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The notion of Hochschild homology of a dg algebra admits a natural dualization, the coHochschild homology of a dg coalgebra, introduced in arXiv:0711.1023 by Hess, Parent, and Scott as a tool to study free loop spaces. In this article we prove "agreement" for coHochschild homology, i.e., that the coHochschild homology of a dg coalgebra $C$ is isomorphic to the Hochschild homology of the dg category of appropriately compact $C$-comodules, from which Morita invariance of coHochschild homology follows. Generalizing the dg case, we define the topological coHochschild homology (coTHH) of coalgebra spectra, of which suspension spectra are the canonical examples, and show that coTHH of the suspension spectrum of a space $X$ is equivalent to the suspension spectrum of the free loop space on $X$, as long as $X$ is a nice enough space (for example, simply connected.) Based on this result and on a Quillen equivalence established by the authors in arXiv:1402.4719, we prove that "agreement" holds for coTHH as well., To appear in JPAA

We develop a new spectral sequence in order to calculate Hochschild homology of smash biproducts (also called twisted tensor products) of unital associative algebras $A\# B$ provided one of $A$ or $B$ has Hochschild dimension less than 2. We use this spectral sequence to calculate Hochschild homology of quantum tori, multiparametric quantum affine spaces, quantum complete intersections, quantum Weyl algebras, deformed completed Weyl algebras, and finally the algebra $M_q(2)$ of quantum $2\times 2$-matrices.

We consider the Gorenstein condition for topological Hochschild homology, and show that it holds remarkably often [7] . More precisely, if R is a commutative ring spectrum and R ⟶ k is a map to a field of characteristic p then, provided k is small as an R -module, T H H ( R ; k ) is Gorenstein in the sense of [11] . In particular, this holds if R is a (conventional) regular local ring with residue field k of characteristic p . Using only Bokstedt's calculation of T H H ( k ) , this gives a non-calculational proof of dualities visible in calculations of Bokstedt [9] , Ausoni [3] , Lindenstrauss and Madsen [17] , Angeltveit and Rognes [2] and others.

The main purpose of this paper is to classify the localising subcategories of the derived category D ( C ∗ ( B G ; k ) ) where G is a compact Lie group and k is a field. We also prove a version of Chouinard’s theorem for D ( C ∗ ( B G ; k ) ) , we describe the relationship between induction and coinduction for a closed subgroup of G , and we use this to describe the relationship between Hochschild homology and cohomology of C ∗ ( B G ; k ) .

Let B⊆A be an H-Galois extension, where H is a Hopf algebra over a field K. If M is a Hopf bimodule then HH∗(A,M), the Hochschild homology of A with coefficients in M, is a right comodule over the coalgebra CH=H/[H,H]. Given an injective left CH-comodule V, our aim is to understand the relationship between HH∗(A,M)□CHV and HH∗(B,M□CHV). The roots of this problem can be found in Lorenz (1994) [15], where HH∗(A,A)G and HH∗(B,B) are shown to be isomorphic for any centrally G-Galois extension. To approach the above mentioned problem, in the case when A is a faithfully flat B-module and H satisfies some technical conditions, we construct a spectral sequence TorpRH(K,HHq(B,M□CHV))⟹HHp+q(A,M)□CHV, where RH denotes the subalgebra of cocommutative elements in H. We also find conditions on H such that the edge maps of the above spectral sequence yield isomorphisms K⊗RHHH∗(B,M□CHV)≅HH∗(A,M)□CHV. In the last part of the paper we define centrally Hopf–Galois extensions and we show that for such an extension B⊆A, the RH-action on HH∗(B,M□CHV) is trivial. As an application, we compute the subspace of H-coinvariant elements in HH∗(A,M). A similar result is derived for HC∗(A), the cyclic homology of A.

The Hochschild homology/cohomology an associative algebra, together with the Connes differential, the contraction map and the Lie derivative, forms the structure of calculus. In this paper we compute explicitely the calculus structure of preprojective algebras of Dynkin quivers over a field of characteristic zero. This also completes the work in math.AG/0502301, where the Batalin-Vilkovisky structure of the Hochschild cohomology of preprojective algebras of non-Dynkin quivers are computed and the calculus can be easily computed from that., Comment: 30 pages

In this paper we use May's algebraic approach to Steenrod operation in conjunction with Munkholm's notion of shc-algebra in order to introduce what we call a π-shc-algebra. (π is a cyclic group of order a fixed prime p.) We prove that the homology and the Hochschild homology of a π-shc-algebra have natural Steenrod operations. Our main topological example is the free loop space. We prove that the Jones’ isomorphism preserves Steenrod operations.

We study Hochschild homology and cohomology for a class of noncommutative polynomial algebras which are both quantum (in the sense that they contain some copies of Manin's quantum plane as subalgebras) and classical (in the sense that they also contain some copies of the Weyl algebra A1). We obtain explicit computations for some significant families of such algebras. In particular, we prove that the algebra of twisted differential operators on a quantum affine space (simple quantum Weyl algebra) has the same Hochschild homology, and satisfies the same duality relation, as the classical Weyl algebra does.

In this paper we use Quillen–Barr–Beck's theory of (co-) homology of algebras in order to define (co-) homology for the category RLie of restricted Lie algebras over a field k of characteristic p≠0. In contrast with the cases of groups, associative algebras and Lie algebras we do not obtain Hochschild (co-) homology shifted by 1.Precisely, we determine for L∈RLie the category of Beck L-modules and the group of Beck derivations of g∈RLie/L to a Beck L-module M. Moreover, we prove a classification theorem which gives a one-to-one correspondence between the one cohomology and the set of equivalent classes of p-extensions. Finally, a universal coefficient theorem is proved, relating the homology to the Hochschild homology via a short exact sequence. This shows that the new homology determines the Hochschild homology.

We derive an \'etale descent formula for topological Hochschild homology and prove a HKR theorem for smooth S-algebras., Comment: 23 pages

We investigate the properties of the iterated bar construction and generalize to it well-known results of Quillen, Gerstenhaber and Schack on the bar construction. In particular, we show that the iterated bar construction and the associated iterated Hochschild homology theories split canonically over any field, refining and extending previous results of Pirashvili. For that purpose, we introduce and study algebraic structures such as algebras, coalgebras or Hopf algebras in the Eilenberg–MacLane “category of constructions”. These results on constructions should be of interest on their own.

We give a natural decomposition of a connected commutative differential graded bi-algebra over a commutative algebra in the case of characteristic zero. This gives the ordinary Hodge decomposition of the Hochschild homology when we apply this natural decomposition to the cyclic bar complex of a commutative algebra. In the case of characteristic p>0, we show that, in the spectral sequence induced by the augmentation ideal filtration of the cyclic bar complex of a commutative algebra, the only possible non-trivial differentials are dk(p−1) for k≥1. Also we show that the spectral sequence which converges to the Hochschild cohomology is multiplicative with respect to the Gerstenhaber brackets and the cup products.

In the situation where we have a sequence of two spectrum extensions, we describe a spectral sequence converging to the topological Hochschild homology of the top spectrum over the base spectrum, in terms of invariants of the intermediate extensions. The main application is to the case of topological Hochschild homology of a ring extension, where the base spectrum is taken to be the sphere spectrum. We get a spectral sequence starting with appropriate algebraic Tor groups of the topological Hochschild homology groups of a ring, converging to the topological Hochschild homology of an algebra over it.

This paper gives a computation of topological Hochschild homology for Z /p n for any prime p and any number n≥2 based on Bokstedt's computation of topological Hochschild homology for Z /p . The most important idea in the computation is to consider Z /p n as a filtered ring, and to construct a spectral sequence for topological Hochschild homology of a filtered ring. The product on topological Hochschild homology is used extensively, and a direct description of this product is given.

We show that the Adams operations on the algebraic K -theory of a commutative unital Q -algebra correspond to Adams operations on Hochschild homology via the Dennis trace map. This establishes a conjecture of Geller and Weibel, Hodge decomposition of Loday symbols in K -theory and cyclic homology, K -theoty 8 (1994) 587–632, that the Dennis trace map preserves the Hodge decomposition.

We consider the topological Hochschild homology (THH) of a group ring R[G], and calculate the restriction map (or transfer) associated with a subgroup K ⊆ G of finite index in terms of ordinary group homology transfers. This gives information on the corresponding restriction map in Quillen's K-theory via the topological Dennis trace tr:K(R[G]) → THH(R[G]). More generally, we consider group rings for “rings up to homotopy” (FSP's) and calculate the THH-rcstriction map in terms of transfers in generalized homology theories.

For a type of quantum algebras we obtain a chain complex, simpler than the canonical one, whose homology is the Hochschild homology of the algebra. We applied this result to some concrete examples, as U q ( sl (2, k )) and O q ( M (2, k )).

We prove that the topological Hochschild homology spectrum THH ( R ) of an E ∞ spectrum R is the S 1 -indexed sum of copies R in the category of E ∞ , ring spectra. As a consequence, we obtain a natural S 1 -action and compatible power operations on THH ( R ). In addition, THH ( R ) admits an A ∞ -comultiplication making it an A ∞ Hopf algebra spectrum over R .

The quantum multiparametric torus is the algebra generated over a field $k$ by the $2N$ variables $x_1,\ldots,x_N$ and $x_1^{-1},\ldots,x_N^{-1}$ and the relations $ x_ix_i^{-1}=1=x_i^{-1} x_i$ and $x_ix_j=q_{ij}x_jx_i$ for every $1\le i,j\le N$ and where $\{q_{ij}\}_{1\le i,j\le N}$ is a family of non-zero scalars of $k$ satisfying the relations $q_{ii}=1$ and $q_{ij}q_{ji}=1$ for every $1\le i,j,\le N$. We explicitly compute its Hochschild homology groups, using previously constructed ``quantum Koszul complexes''. We deduce the corresponding cyclic homology groups.

Let A be an associative k-algebra with involution, where k is a commutative ring of characteristic not equal to two. Then the dihedral groups act on the Hochschild complex and, following Loday, a new homological theory, called dihedral homology, can be defined generalizing the notion of cyclic homology defined by Connes. Here we give a model to compute dihedral homology of a commutative algebra over a characteristic zero field. As, for an involutive algebra, we have a decomposition of Hochschild homology into a direct sum of two k-modules: Z2-equivariant and skew Z2-equivariant Hochschild homologies, we give smoothness criteria in terms of vanishing of some Z2-equivariant Hochschild homology groups.

We develop a theory for Morita equivalence of Banach algebras with bounded approximate identities and categories of essential modules, using functors compatible with the topology. Many aspects of discrete theory are carried over. Most importantly, the Eilenberg-Watts theorem holds, so that equivalence functors are representable as tensor functors. This enables us to determine how Banach algebras which are Morita equivalent to a given Banach algebra are constructed. It also leads to Morita invariance of bounded Hochschild homology, thus providing an efficient tool for computation of homology groups.

McCarthy, R., The cyclic homology of an exact category, Journal of Pure and Applied Algebra 93 (1994) 251-296. We define Hochschild and cyclic homology groups for an exact category which generalize the usual definitions when one considers finitely generated projective modules. They satisfy additivity as well as many of the usual properties one expects from the homology groups of an algebra. The Dennis trace and its lift to negative homology are also (multiplicatively) generalized to this setting. We use the S construction of Waldhausen and a formal generalization of the usual cyclic complex from Hochschild homology for our definition.

The topological Hochschild homology of a discrete ring is shown to agree with the MacLane homology of that ring.

We begin by briefly recalling some of our previous results on the descent algebra of the hyperoctahedral groups Bn. From this we construct a ‘nice’ expression for the generating function of a family of orthogonal idempotents ϱkn. More precisely, ∑ k=1 n ρ k n x k = 1 2 n n! ∑ π∈B n (x−2d(π))↑ B n π , where d(π) stands for the number of descents of π and (x)↑ B n = (x + 1)(x + 3)(x + 5)⋯(x + 2n −1 ) . We show that the dimension of the right ideal Q [Bn]ϱkn is given by the number c(n,k) of elements of Bn having k positive cycles. Thus the dimension is given by an analogue of the Stirling numbers of the first kind. We also show that the algebra Λ generated by the classes of elements of Bn having equal numbers of descents is commutative. This relates to Loday's work on cyclic homology and Hochschild homology. In fact we show that we can define some λ operators on Λ which commute with the Hochschild boundary operator. This gives a decomposition for commutative hyperoctahedral algebra homology. We conclude our paper by presenting results about the hyperoctahedral shuffle algebra which are extensions of aspects of the work of Aldous, Bayer and Diaconis related to the shuffle algebra.

If A is an algebra with unit, M an A -bimodule, the Hochschild homology groups H i ( A , M ) are modules over the center Z ( A ). We show that these homology groups behave well with respect to localization in the commutative ring Z ( A ). We deduce a Mayer-Vietoris principle, and give some examples (commutative algebras, enveloping algebras, crossed–product algebras).

Let k be an arbitrary commutative ring, A a smooth k-algebra and {a1,…,am}⊆A a regular sequence. We compute the Hochschild (co)-homology of the k-algebra A|〈a1,…,am〉. As an application we obtain some criteria of smoothness for this type of algebras.

We compute the topological Hochschild homology modules of finitely generated commutative algebras over finite fields provided their singularities are sufficiently mild.

We compute the natural multiplication on mod 4 homotopy for the ring spectrum T( Z ) = THH( Z ) , and its associated natural module action upon the mod 2 homotopy.

The finite dimensional complex truncated tensor algebras have a natural module structure over the complex general linear group. This structure is inherited by the Hochschild homology of these algebras. In this paper we determine this module structure by combining techniques from homological algebra and representation theory, such as the Schur duality theorem.

In this note, the cyclic homology of quiver algebras with quadratic monomial relations over a field are computed. This is done by using a mixed complex due to Cibils. An associated spectral sequence is considered, which converges to the cyclic homology. The E 1 -term of this is calculated with the aid of the author's earlier work on the Hochschild homology of these algebras. The E 2 -term is then calculated, and since all higher differentials in the spectral sequence vanishes, the homology spaces can be determined from it.

In analogy with Hochschild-Mitchell homology for linear categories topological Hochschild and cyclic homology (THH and TC) are defined for ring functors on a category β. Fundamental properties of THH and TC are proven and some examples are analyzed. A special case of a ring functor on an exact category is treated separately, and is compared with algebraic K-theory via a Dennis-Bokstedt trace map. Calling THH and TC applied to these ring functors simply THH( ) and TC( ), we get that the iteration of Waldhausen's S construction yields spectra {THH(S(n) )} and {TC(S(n) )}, and the maps from K-theory become maps of spectra. If is split exact, the THH and TC spectra are Ω-spectra. The inclusion by degeneracies THH0(S(n) ) ⊆ THH(S(n) ) is a stable equivalence, and it is shown how this leads to a weak resolution theorem for THH. If ℘A is the category of finitely generated projective modules over a unital and associative ring A, we get that THH(A) THH(℘A) and TC(A) TC(℘A).

We compute the cyclic homology of D[X] n > , D[X] and D[X],X -1 ] in terms of cyclic and Hochschild homology of D , for a k -algebra D , where k is a commutative ring containing a field.

Let p be any prime. We consider Bo¨kstedt's topological refinement K (ℤ) → T (ℤ) = THH(ℤ)of the Dennis trace map from algebraic K -theory of the integers to topological Hochschild homology of the integers. This trace map is shown to induce a surjection on homotopy in degree2 p − 1, onto the first P -torsion in the target. Furthermore, Bo¨kstedt's map factors through the S 1 -homotopy fixed points T (ℤ) hS 1 of T (ℤ), and it is shown that the first p -torsion element in degree2 p − 3 of the stable homotopy groups of spheres is detected in the homotopy of T (ℤ) hS 1 . Both results are due to Bo¨kstedt, but have remained unpublished.

Let k be an arbitrary commutative ring. We compute the Hochschild homology HH∗(A, A) of the k-algebra A = k[x1,…,xn]/〈ƒ1,…,ƒr〉, when ƒ1,…,ƒr is a regular sequence.

We introduce the notion of non commutative truncated polynomial extension of an algebra A. We study two families of these extensions. For the first one we obtain a complete classification and for the second one, which we call upper triangular, we find that the obstructions to inductively construct them, lie in the Hochschild homology of A, with coefficients in a suitable A-bimodule., Comment: 29 pages, We add some results about simplicirty (Prop. 1.11 and 3.6) and a result about rigidity (Th. 4.17)

A systematic study of the homological behavior of finitely and linearly presented modules over Koszul algebras is started. In particular, it is shown that the generating series of the linear Betti numbers of these modules over some precise special Koszul algebras are rationally related to the generating series for the Betti numbers of all local commutative noetherian rings. It is also shown that there are very small commutative Koszul algebras (4 generators and 4, 5 or 6 relations) which are bad in the sense that rational generating series for Betti numbers of linearly presented modules over them cannot be put on a common denominator. Finally, we do some explicit calculations that indicate that if a Koszul algebra is not a local complete intersection, then the generating series for its Hochschild (and also cyclic homology) is an irrational function. This supports the conjecture that the answer to our Question 1 on pp. 185–186 of Roos (Varna (1986) 173–189; Lecture Notes in Mathematics, vol. 1352, Springer, Berlin, 1988) is positive.