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Consider a reductive p-adic group G, its (complex-valued) Hecke algebra H(G), and the Harish-Chandra-Schwartz algebra S(G). We compute the Hochschild homology groups of H(G) and of S(G), and we describe the outcomes in several ways. Our main tools are algebraic families of smooth G-representations. With those we construct maps from HHn(H(G)) and HHn(S(G)) to modules of differential n-forms on affine varieties. For n=0, this provides a description of the cocentres of these algebras in terms of nice linear functions on the Grothendieck group of finite length (tempered) G-representations. It is known from [J. Algebra 606 (2022), 371-470] that every Bernstein ideal H(G)s of H(G) is closely related to a crossed product algebra of the form O(T)⋊W. Here O(T) denotes the regular functions on the variety T of unramified characters of a Levi subgroup L of G, and W is a finite group acting on T. We make this relation even stronger by establishing an isomorphism between HH∗(H(G)s) and HH∗(O(T)⋊W), although we have to say that in some cases it is necessary to twist C[W] by a 2-cocycle. Similarly, we prove that the Hochschild homology of the two-sided ideal S(G)s of S(G) is isomorphic to HH(C∞(Tu)⋊W), where Tu denotes the Lie group of unitary unramified characters of L. In these pictures of HH*(H(G)) and HH∗(S(G)), we also show how the Bernstein centre of H(G) acts. Finally, we derive similar expressions for the (periodic) cyclic homology groups of H(G) and of S(G) and we relate that to topological K-theory. [ABSTRACT FROM AUTHOR]

Given a group G and a partial factor set σ of G, we introduce the twisted partial group algebra κ par σ G , which governs the partial projective σ -representations of G into algebras over a field κ. Using the relation between partial projective representations and twisted partial actions we endow κ par σ G with the structure of a crossed product by a twisted partial action of G on a commutative subalgebra of κ par σ G. Then, we use twisted partial group algebras to obtain a first quadrant Grothendieck spectral sequence converging to the Hochschild homology of the crossed product A ∗ Θ G , involving the Hochschild homology of A and the partial homology of G, where Θ is a unital twisted partial action of G on a κ -algebra A with a κ -based twist. An analogous third quadrant cohomological spectral sequence is also obtained. [ABSTRACT FROM AUTHOR]

In this paper, we show that higher Hochschild homology groups do not vanish for a class of Hochschild extension algebras of truncated quiver algebras by the standard duality module. Consequently, this result extends the result of the trivial extension algebra by the standard duality module for truncated quiver algebras. Moreover, as an application, we show that for any Hochschild extension algebra of selfinjective Nakayama algebras by the standard duality module, its Hochschild homology dimension is infinite. [ABSTRACT FROM AUTHOR]

Given a cocommutative Hopf algebra H over a commutative ring K and a symmetric partial action of H on a K -algebra A , we obtain a first quadrant Grothendieck spectral sequence converging to the Hochschild homology of the smash product A # H , involving the Hochschild homology of A and the partial homology of H. An analogous third quadrant cohomological spectral sequence is also obtained. The definition of the partial (co)homology of H under consideration is based on the category of the partial representations of H. A specific partial representation of H on a subalgebra B of the partial "Hopf" algebra H p a r is involved in the definition and we construct a projective resolution of B. [ABSTRACT FROM AUTHOR]

Using Chern character, we construct a natural transformation from the local Hilbert functor to a functor of Artin rings defined from Hochschild homology. This enables us to realize (after slight modification) the infinitesimal Abel-Jacobi map as a morphism between tangent spaces of two functors of Artin rings and also enables us to reconstruct the semi-regularity map together with giving a different proof of a theorem of Bloch stating that the semi-regularity map annihilates certain obstructions to embedded deformations of a closed subvariety which is a locally complete intersection. [ABSTRACT FROM AUTHOR]

This work is part of a research program to compute the Hochschild homology groups HH ∗ (C [ x 1 , ... , x d ] / (x 1 , ... , x d) 3 ; C) in the case d = 2 through a lesser-known invariant called Coxeter cohomology, motivated by the isomorphism HH i (C [ x 1 , ... , x d ] / (x 1 , ... , x d) 3 ; C) ≅ ∑ 0 ≤ j ≤ i H C j S i + j , V ⊗ (i + j) provided by Larsen and Lindenstrauss. Here, H C ∗ denotes Coxeter cohomology, S i + j denotes the symmetric group on i + j letters, and V is the standard representation of GL d (C) on C d . We compute the Euler characteristic of the Coxeter cohomology (the alternating sum of the ranks of the Coxeter cohomology groups) of several representations of S n . In particular, the aforementioned tensor representation, and also several classes of irreducible representations of S n . Although the problem and its motivation are algebraic and topological in nature, the techniques used are largely combinatorial. [ABSTRACT FROM AUTHOR]

We compute the Hochschild homology and cohomology of A (1) , the subalgebra of the 2-primary Steenrod algebra generated by the first two Steenrod squares, Sq 1 , Sq 2 . The computation is accomplished using several May-type spectral sequences. [ABSTRACT FROM AUTHOR]

In this paper, we discuss Hochschild chain models for some of the string topology operations. We use these models to simplify the proofs and computations of some of the results in string topology. Along the way we also make some new observations. We further discuss how nonnilpotent local level homology classes with respect to the Chas–Sullivan and the Goresky–Hingston product detect closed geodesics with optimal index growth rates. [ABSTRACT FROM AUTHOR]

The cohomology algebra of the space H∗ (X) defines neither cohomology modules of the loop space H∗ (ΩX) nor cohomologies of the free loop space H∗ (ΛX). But by the author's minimality theorem, there exists a structure of A(∞)-algebra (H∗ (X), {mi}) on H∗ (X), which determines H∗ (ΩX). Here will be shown that the same A(∞)-algebra (H∗(X), {mi}) determines also cohomology modules H∗ (ΛX). [ABSTRACT FROM AUTHOR]

We describe two major string topology operations, the Chas–Sullivan product and the Goresky–Hingston coproduct, from geometric and algebraic perspectives. The geometric construction uses Thom–Pontrjagin intersection theory while the algebraic construction is phrased in terms of Hochschild homology. We give computations of products and coproducts on lens spaces via geometric intersection, and deduce that the coproduct distinguishes 3-dimensional lens spaces. Algebraically, we describe the structure these operations define together on the Tate–Hochschild complex. We use rational homotopy theory methods to sketch the equivalence between the geometric and algebraic definitions for simply-connected manifolds and real coefficients, emphasizing the role of configuration spaces. Finally, we study invariance properties of the operations, both algebraically and geometrically. [ABSTRACT FROM AUTHOR]

In this paper, we study a generalization of Kähler differentials, which correspond to the secondary Hochschild homology associated to a triple. We establish computations in low dimension, while also showing how this connects with the kernel of a multiplication map. [ABSTRACT FROM AUTHOR]

We study the Hochschild homology of the convolution algebra of a proper Lie groupoid by introducing a convolution sheaf over the space of orbits. We develop a localization result for the associated Hochschild homology sheaf, and we prove that the Hochschild homology sheaf at each stalk is quasi-isomorphic to the stalk at the origin of the Hochschild homology of the convolution algebra of its linearization, which is the transformation groupoid of a linear action of a compact isotropy group on a vector space. We then explain Brylinski's ansatz to compute the Hochschild homology of the transformation groupoid of a compact group action on a manifold. We verify Brylinski's conjecture for the case of smooth circle actions that the Hochschild homology is given by basic relative forms on the associated inertia space. [ABSTRACT FROM AUTHOR]

A typical example of a Cartan calculus consists of the Lie derivative and the contraction with vector fields of a manifold on the derivation ring of the de Rham complex. In this manuscript, a second stage of the Cartan calculus is investigated. In a general setting, the stage is formulated with operators obtained by the André–Quillen cohomology of a commutative differential graded algebra A on the Hochschild homology of A in terms of the homotopy Cartan calculus in the sense of Fiorenza and Kowalzig. Moreover, the Cartan calculus is interpreted geometrically with maps from the rational homotopy group of the monoid of self-homotopy equivalences on a space M to the derivation ring on the loop cohomology of M. We also give a geometric description to Sullivan's isomorphism, which relates the geometric Cartan calculus to the algebraic one, via the Γ 1 map due to Félix and Thomas. [ABSTRACT FROM AUTHOR]

The Hochschild homology of the ring k [ x 1 , x 2 , ... , x d ] / (x 1 , x 2 , ... , x d) 2 has been known and calculated several ways. This paper uses those calculations to calculate cyclic, negative cyclic, and periodic cyclic homology of k [ x 1 , x 2 , ... , x d ] / (x 1 , x 2 , ... , x d) 2 over k for k = Z. These calculations have been known for k = Q or fields containing it, but this paper studies the torsion which one gets for k = Z and through it, via the universal coefficient theorem, for general k. The computations are related to homology, cohomology, and Tate cohomology of cyclic groups. [ABSTRACT FROM AUTHOR]

We prove a conjecture of Shklyarov concerning the relationship between K. Saito's higher residue pairing and a certain pairing on the periodic cyclic homology of matrix factorization categories. Along the way, we give new proofs of a result of Shklyarov (Corollary 2 of [Adv. Math. 292 (2016), 181--209]) and Polishchuk--Vaintrob's Hirzebruch--Riemann--Roch formula for matrix factorizations (Theorem 4.1.4 (i) of [Duke Math. J. 161 (2012), 1863--1926]). [ABSTRACT FROM AUTHOR]

These are slightly expanded notes of lectures given in April 2019 at the Isfahan school and conference on representations of algebras. We recall the formalism of derived categories and functors and survey invariance results for the Hochschild (co)homology of differential graded algebras and categories. [ABSTRACT FROM AUTHOR]

For a commutative ring R and an ADE Dynkin quiver Q , we prove that the multiplicative preprojective algebra of Crawley-Boevey and Shaw, with parameter q = 1 , is isomorphic to the (additive) preprojective algebra as R -algebras if and only if the bad primes for Q – 2 in type D, 2 and 3 for Q = E 6 , E 7 and 2, 3 and 5 for Q = E 8 – are invertible in R. We construct an explicit isomorphism over Z [ 1 / 2 ] in type D, over Z [ 1 / 2 , 1 / 3 ] for Q = E 6 , E 7 and over Z [ 1 / 2 , 1 / 3 , 1 / 5 ] for Q = E 8. Conversely, if some bad prime is not invertible in R , we show that the additive and multiplicative preprojective algebras differ in zeroth Hochschild homology, and hence are not isomorphic. In fact, one only needs the vanishing of certain classes in zeroth Hochschild homology of the multiplicative preprojective algebra, utilizing a rigidification argument for isomorphisms that may be of independent interest. In the setting of Ginzburg dg-algebras, our obstructions are new in type E and give a more elementary proof of the negative result of Etgü–Lekili [5, Theorem 13] in type D. Moreover, the zeroth Hochschild homology of the multiplicative preprojective algebra, computed in Section 4, can be interpreted as the space of unobstructed deformations of the multiplicative Ginzburg dg-algebra by Van den Bergh duality. Finally, we observe that the multiplicative preprojective algebra is not symmetric Frobenius if Q ≠ A 1 , a departure from the additive preprojective algebra in characteristic 2 for Q = D 2 n , n ≥ 2 and Q = E 7 , E 8. [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]