kernel, cokernel and operators, and for each case, calculate the second cohomology group. By comparing resolutions, we get an explicit correspondence between the second cohomology group and the group of congruence classes of extensions. Using this construction, we determine (up to congruence) the extensions for the different combinations.

We generalise the notion of a Barge-Diamond complex, in the one-dimensional case, to a mixed system of tiling substitutions. This gives a way of describing the associated tiling space as an inverse limit of Barge-Diamond complexes. We give an effective method for calculating the \v{C}ech cohomology of the tiling space via an exact sequence relating the associated sequence of substitution matrices and certain subcomplexes appearing in the approximants. As an application, we show that there exists a system of three substitutions on two letters which exhibit an uncountable collection of minimal tiling spaces with distinct isomorphism classes of \v{C}ech cohomology., Comment: 22 pages

By comparing Deligne complex and Aeppli–Bott-Chern complex, we construct a differential cohomology \(\widehat{H}^*(X, *, *)\) that plays the role of Harvey–Lawson spark group \(\widehat{H}^*(X, *)\), and a cohomology \(H^*_{\mathrm{ABC}}(X; \mathbb Z(*, *))\) that plays the role of Deligne cohomology \(H^*_{\mathcal {D}}(X; \mathbb Z(*))\) for every complex manifold X. They fit in the short exact sequence $$\begin{aligned} 0\rightarrow H^{k+1}_{\mathrm{ABC}}(X; \mathbb Z(p, q)) \rightarrow \widehat{H}^k(X, p, q) \overset{\delta _1}{\rightarrow } Z^{k+1}_I(X, p, q) \rightarrow 0 \end{aligned}$$ and \(\widehat{H}^{\bullet }(X, \bullet , \bullet )\) possess ring structure and refined Chern classes, acted by the complex conjugation, and if some primitive cohomology groups of X vanish, there is a Lefschetz isomorphism. Furthermore, the ring structure of \(H^{\bullet }_{\mathrm{ABC}}(X; \mathbb Z(\bullet , \bullet ))\) inherited from \(\widehat{H}^{\bullet }(X, \bullet , \bullet )\) is compatible with the one of the analytic Deligne cohomology \(H^{\bullet }(X; \mathbb Z(\bullet ))\). We compute \(\widehat{H}^*(X, *, *)\) for X the Iwasawa manifold and its small deformations and get a refinement of the classification given by Nakamura.

For any hyperelliptic curve X, we give an explicit basis of the first de-Rham cohomology of X in terms of ��ech cohomology. We use this to produce a family of curves in characteristic p>2 for which the Hodge-de-Rham short exact sequence does not split equivariantly; this generalises a result of Hortsch. Further, we use our basis to show that the hyperelliptic involution acts on the first de-Rham cohomology by multiplication by -1, i.e., acts as the identity when p=2., 23 pages; v2: references added; v3: comments of referees addressed, to appear in Res. Number Theory

We explicitly calculate an arithmetic adelic quotient group for a locally free sheaf on an arithmetic surface when the fiber over the infinite point of the base is taken into account. The calculations are presented via a short exact sequence. We relate the last term of this short exact sequence with the projective limit of groups which are finite direct products of copies of one-dimensional real torus and are connected with first cohomology groups of locally free sheaves on the arithmetic surface., 21 pages; minor changes; to appear in Izvestiya: Mathematics

Stable cohomology is a generalization of Tate cohomology to associative rings, first defined by Pierre Vogel. For a commutative local ring R with residue field k, stable cohomology modules Ext ˆ R n ( k , k ) , defined for n ∈ Z , have been studied by Avramov and Veliche. Stable cohomology carries a structure of Z -graded k-algebra. One of the main goals of this paper is to prove that, for a class of Gorenstein rings, this algebra is a trivial extension of absolute cohomology Ext R ( k , k ) and a shift of Hom k ( Ext R ( k , k ) , k ) . We use this information to characterize the rings R for which stable cohomology is graded-commutative. Stable cohomology is connected through an exact sequence to bounded cohomology. We use this connection to understand the algebra structure of Ext ˆ R ( k , k ) by investigating the structure of bounded cohomology Ext ‾ R ( k , k ) as a graded Ext R ( k , k ) -bimodule.

Some properties of the set of cocyclic element preserving maps are studied for cocyclic elements in cohomology groups defined by Haslam and Yoon. Some subsets of self-homotopy sets and their monoid structures are defined making use of the cocyclic element preserving self maps. Cocyclic element preserving pair maps are examined to obtain further results. Fibration sequences are studied as examples, and non-trivial examples are obtained by making use of Nomura's exact sequence involving groups of self-homotopy equivalences.

We investigate the relative and Tate cohomology theories with respect to Ding modules and complexes, consider their relations with classical and Gorenstein cohomology theories. As an application, the Avramov-Martsinkovsky type exact sequence of Ding modules is obtained.

The aim of this paper is to study the consequences on the non– abelian 2–cohomology of the vanishing on the elementary obstruction described by Colliot-Thelene and Sansuc. Let k be an arbitrary field, let X be an algebraic smooth, proper and geometrically integral k-variety and finally let G be an k–scheme of reductive groups. We show that, when the elementary obstruction ob(X) associated to X vanishes, i.e. when the exact sequence 1 −→ k −→ k(X)∗ −→ k(X)∗/k −→ 1 is split, then the map H1 et(X,G) −→ H0(k,H1 et(X,G)) is surjective, in other words, if k is the field of moduli of a X-fibration of group G, (X = X⊗k k, G = G⊗k k), the field k is then also its field of definition. To do so, we reduce the 2–cohomology of X with value in G to the 2–cohomology of X with value in a maximal X–torus of G.

Let $$h^{\bullet }$$ be a cohomology theory and $${\hat{h}}^{\bullet }$$ the natural differential refinement, as defined by Hopkins and Singer. We consider the possible definitions of the relative differential cohomology groups, generalizing the analogous picture for the Deligne cohomology, and we show the corresponding long exact sequence in each case.

Classifying obstructions to the problem of finding extensions between two fixed modules goes back at least to L. Illusie's thesis. Our approach, following in the footsteps of J. Wise, is to introduce an analogous Grothendieck Topology on the category $A$-mod of modules over a fixed ring $A$ in a topos $E$. The problem of finding extensions becomes a banded gerbe and furnishes a cohomology class on the site $A$-mod. We compare our obstruction and that coming from Illusie's work, giving another construction of the exact sequence Illusie used to obtain his obstruction. Our work circumvents the cotangent complex entirely and answers a question posed by llusie., Comment: 29 pages; comments welcome

The mapping class group $\Gamma ^k(N_g)$ of a non-orientable surface with punctures is studied via classical homotopy theory of configuration spaces. In particular, we obtain a non-orientable version of the Birman exact sequence. In the case of ${\mathbb{R}} \textrm{P}^2$, we analyze the Serre spectral sequence of a fiber bundle $F_k({\mathbb{R}}{\textrm{P}}^{2}) / \Sigma _k \to X_k \to BSO(3)$ where $X_k$ is a $K(\Gamma ^k({\mathbb{R}} \textrm{P}^2),1)$ and $F_k({\mathbb{R}}{\textrm{P}}^{2}) / \Sigma _k$ denotes the configuration space of unordered $k$-tuples of distinct points in ${\mathbb{R}} \textrm{P}^2$. As a consequence, we express the mod-2 cohomology of $\Gamma ^k({\mathbb{R}} \textrm{P}^2)$ in terms of that of $F_k({\mathbb{R}}{\textrm{P}}^{2}) / \Sigma _k$.

Let \(D\) be a Dedekind scheme with the characteristic of all residue fields not equal to 2. To every tame cover \(C\to D\) with only odd ramification we associate a second Stiefel-Whitney class in the second cohomology with mod 2 coefficients of a certain tame orbicurve \([D]\) associated to \(D\). This class is then related to the pull-back of the second Stiefel-Whitney class of the push-forward of the line bundle of half of the ramification divisor. This shows (indirectly) that our Stiefel-Whitney class is the pull-back of a sum of cohomology classes considered by Esnault, Kahn and Viehweg in ‘Coverings with odd ramification and Stiefel-Whitney classes’. Perhaps more importantly, in the case of a proper and smooth curve over an algebraically closed field, our Stiefel-Whitney class is shown to be the pull-back of an invariant considered by Serre in ‘Revetements a ramification impaire et theta-caracteristiques’, and in this case our arguments give a new proof of the main result of that article.

It is well known that the validity of the so called Lenard–Magri scheme of integrability of a bi-Hamiltonian PDE can be established if one has some precise information on the corresponding 1st variational Poisson cohomology for one of the two Hamiltonian operators. In the first part of the paper we explain how to introduce various cohomology complexes, including Lie superalgebra and Poisson cohomology complexes, and basic and reduced Lie conformal algebra and Poisson vertex algebra cohomology complexes, by making use of the corresponding universal Lie superalgebra or Lie conformal superalgebra. The most relevant are certain subcomplexes of the basic and reduced Poisson vertex algebra cohomology complexes, which we identify (non-canonically) with the generalized de Rham complex and the generalized variational complex. In the second part of the paper we compute the cohomology of the generalized de Rham complex, and, via a detailed study of the long exact sequence, we compute the cohomology of the generalized variational complex for any quasiconstant coefficient Hamiltonian operator with invertible leading coefficient. For the latter we use some differential linear algebra developed in the Appendix., National Science Foundation (U.S.), ERC (Advanced Grant)

In this monograph, we extend S. Schwede's exact sequence interpretation of the Gerstenhaber bracket in Hochschild cohomology to certain exact and monoidal categories. Therefore we establish an explicit description of an isomorphism by A. Neeman and V. Retakh, which links $\mathrm{Ext}$-groups with fundamental groups of categories of extensions and relies on expressing the fundamental group of a (small) category by means of the associated Quillen groupoid. As a main result, we show that our construction behaves well with respect to structure preserving functors between exact monoidal categories. We use our main result to conclude, that both the Lie bracket and the squaring map in Hochschild cohomology are invariants under Morita equivalence. For quasi-triangular bialgebras, we further determine a significant part of the Lie bracket's kernel, and thereby prove a conjecture by L. Menichi. Along the way, we introduce $n$-extension closed and entirely extension closed subcategories of abelian categories, and study some of their properties., Comment: Modified version of author's PhD thesis (Bielefeld University, December 2013). 159 pages. --- Final version, to appear in "Memoirs of the American Mathematical Society". Corrected a mistake in Section 6 (the main results are not affected) and made minor changes according to the suggestions of the referees

We state the definition and general properties of sheaves from presheaves to espaces etales. We introduce thoroughly the sheaves we encounter in the next chapters. These are mostly sheaves on the circle S1 of directions about 0 in \( {\Bbb C}\). We reformulate the Borel-Ritt theorem in terms of sheaves. In the second part of the chapter we introduce the Cech cohomology for sheaves of groups, both in the abelian and the non abelian case; and we reformulate the Borel-Ritt and the Cauchy-Heine theorems in the cohomological language. For more precisions on sheaves and cohomology we refer to the classical literature (cf. [God58], [Ten75], [Ive86] for instance).

This chapter develops the rudiments of the subject of homological algebra, which is an abstraction of various ideas concerning manipulations with homology and cohomology. Sections 1–7 work in the context of good categories of modules for a ring, and Section 8 extends the discussion to abelian categories. Section 1 gives a historical overview, defines the good categories and additive functors used in most of the chapter, and gives a more detailed outline than appears in this abstract. Section 2 introduces some notions that recur throughout the chapter—complexes, chain maps, homotopies, induced maps on homology and cohomology, exact sequences, and additive functors. Additive functors that are exact or left exact or right exact play a special role in the theory. Section 3 contains the first main theorem, saying that a short exact sequence of chain or cochain complexes leads to a long exact sequence in homology or cohomology. This theorem sees repeated use throughout the chapter. Its proof is based on the Snake Lemma, which associates a connecting homomorphism to a certain kind of diagram of modules and maps and which establishes the exactness of a certain 6-term sequence of modules and maps. The section concludes with proofs of the crucial fact that the Snake Lemma and the first main theorem are functorial. Section 4 introduces projectives and injectives and proves the second main theorem, which concerns extensions of partial chain and cochain maps and also construction of homotopies for them when the complexes in question satisfy appropriate hypotheses concerning exactness and the presence of projectives or injectives. The notion of a resolution is defined in this section, and the section concludes with a discussion of split exact sequences. Section 5 introduces derived functors, which are the basic mathematical tool that takes advantage of the theory of homological algebra. Derived functors of all integer orders $\geq0$ are defined for any left exact or right exact additive functor when enough projectives or injectives are present, and they generalize homology and cohomology functors in topology, group theory, and Lie algebra theory. Section 6 implements the two theorems of Section 3 in the situation in which a left exact or right exact additive functor is applied to an exact sequence. The result is a long exact sequence of derived functor modules. It is proved that the passage from short exact sequences to long exact sequences of derived functor modules is functorial. Section 7 studies the derived functors of $\mathrm{Hom}$ and tensor product in each variable. These are called $\mathrm{Ext}$ and $\mathrm{Tor}$, and the theorem is that one obtains the same result by using the derived functor mechanism in the first variable as by using the derived functor mechanism in the second variable. Section 8 discusses the generalization of the preceding sections to abelian categories, which are abstract categories satisfying some strong axioms about the structure of morphisms and the presence of kernels and cokernels. Some generalization is needed because the theory for good categories is insufficient for the theory for sheaves, which is an essential tool in the theory of several complex variables and in algebraic geometry. Two-thirds of the section concerns the foundations, which involve unfamiliar manipulations that need to be internalized. The remaining one-third introduces an artificial definition of “member” for each object and shows that familiar manipulations with members can be used to verify equality of morphisms, commutativity of square diagrams, and exactness of sequences of objects and morphisms. The consequence is that general results for categories of modules in homological algebra requiring such verifications can readily be translated into results for general abelian categories. The method with members, however, does not provide for constructions of morphisms member by member. Thus the construction of the connecting homomorphism in the Snake Lemma needs a new proof, and that is given in a concluding example.

We prove that any abelian surface admits a rank 2 Ulrich bundle., Small inaccuracy corrected

The problem of the representation of an action of a Lie–Rinehart algebra on a Lie 𝖠-algebra by means of a homomorphism of Lie–Rinehart algebras is studied. An eight-term exact sequence associated to an epimorphism of Lie–Rinehart algebras for the cohomology of Lie–Rinehart algebras developed by Casas, Ladra and Pirashvili is obtained. This sequence is applied to study the obstruction theory of Lie–Rinehart algebra extensions.

For odd dimensional Poincare-Einstein manifolds $(X^{n+1},g)$, we study the set of harmonic $k$-forms (for $k

We compare three different ways of defining group cohomology with coefficients in a crossed module: (1) explicit approach via cocycles; (2) geometric approach via gerbes; (3) group theoretic approach via butterflies. We discuss the case where the crossed module is braided and the case where the braiding is symmetric. We prove the functoriality of the cohomologies with respect to weak morphisms of crossed modules and also prove the ‘long’ exact cohomology sequence associated to a short exact sequence of crossed modules and weak morphisms.

Let □ = l 1 (ℬ) be the semigroup algebra of ℬ, the bicyclic semigroup. We give a resolution of l ∞ (ℬ) which simplifies the computation of the cohomology of l 1 (ℬ) dual bimodules. We apply this to the dual module l ∞ (ℬ) and show that the simplicial cohomology groups ℋ n (□, □′) vanish for n ≥ 2. Using the Connes–Tzygan exact sequence, these results are used to show that the cyclic cohomology groups ℋ□ n (□, □′) vanish when n is odd and are one-dimensional when n is even (n ≥ 2).

The Rabinowitz–Floer homology of a Liouville domain W is the Floer homology of the Rabinowitz free period Hamiltonian action functional associated to a Hamiltonian whose zero energy level is the boundary of W. This invariant has been introduced by K. Cieliebak and U. Frauenfelder and has already found several applications in symplectic topology and in Hamiltonian dynamics. Together with A. Oancea, the same authors have recently computed the Rabinowitz–Floer homology of the cotangent disk bundle D* M of a closed Riemannian manifold M, by means of an exact sequence relating the Rabinowitz–Floer homology of D* M with its symplectic homology and cohomology. The first aim of this paper is to present a chain level construction of this exact sequence. In fact, we show that this sequence is the long homology sequence induced by a short exact sequence of chain complexes, which involves the Morse chain complex and the Morse differential complex of the energy functional for closed geodesics on M. These chain maps are defined by considering spaces of solutions of the Rabinowitz–Floer equation on half-cylinders, with suitable boundary conditions which couple them with the negative gradient flow of the geodesic energy functional. The second aim is to generalize this construction to the case of a fiberwise uniformly convex compact subset W of T* M whose interior part contains a Lagrangian graph. Equivalently, W is the energy sublevel associated to an arbitrary Tonelli Lagrangian L on TM and to any energy level which is larger than the strict Mañé critical value of L. In this case, the energy functional for closed geodesics is replaced by the free period Lagrangian action functional associated to a suitable calibration of L. An important issue in our analysis is to extend the uniform estimates for the solutions of the Rabinowitz–Floer equation — both on cylinders and on half-cylinders — to Hamiltonians which have quadratic growth in the momenta. These uniform estimates are obtained by the Aleksandrov integral version of the maximum principle. In the case of half-cylinders, they are obtained by an Aleksandrov-type maximum principle with Neumann conditions on part of the boundary.

Suppose B is an algebra with a stratifying ideal B e B generated by an idempotent e . We will establish long exact sequences relating the Hochschild cohomology groups of the three algebras B , B / B e B and e B e . This provides a common generalization of various known results, all of which extend Happel’s long exact sequence for one-point extensions. Applying one of these sequences to Hochschild cohomology algebras modulo the ideal generated by homogeneous nilpotent elements shows, in some cases, that these algebras are finitely generated.