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LetAbe a (not necessarily commutative) monoid object in an abelian symmetric monoidal category (C, ⊗,1) satisfying certain conditions. In this paper, we continue our study of the localizationMSof anyA-moduleMwith respect to a subsetS ⊆ HomA−Bimod(A,A) that is closed under composition. In particular, we prove the following theorem: ifPis anA-bimodule such thatPis symmetric as a bimodule over the centerZ(A) ofA, we have isomorphismsHH*(A,P)S ≅ HH*(A,PS) ≅ HH*(AS,PS) of Hochschild homology groups. [ABSTRACT FROM AUTHOR]

In this paper we prove a version of curved Koszul duality for $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}\mathbb{Z}/2\mathbb{Z}$-graded curved coalgebras and their cobar differential graded algebras. A curved version of the homological perturbation lemma is also obtained as a useful technical tool for studying curved (co)algebras and precomplexes. The results of Koszul duality can be applied to study the category of matrix factorizations $\mathsf{MF}(R,W)$. We show how Dyckerhoff’s generating results fit into the framework of curved Koszul duality theory. This enables us to clarify the relationship between the Borel–Moore Hochschild homology of curved (co)algebras and the ordinary Hochschild homology of the category $\mathsf{MF}(R,W)$. Similar results are also obtained in the orbifold case and in the graded case. [ABSTRACT FROM AUTHOR]

We prove an analog of the holomorphic Lefschetz formula for endofunctors of smooth compact dg-categories. We deduce from it a generalization of the Lefschetz formula of Lunts (J Algebra 356:230-256, ) that takes the form of a reciprocity law for a pair of commuting endofunctors. As an application, we prove a version of Lefschetz formula proposed by Frenkel and Ngô (Bull Math Sci 1(1):129-199, ). Also, we compute explicitly the ingredients of the holomorphic Lefschetz formula for the dg-category of matrix factorizations of an isolated singularity $${\varvec{w}}$$ . We apply this formula to get some restrictions on the Betti numbers of a $${\mathbb Z}/2$$ -equivariant module over $$k[[x_1,\ldots ,x_n]]/({\varvec{w}})$$ in the case when $${\varvec{w}}(-x)={\varvec{w}}(x)$$ . [ABSTRACT FROM AUTHOR]

Abstract: Fix a noetherian scheme S. For any flat map of separated essentially-finite-type perfect S-schemes we define a canonical derived-category map , the fundamental class of f, where is the (pre-)Hochschild complex of an S-scheme Z and is the twisted inverse image coming from Grothendieck duality theory. When and f is essentially smooth of relative dimension n, this gives an isomorphism . We focus mainly on transitivity of vis-à-vis compositions , and on the compatibility of with flat base change. These properties imply that orients the flat maps in the bivariant theory of part I [1], compatibly with essentially étale base change. Furthermore, leads to a dual oriented bivariant theory, whose homology is the classical Hochschild homology of flat S-schemes. When , is used to define a duality map , an isomorphism if f is essentially smooth. These results apply in particular to flat essentially-finite-type maps of noetherian rings. [Copyright &y& Elsevier]

From any algebra A defined by a single non-degenerate homogeneous quadratic relation f, we prove that the quadratic algebra B defined by the potential w = fz is 3-Calabi-Yau. The algebra B can be viewed as a 3-Calabi-Yau completion of Keller. The algebras A and B are both Koszul. The classification of the algebras B in three generators, i.e., when A has two generators, leads to three types of algebras. The second type (the most interesting one) is viewed as a deformation of a Poisson algebra S whose Poisson bracket is non-diagonalizable quadratic. Although the potential of S has non-isolated singularities, the homology of S is computed. Next the Hochschild homology of B is obtained. [ABSTRACT FROM AUTHOR]

We give the dimension formula of the cyclic homology of truncated quiver algebras over a field of positive characteristic. This is done by using a mixed complex due to Cibils. We consider a spectral sequence associated with the Cibils' mixed complex, which converges to the cyclic homology. The -term of this spectral sequence, that is the Hochschild homology, is calculated by Sköldberg. In this paper, by means of chain maps between the projective resolution constructed by Sköldberg and one by Cibils, we calculate the -term, and we have the dimension formula of the cyclic homology. [ABSTRACT FROM AUTHOR]

We study the interaction between various analytification functors, and a class of morphisms of rings, called homotopy epimorphisms. An analytification functor assigns to a simplicial commutative algebra over a ring R , along with a choice of Banach structure on R , a commutative monoid in the symmetric monoidal model category of simplicial ind-Banach R -modules. We show that several analytifications relevant to analytic geometry - such as Tate, overconvergent, Stein analytification, and formal completion - are homotopy epimorphisms. Another class of examples of homotopy epimorphisms arises from Weierstrass, Laurent and rational localizations in derived analytic geometry. As applications of this result, we prove that Hochschild homology and the cotangent complex are computable for analytic rings, and the computation relies only on known computations of Hochschild homology for polynomial rings. We show that in various senses, Hochschild homology as we define it commutes with localizations, analytifications and completions. [ABSTRACT FROM AUTHOR]

Abstract: In this paper we construct “structural” pre-braidings characterizing different algebraic structures: a rack, an associative algebra, a Leibniz algebra and their representations. Some of these pre-braidings seem original. On the other hand, we propose a general homology theory for pre-braided vector spaces and braided modules, based on the quantum co-shuffle comultiplication. Applied to the structural pre-braidings above, it gives a generalization and a unification of many known homology theories. All the constructions are categorified, resulting in particular in their super- and co-versions. Lodayʼs hyper-boundaries, as well as certain homology operations are efficiently treated using the “shuffle” tools. [Copyright &y& Elsevier]

Abstract: We construct an exceptional sequence of length 11 on the classical Godeaux surface which is the -quotient of the Fermat quintic surface in . This is the maximal possible length of such a sequence on this surface which has Grothendieck group . In particular, the result answers Kuznetsov’s Nonvanishing Conjecture, which concerns Hochschild homology of an admissible subcategory, in the negative. The sequence carries a symmetry when interpreted in terms of the root lattice of the simple Lie algebra of type . We also produce explicit nonzero objects in the (right) orthogonal to the exceptional sequence. [Copyright &y& Elsevier]

In classical differential topology and geometry, the double Čech-de Rham complex for a good covering U of M corresponds to the single de Rham complex. When passing to non-commutative geometry on smooth manifolds (i. e. to noncommutative geometry of the algebra C∞ (M) of smooth functions on M), the single Connes complex is the counterpart of the single de Rham complex. We can ask: which bicomplex corresponds to the single Connes complex in noncommutative geometry of smooth manifolds? We give an answer to this question defining the Cyclic Čech-Hochschild bicomplex for a good covering of a smooth manifold. We pose fundamental problems on the total homology of this bicomplex. [ABSTRACT FROM AUTHOR]

Abstract: Stable equivalences of Morita type preserve many interesting properties and are proved to be the appropriate concept for studying equivalences between stable categories. Recently the singularity category attained much attraction and Xiao-Wu Chen and Long-Gang Sun gave an appropriate definition of singular equivalence of Morita type. We shall show that under some conditions singular equivalences of Morita type have some biadjoint functor properties and preserve positive degree Hochschild homology. [Copyright &y& Elsevier]

Abstract: Let be an affine Hecke algebra with complex, possibly unequal parameters q, which are not roots of unity. We compute the Hochschild and the cyclic homology of . It turns out that these are independent of q and that they admit an easy description in terms of the extended quotient of a torus by a Weyl group, both of which are canonically associated to the root datum . For positive q we also prove that the representations of the family of algebras , come in families which depend analytically on ϵ. Analogous results are obtained for graded Hecke algebras and for Schwartz completions of affine Hecke algebras. [Copyright &y& Elsevier]

We show that if $A$ and $H$ are Hopf algebras that have equivalent tensor categories of comodules, then one can transport what we call a free Yetter–Drinfeld resolution of the counit of $A$ to the same kind of resolution for the counit of $H$, exhibiting in this way strong links between the Hochschild homologies of $A$ and $H$. This enables us to obtain a finite free resolution of the counit of $\mathcal {B}(E)$, the Hopf algebra of the bilinear form associated with an invertible matrix $E$, generalizing an earlier construction of Collins, Härtel and Thom in the orthogonal case $E=I_n$. It follows that $\mathcal {B}(E)$ is smooth of dimension 3 and satisfies Poincaré duality. Combining this with results of Vergnioux, it also follows that when $E$ is an antisymmetric matrix, the $L^2$-Betti numbers of the associated discrete quantum group all vanish. We also use our resolution to compute the bialgebra cohomology of $\mathcal {B}(E)$in the cosemisimple case. [ABSTRACT FROM AUTHOR]

We adapt to the case of deformation quantization modules a formula of Lunts (Lefschetz fixed point theorems for Fourier-Mukai functors and DG algebras. arXiv:1102.2884. ArXiv e-prints, ) who calculates the trace of a kernel acting on the Hochschild homology of a DQ-algebroid. [ABSTRACT FROM AUTHOR]

We study the Hochschild (co)homology of curved A∞ alge- bras that arise in the study of Landau-Ginzburg (LG) models in physics. We show that the ordinary Hochschild homology and cohomology of these algebras vanish. To correct this we introduce modified versions of these theories, Borel-Moore Hochschild homology and compactly sup- ported Hochschild cohomology. For LG models the new invariants yield the answer predicted by physics, shifts of the Jacobian ring. We also study the relationship between graded LG models and the geometry of hypersurfaces. We prove that Orlov's derived equivalence descends from an equivalence at the differential graded level, so in particular the CY/LG correspondence is a dg equivalence. This leads us to study the equivariant Hochschild homology of orbifold LG models. The results we get can be seen as noncommutative analogues of the Lefschetz hyperplane and Griffiths transversality theorems. [ABSTRACT FROM AUTHOR]

Let X be a smooth projective complex variety. The Hochschild homology HH( X) of X is an important invariant of X, which is isomorphic to the Hodge cohomology of X via the Hochschild-Kostant-Rosenberg isomorphism. On HH( X), one has the Mukai pairing constructed by Caldararu. An explicit formula for the Mukai pairing at the level of Hodge cohomology was proven by the author in an earlier work (following ideas of Markarian). This formula implies a similar explicit formula for a closely related variant of the Mukai pairing on HH( X). The latter pairing on HH( X) is intimately linked to the study of Fourier-Mukai transforms of complex projective varieties. We give a new method to prove a formula computing the aforementioned variant of Caldararu's Mukai pairing. Our method is based on some important results in the area of deformation quantization. In particular, we use part of the work of Kashiwara and Schapira on Deformation Quantization modules together with an algebraic index theorem of Bressler, Nest and Tsygan. Our new method explicitly shows that the 'Noncommutative Riemann-Roch' implies the classical Riemann-Roch. Further, it is hoped that our method would be useful for generalization to settings involving certain singular varieties. [ABSTRACT FROM AUTHOR]

Abstract: We consider the space of polydifferential operators on n functions on symplectic manifolds invariant under symplectic automorphisms, whose study was initiated by Mathieu in 1995. Permutations of inputs yield an action of , which extends to an action of . We study this structure viewing n as a parameter, in the sense of Deligneʼs category. For manifolds of dimension 2d, we show that the isotypic part of this space of -th tensor powers of the reflection representation of is spanned by Poisson polynomials. We also prove a partial converse, and compute explicitly the isotypic part of ⩽4-th tensor powers of the reflection representation. We give generating functions for the isotypic parts corresponding to Young diagrams which only differ in the length of the top row, and prove that they are rational fractions whose denominators are related to hook lengths of the diagrams obtained by removing the top row. This also gives such a formula for the same isotypic parts of induced representations from to where n is viewed as a parameter. We show that the space of invariant operators of order 2m has polynomial dimension in n of degree equal to 2m, while the part not coming from Poisson polynomials has polynomial dimension of degree . We use this to compute asymptotics of the dimension of invariant operators. We also give new bounds on the order of invariant operators for a fixed n. We apply this to the Poisson and Hochschild homology associated to the singularity . Namely, the canonical surjection from to (the Brylinski spectral sequence in degree zero) restricts to an isomorphism in the aforementioned isotypic part , and also in . We prove a partial converse. Finally, the kernel of the entire surjection has dimension on the order of times the dimension of the homology group. [Copyright &y& Elsevier]

Przytycki has established a connection between the Hochschild homology of an algebra A and the chromatic graph homology of a polygon graph with coefficients in A. In general the chromatic graph homology is not defined in the case where the coefficient ring is a non-commutative algebra. In this paper we define a new homology theory for directed graphs which takes coefficients in an arbitrary A-A bimodule, for A possibly non-commutative, which on polygons agrees with Hochschild homology through a range of dimensions. [ABSTRACT FROM AUTHOR]

We study algebras having 2-truncated cycles and show that these algebras have infinitely many nonzero Hochschild homology groups. Consequently, algebras of finite global dimension have no 2-truncated cycles and therefore satisfy a higher version of the "no loops conjecture". [ABSTRACT FROM AUTHOR]

We prove that for an inclusion of unital associative but not necessarily commutative k-algebras B ⊆ A we have long exact sequences in Hochschild homology and cyclic (co)homology akin to the Jacobi-Zariski sequence in André-Quillen homology, provided that the quotient B-module A=B is flat. We also prove that for an arbitrary r-flat morphism φ : B → A with an H-unital kernel, one can express the Wodzicki excision sequence and our Jacobi-Zariski sequence in Hochschild homology and cyclic (co)homology as a single long exact sequence. [ABSTRACT FROM AUTHOR]

Abstract: The strong no loop conjecture states that a simple module of finite projective dimension over an artin algebra has no non-zero self-extension. The main result of this paper establishes this well known conjecture for finite dimensional algebras over an algebraically closed field. [Copyright &y& Elsevier]

This work establishes a comparison between functions on derived loop spaces (Toën and Vezzosi, Chern character, loop spaces and derived algebraic geometry, in Algebraic topology: the Abel symposium 2007, Abel Symposia, vol. 4, eds N. Baas, E. M. Friedlander, B. Jahren and P. A. Østvær (Springer, 2009), ISBN:978-3-642-01199-3) and de Rham theory. If A is a smooth commutative k-algebra and k has characteristic 0, we show that two objects, S1⊗A and ϵ(A), determine one another, functorially in A. The object S1⊗A is the S1-equivariant simplicial k-algebra obtained by tensoring A by the simplicial group S1 :=Bℤ, while the object ϵ(A) is the de Rham algebra of A, endowed with the de Rham differential, and viewed as a ϵ-dg-algebra (see the main text). We define an equivalence φ between the homotopy theory of simplicial commutative S1-equivariant k-algebras and the homotopy theory of ϵ-dg-algebras, and we show the existence of a functorial equivalence ϕ(S1 ⊗A)∼ϵ(A) . We deduce from this the comparison mentioned above, identifying the S1-equivariant functions on the derived loop space LX of a smooth k-scheme X with the algebraic de Rham cohomology of X/k. As corollaries, we obtain functorial and multiplicative versions of decomposition theorems for Hochschild homology (in the spirit of Hochschild–Kostant–Rosenberg) for arbitrary semi-separated k-schemes. By construction, these decompositions are moreover compatible with the S1-action on the Hochschild complex, on one hand, and with the de Rham differential, on the other hand. [ABSTRACT FROM PUBLISHER]

Abstract: We prove that the algebra of chains on the based loop space recovers the derived (wrapped) Fukaya category of the cotangent bundle of a closed smooth oriented manifold. The main new idea is the proof that a cotangent fibre generates the Fukaya category using a version of the map from symplectic cohomology to the homology of the free loop space introduced by Cieliebak and Latschev. [Copyright &y& Elsevier]

We prove that E-homology of non-unital commutative algebras can be described as functor homology when one considers functors from a certain category of planar trees with n levels. For different n these homology theories are connected by natural maps, ranging from Hochschild homology and its higher order versions to Gamma homology. [ABSTRACT FROM AUTHOR]

This research announcement continues the study of the symplectic homology of Weinstein manifolds undertaken by the authors [Bourgeois F, Ekholm T, Eliashberg Y (2009) arXiv:0911.0026] where the symplectic homology, as a vector space, was expressed in terms of the Legendrian homology algebra of the attaching spheres of critical handles. Here, we express the product and Batalin-Vilkovisky operator of symplectic homology in that context. [ABSTRACT FROM AUTHOR]

Let $${\mathbb{K}}$$ be a field of characteristic p > 0 and S the unit circle. We construct a model for the negative cylic homology of a commutative cochain algebra with two stages Sullivan minimal model. Using the notion of shc-formality introduced in Bitjong and Thomas (Topology 41:85-106), the main result of Bitjong and El Haouari (Math Ann 338:347-354) and techniques of Vigué-Poirrier (J Pure Appl Algebra 91:347-354) we compute the S-equivariant cohomology algebras of the free loop spaces of the infinite complex projective space $${\mathbb{CP}(\infty)}$$ and the odd spheres S. [ABSTRACT FROM AUTHOR]

Abstract: We study the Hochschild homology of algebras related via split pairs, and apply this to fiber products, trivial extensions, monomial algebras, graded-commutative algebras and quantum complete intersections. In particular, we compute lower bounds for the dimensions of both the Hochschild homology and cohomology groups of quantum complete intersections. [ABSTRACT FROM AUTHOR]

It is well known that the Fano scheme of lines on a cubic 4-fold is a symplectic variety. We generalize this fact by constructing a closed (2n - 4)-form on the Fano scheme of lines on a (2n - 2)-dimensional hypersurface Yn of degree n. We provide several definitions of this form — via the Abel–Jacobi map, via Hochschild homology, and via the linkage class — and compute it explicitly for n = 4. In the special case of a Pfaffian hypersurface Yn we show that the Fano scheme is birational to a certain moduli space of sheaves of a (2n - 4)-dimensional Calabi–Yau variety X arising naturally in the context of homological projective duality, and that the constructed form is induced by the holomorphic volume form on X. This remains true for a general non-Pfaffian hypersurface but the dual Calabi–Yau becomes noncommutative. [ABSTRACT FROM AUTHOR]

We describe explicitly the continuous Hochschild and cyclic cohomology groups of certain tensor products of ⊗^-algebras which are Fréchet spaces or nuclear DF-spaces. To this end we establish the existence of topological isomorphisms in the Künneth formula for the cohomology of complete nuclear DF-complexes and in the Künneth formula for continuous Hochschild cohomology of nuclear ⊗^-algebras which are Fréchet spaces or DF-spaces for which all boundary maps of the standard homology complexes have closed ranges. [ABSTRACT FROM AUTHOR]

Abstract: Let be a graded Hecke algebra with complex deformation parameters and Weyl group W. We show that the Hochschild, cyclic and periodic cyclic homologies of are all independent of the parameters, and compute them explicitly. We use this to prove that, if the deformation parameters are real, the collection of irreducible tempered -modules with real central character forms a -basis of the representation ring of W. Our method involves a new interpretation of the periodic cyclic homology of finite type algebras, in terms of the cohomology of a sheaf over the underlying complex affine variety. [Copyright &y& Elsevier]

We consider a complex simple Lie algebra $${\mathfrak{g}}$$, with the action of its adjoint group. Among the three canonical nilpotent orbits under this action, the minimal orbit is the non zero orbit of smallest dimension. We are interested in equivariant deformation quantization: we construct $${\mathfrak{g}}$$-invariant star-products on the minimal orbit and on its closure, a singular algebraic variety. We shall make use of Hochschild homology and cohomology, of some results about the invariants of the classical groups, and of some interesting representations of simple Lie algebras. To the minimal orbit is associated a unique, completely prime two-sided ideal of the universal enveloping algebra $${{\rm U}(\mathfrak{g})}$$. This ideal is primitive and is called the Joseph ideal. We give explicit expressions for the generators of the Joseph ideal and compute the infinitesimal characters. [ABSTRACT FROM AUTHOR]

In this paper, we compute the Hochschild homology of elliptic Sklyanin algebras. These algebras are deformations of polynomial algebra with a Poisson bracket called the Sklyanin Poisson bracket. [ABSTRACT FROM AUTHOR]

In this paper we prove that the Steenrod operations act naturally on the negative cyclic homology of a differential graded algebra A over the prime field Fp satisfying some extra conditions. When A denotes the singular cochains with coefficients in Fp of a 1-connected space X, these extra conditions are satisfied. The Jones isomorphism identifies these Steenrod operations with the usual ones on the S¹-equivariant cohomology of the free loop space on X with coefficients in Fp. We conclude by performing some calculations on the negative cyclic homology. [ABSTRACT FROM AUTHOR]

Abstract: We study the Hochschild and cyclic homology of noncommutative monogenic extensions. As an application we compute the Hochschild and cyclic homology of the rank 1 Hopf algebras introduced in [L. Krop, D. Radford, Finite dimensional Hopf algebras of rank 1 in characteristic 0, Journal of Algebra 302 (1) (2006) 214–230]. [Copyright &y& Elsevier]

Let  be a simple Lie algebra of rank l,  a Cartan subalgebra and W the corresponding Weyl group. The space ⊕* is naturally equipped with a symplectic structure and the invariant algebra S(⊕*)W inherits a Poisson structure which can be deformed by the invariant algebra [image omitted] where Al denotes the Weyl algebra of rank l. The purpose of this article is to compute the Poisson homology group of degree zero of S(⊕*)W and to compare it to the Hochschild homology group of degree zero of [image omitted]. These two groups have the same dimensions in rank 2 which are equal respectively to 1.2 and 3 in the types A2, B2 and G2. [ABSTRACT FROM AUTHOR]