Lecture 7 hochschild cohomology and deformations lectures. Moreover, we give a lot of examples of complexes arising in di erent areas of mathematics giving di erent cohomology theories. I will describe a short proof of this theorem using the theory of topological hochschild homology, which provides a canonical oneparameter deformation of hochschild homology in characteristic p. The reader can refer to h for a summary of the work in this area.
Groups can be studied homologically through their associated group algebras, and in turn this can be connected to the geometry of cer. Topological hochschild cohomology and generalized morita. Browse other questions tagged algebraic topology homology cohomology homological algebra dualspaces loopspaces or ask your own question. In section 2 we prove that the hochschild homology of finite type algebras be. A relationship between hochschild cohomology and the goldman. Thus its hochschild cohomology ring modulo nilpotents is not. In this paper we present a direct proof of what is suggested by holms results t.
Cartan equation, poisson structure, double poisson bracket, hochschild cohomology, lin nity structure, formality. On a cohomology of digraphs and hochschild cohomology. To any ring a one may associated its topological hochschild homology. The dimensions of the hochschild cohomology groups are calculated, and a description of the hochschild cohomology algebra in terms of generators with relations is obtained for the algebras under. To achieve this, one needs to work with homotopy algebras, homotopy bimodules and homotopy maps between these structures, even in the most simple cases. On the hochschild and harrison cohomology of algebras. This cohomology was popularized in the many works of gerstenhaber see e. Hochschild cohomology of associative algebras is important in many areas of mathematics, such as ring theory, commutative algebra, geometry, noncommutative geometry, representation theory, group theory, mathematical physics, homotopy theory and topology.
Theorem 22 comes from a general result propositions 11 and 12. The string topology bv algebra, hochschild cohomology and. Charles weibel, an introduction to homological algebra. November 17, 2009 communicated by wolfgang soergel abstract.
The formal here refers to the fact that for the large topological algebras we shall. Spectral sequence string topology commutative algebra algebra structure hochschild cohomology these keywords were added by machine and not by the authors. For a flat commutative k algebra a such that the enveloping algebra a. I am aware of many places where the result is proven for hochschild homology i. The product on topological hochschild homology of the integers with mod 4 coefficients dvi. Let k be a field, a an associative k algebra, and m an abimodule. Browse other questions tagged algebraic geometry referencerequest homological algebra noncommutative geometry or ask your own question. Generalities on hochschild cohomology in this paper, we deal with hochschild cohomology of a. This structure is present in many di erent areas, including topology of compact. In 1 the authors associate to each ideal triangulation t of s,m a gentle algebra. A solution of delignes hochschild cohomology conjecture. X of the space of free loops on a closed oriented smooth manifold x has a rich algebraic structure called string topology. Holm, the hochschild cohomology ring of a modular group algebra. Cartan equation, poisson structure, double poisson bracket, hochschild cohomology, l in nity structure, formality.
Some versions of cohomology arise by dualizing the construction of homology. Besides string topology there are other reasons to study c 1algebras and co. X y determines a homomorphism from the cohomology ring of y to that of x. We give a new computation of hochschild cohomology of the exterior algebra, together with algebraic structures, by direct comparison with the symmetric algebra. Vanishing of hochschild cohomology for affine group schemes. Hochschild cohomology is fundamental in that it is the cohomology associated to a dg algebra awhich \controls the formal deformation. Another example of a gerstenhaber algebra is the homology of a 2fold loop space. In this paper we will study the hochschild cohomology of a triangular matrix algebra b z. Mccarthy, relative algebraic ktheory and topological cyclic homology, acta math. Hochschild cohomology encodes information on deformations, smoothness and representations of algebras, among others. Thanks for contributing an answer to mathematics stack exchange. The enalgebra structure on enhochschild cohomology hhen a,b 52. Hochschild cohomology of triangular matrix algebras.
We prove the existence of two long exact sequences of kvector spaces relating the hochschild cohomology of a, r, and b. Is there a place where there is a full proof of the hochschild kostantrosenberg theorem for hochschild cohomology. Hochschild complex which allows to interpret derived closed forms in symplectic and poisson derived geometry ptvv, cptvv. For a flat commutative kalgebra a such that the enveloping algebra a. Proceeding from the view of topology as a form of geometry, wallace emphasizes geometrical motivations and interpretations. Singular hochschild cohomology and gerstenhaber algebra. Thus the factor hh1 z1b1 inherits a liealgebra structure. On the hochschild and harrison cohomology of algebras and. In algebraic topology, hochschild homology has been used intensively as a model for the forms or cochains on free or based depending on the module. But avoid asking for help, clarification, or responding to other answers. Calculations of iterated topological hochschild homology use higher thh as an important ingredient. Formality theorems and deformation theory 16 appendix a.
Feb 20, 2016 hochschild cohomology in algebra, geometry, and topology. Once beyond the singular homology groups, however, the author advances an understanding of the subjects algebraic patterns, leaving geometry aside in order to study these patterns as pure algebra. Let a be a zgraded algebra and m be a zgraded abimodule meaning that the structure map a. In this chapter we introduce basic notions of homological algebra such as complexes and cohomology. By specializing this gives a notion of derived centre of an associative algebra, i. Sand by c 1 the number of those components with exactly. The dimensions of the hochschild cohomology groups are calculated, and a description of the hochschild cohomology algebra in terms of generators with relations is. However, the actual calculations of hochschild cohomology groups have been fairly limited. In section 3 we consider the lower cohomology groups, that is, h i. The enveloping algebra of a is the tensor product a e a. In contrast, the notion of hochschild cohomology, while making appearances in diverse algebraic settings, has not been given such a uniform treatment.
Singular cohomology is a powerful invariant in topology, associating a gradedcommutative ring to any topological space. On the hochschild cohomology ring of tensor products of. In mathematics, specifically in homology theory and algebraic topology, cohomology is a general term for a sequence of abelian groups associated to a topological space, often defined from a cochain complex. Hochschild cohomology commutes with adic completion. On the hilbert series of hochschild cohomology of block. Gerhard hochschild, on the cohomology groups of an associative algebra. This process is experimental and the keywords may be updated as the learning algorithm improves. Free algebraic topology books download ebooks online. Notes on factorization algebras, factorization homology and. This structure is present in many di erent areas, including topology. Vanishing of hochschild cohomology for affine group schemes and rigidity of homomorphisms between algebraic groups benedictus margaux received.
Let a be a z graded algebra and m be a z graded a bimodule meaning that the structure map a. Hochschild cohomology of finitedimensional algebras. On the hochschild cohomology ring of tensor products of algebras. The second proposition is the decomposition theorem of the hochschild cohomology of the twisted group algebra, and the third proposition says that one can express the hochschild homology of the quantum algebra awith values in the bimodule agtwisted by the. Does hochschild cohomology have any geometric meaning say, hochschild homology is connected with ktheory, what about cohomology. In 1, we give the classic definition of the hochschild cohomology hha.
A geometric approach to hochschild cohomology of the exterior algebra michael wong department of mathematics, the university of texas at austin, 2515 speedway, austin, tx 78712 abstract. If one forgets about the chosen points,it can be shown that the. R 0, arma where a and r are finite dimensional algebras over an algebraically closed field k and m is an arbimodule. Is there some structure on it that corresponds to the gerstenhaber bracket on usual hochschild cohomology. Bimodules over a are essentially the same as modules over the enveloping algebra of a, so in particular a and m can be considered as a emodules. Higher hochschild cohomology, brane topology and centralizers. Rognes, algebraic and geometric topology 5 2005 653690.
Delignes hochschild cohomology conjecture kathryn hess the conjecture a tale of many proofs yet another proof. Cohomology can be viewed as a method of assigning richer algebraic invariants to a space than homology. Aws 2019 topological hochschild homology for shifting complexes we follow the convention c1n. Bimodules over a are essentially the same as modules over the enveloping algebra of a, so in particular a and m can be considered as a e. We show that the degrees and relations of the hochschild cohomology of a pblock algebra of a finite group over an algebraically closed field of prime characteristic p are bounded in terms of the defect groups of the block and that for a fixed defect d, there are only finitely many hilbert series of hochschild cohomology algebras of blocks of defect d. Mathematisches forschungsinstitut oberwolfach hochschild. Hochschild cohomology of ordinary algebras was introduced in. Aws 2019 topological hochschild homology in arithmetic. The term \quillen, or \andr equillen cohomology has a uniform meaning.
We note that the category algebra kc is neither a hopf nor a selfinjective algebra, in contrast to the fact that the hochschild cohomology ring of a. Grgw, gw and also beautiful applications in algebraic geometry and geometric. On higher topological hochschild homology of rings of integers. For a graded vector space v, we use the grading convention v n v.
Aws 2019 topological hochschild homology in arithmetic geometry. Avramov, lincoln ragnarolaf buchweitz, toronto wendy lowen, antwerpen. If g is a linearly reductive kgroup and h is a smooth algebraic kgroup. This algebra is often called outa,the space of outer derivations ofa. On the higher topological hochschild homology of fp f p and. Hochschild cohomology of quantized symplectic orbifolds. Vanishing of hochschild cohomology for affine group. For instance, we discuss simplicial cohomology, cohomology of sheaves, group cohomology, hochschild cohomology, di. This is a basic note in algebraic topology, it introduce the notion of fundamental groups, covering spaces, methods for computing fundamental groups using seifert van kampen theorem and some applications such as the brouwers fixed point theorem, borsuk ulam theorem, fundamental theorem of algebra. Wolfgang the decomposition of hochschild cohomology and gerstenhaber operations, j. Free algebraic topology books download ebooks online textbooks. Uniqueness of a structures and hochschild cohomology. Hochschild cohomology of a graded algebra 20 references 23 email address.
Daniel joseph vera, topological hochschild homology of twisted group algebra, mit ph. Lecture notes on algebraic topology pdf 169p download book. Homology of free loop space and hochschild cohomology. R 0 where a, r are arbitrary finite dimensional arma kalgebras and m is a finitely. Let k be a field, a an associative kalgebra, and m an abimodule. A textbook discussion is for instance in chapter 9 of. Hochschild cohomology in algebra, geometry, and topology mfo. This set up is equivalent to that of grmodules as we now explain. The string topology bv algebra, hochschild cohomology and the. Staffeldt, on the topological hochschild homology of b u b u, i, pdf.
Hochschild cohomology in algebra, geometry, and topology. Bv structure on the hochschild cohomology of sullivan. The etale cohomology of the general linear group over a finite field and the dickson algebra tezuka, michishige and yagita, nobuaki, kyoto journal of mathematics, 2018 deformations with respect to an algebraic group bleher, frauke m. We study the hochschild cohomology of triangular matrix rings b z. In this paper we present our construction of the hochschild and cyclic homology of a algebra a. Using relative homological algebra in hochschild cohomology. A relationship between hochschild cohomology and the. Hodge ltration and operations in higher hochschild co. This construction is a generalization of the hochschild homology of an arbitrary associative algebra from 11 and also is a generalization of the hochschild.
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