Lecture notes on algebraic topology pdf 169p download book. This cohomology was popularized in the many works of gerstenhaber see e. On the hilbert series of hochschild cohomology of block. Hochschild cohomology is fundamental in that it is the cohomology associated to a dg algebra awhich \controls the formal deformation. On higher topological hochschild homology of rings of integers. Does hochschild cohomology have any geometric meaning say, hochschild homology is connected with ktheory, what about cohomology. Let a be a zgraded algebra and m be a zgraded abimodule meaning that the structure map a. This set up is equivalent to that of grmodules as we now explain. This structure is present in many di erent areas, including topology. Hochschild cohomology in algebra, geometry, and topology mfo. Hochschild cohomology encodes information on deformations, smoothness and representations of algebras, among others. 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. Topological hochschild cohomology and generalized morita.
Hochschild cohomology of ordinary algebras was introduced in. For a flat commutative kalgebra a such that the enveloping algebra a. By specializing this gives a notion of derived centre of an associative algebra, i. Gerhard hochschild, on the cohomology groups of an associative algebra. Free algebraic topology books download ebooks online textbooks. In 1999 chas and sullivan 2 discovered that the homology h. 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. There are several conjectures connecting the string topology bv algebra with algebraic structures on the hochschild cohomology of. Hochschild cohomology of triangular matrix algebras. To any ring a one may associated its topological hochschild homology.
This process is experimental and the keywords may be updated as the learning algorithm improves. The reader can refer to h for a summary of the work in this area. Proceeding from the view of topology as a form of geometry, wallace emphasizes geometrical motivations and interpretations. Lecture 7 hochschild cohomology and deformations lectures. If g is a linearly reductive kgroup and h is a smooth algebraic kgroup. 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. Avramov, lincoln ragnarolaf buchweitz, toronto wendy lowen, antwerpen. Vanishing of hochschild cohomology for affine group schemes and rigidity of homomorphisms between algebraic groups benedictus margaux received. In 1 the authors associate to each ideal triangulation t of s,m a gentle algebra. Rognes, algebraic and geometric topology 5 2005 653690. Homology of free loop space and hochschild cohomology. Sand by c 1 the number of those components with exactly. Moreover, we give a lot of examples of complexes arising in di erent areas of mathematics giving di erent cohomology theories.
Is there some structure on it that corresponds to the gerstenhaber bracket on usual hochschild cohomology. In section 2 we prove that the hochschild homology of finite type algebras be. Vanishing of hochschild cohomology for affine group schemes. Higher hochschild cohomology, brane topology and centralizers. Hochschild cohomology of a graded algebra 20 references 23 email address. A textbook discussion is for instance in chapter 9 of.
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 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. The product on topological hochschild homology of the integers with mod 4 coefficients dvi. Groups can be studied homologically through their associated group algebras, and in turn this can be connected to the geometry of cer. Formality theorems and deformation theory 16 appendix a. 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. R 0, arma where a and r are finite dimensional algebras over an algebraically closed field k and m is an arbimodule. However, the actual calculations of hochschild cohomology groups have been fairly limited. But avoid asking for help, clarification, or responding to other answers. Browse other questions tagged algebraic topology homology cohomology homological algebra dualspaces loopspaces or ask your own question. We study the hochschild cohomology of triangular matrix rings b z. Thus its hochschild cohomology ring modulo nilpotents is not. Is there a place where there is a full proof of the hochschild kostantrosenberg theorem for hochschild cohomology. R 0 where a, r are arbitrary finite dimensional arma kalgebras and m is a finitely.
The dimensions of the hochschild cohomology groups are calculated, and a description of the hochschild cohomology algebra in terms of generators with relations is. November 17, 2009 communicated by wolfgang soergel abstract. Hochschild cohomology of quantized symplectic orbifolds. Cartan equation, poisson structure, double poisson bracket, hochschild cohomology, lin nity structure, formality. Holm, the hochschild cohomology ring of a modular group algebra. The term \quillen, or \andr equillen cohomology has a uniform meaning. Aws 2019 topological hochschild homology for shifting complexes we follow the convention c1n. In contrast, the notion of hochschild cohomology, while making appearances in diverse algebraic settings, has not been given such a uniform treatment. Ocal tamu an approach to hochschild cohomology april 21, 2018 3. Charles weibel, an introduction to homological algebra.
Hochschild cohomology commutes with adic completion. 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. In this paper we present our construction of the hochschild and cyclic homology of a algebra a. We prove the existence of two long exact sequences of kvector spaces relating the hochschild cohomology of a, r, and b. The enalgebra structure on enhochschild cohomology hhen a,b 52. 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. On the hochschild cohomology ring of tensor products of algebras. Calculations of iterated topological hochschild homology use higher thh as an important ingredient. For a graded vector space v, we use the grading convention v n v. Thus the factor hh1 z1b1 inherits a liealgebra structure. If one forgets about the chosen points,it can be shown that the. On the hochschild and harrison cohomology of algebras. Hodge ltration and operations in higher hochschild co.
Another example of a gerstenhaber algebra is the homology of a 2fold loop space. Staffeldt, on the topological hochschild homology of b u b u, i, pdf. Let k be a field, a an associative kalgebra, and m an abimodule. Some versions of cohomology arise by dualizing the construction of homology. The enveloping algebra of a is the tensor product a e a. To achieve this, one needs to work with homotopy algebras, homotopy bimodules and homotopy maps between these structures, even in the most simple cases. Grgw, gw and also beautiful applications in algebraic geometry and geometric.
In this chapter we introduce basic notions of homological algebra such as complexes and cohomology. In 1, we give the classic definition of the hochschild cohomology hha. The string topology bv algebra, hochschild cohomology and the. Feb 20, 2016 hochschild cohomology in algebra, geometry, and topology. This algebra is often called outa,the space of outer derivations ofa. Uniqueness of a structures and hochschild cohomology. Vanishing of hochschild cohomology for affine group. Thanks for contributing an answer to mathematics stack exchange. Cohomology can be viewed as a method of assigning richer algebraic invariants to a space than homology. For a flat commutative k algebra a such that the enveloping algebra a. Relationship between hochschild cohomology and drinfeld. Mathematisches forschungsinstitut oberwolfach hochschild.
Browse other questions tagged algebraic geometry referencerequest homological algebra noncommutative geometry or ask your own question. Singular hochschild cohomology and gerstenhaber algebra. A relationship between hochschild cohomology and the goldman. Let a be a z graded algebra and m be a z graded a bimodule meaning that the structure map a. X of the space of free loops on a closed oriented smooth manifold x has a rich algebraic structure called string topology. Using relative homological algebra in hochschild cohomology. Wolfgang the decomposition of hochschild cohomology and gerstenhaber operations, j. In this paper we will study the hochschild cohomology of a triangular matrix algebra b z. On the hochschild cohomology ring of tensor products of. In this paper we present a direct proof of what is suggested by holms results t. 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.
We give a new computation of hochschild cohomology of the exterior algebra, together with algebraic structures, by direct comparison with the symmetric algebra. A relationship between hochschild cohomology and the. Delignes hochschild cohomology conjecture kathryn hess the conjecture a tale of many proofs yet another proof. 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. Hochschild cohomology of finitedimensional algebras. Hochschild cohomology conjecture kathryn hess the conjecture a tale of many proofs yet another proof. Hochschild complex which allows to interpret derived closed forms in symplectic and poisson derived geometry ptvv, cptvv. A solution of delignes hochschild cohomology conjecture.
I am aware of many places where the result is proven for hochschild homology i. Theorem 22 comes from a general result propositions 11 and 12. On the higher topological hochschild homology of fp f p and. This structure is present in many di erent areas, including topology of compact. 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. This construction is a generalization of the hochschild homology of an arbitrary associative algebra from 11 and also is a generalization of the hochschild. Aws 2019 topological hochschild homology in arithmetic geometry. Generalities on hochschild cohomology in this paper, we deal with hochschild cohomology of a. The string topology bv algebra, hochschild cohomology and. Daniel joseph vera, topological hochschild homology of twisted group algebra, mit ph.
Hochschild cohomology in algebra, geometry, and topology. Singular cohomology is a powerful invariant in topology, associating a gradedcommutative ring to any topological space. 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. Spectral sequence string topology commutative algebra algebra structure hochschild cohomology these keywords were added by machine and not by the authors. Notes on factorization algebras, factorization homology and. 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.
Let k be a field, a an associative k algebra, and m an abimodule. Mccarthy, relative algebraic ktheory and topological cyclic homology, acta math. Free algebraic topology books download ebooks online. Cartan equation, poisson structure, double poisson bracket, hochschild cohomology, l in nity structure, formality. Besides string topology there are other reasons to study c 1algebras and co. 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. X y determines a homomorphism from the cohomology ring of y to that of x. For instance, we discuss simplicial cohomology, cohomology of sheaves, group cohomology, hochschild cohomology, di. On a cohomology of digraphs and hochschild cohomology. The formal here refers to the fact that for the large topological algebras we shall. Aws 2019 topological hochschild homology in arithmetic. In section 3 we consider the lower cohomology groups, that is, h i.