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Topological Hochschild Homology
In mathematics, Topological Hochschild homology is a topological refinement of Hochschild homology which rectifies some technical issues with computations in characteristic p. For instance, if we consider the \mathbb-algebra \mathbb_p then HH_k(\mathbb_p/\mathbb) \cong \begin \mathbb_p & k \text \\ 0 & k \text \end but if we consider the ring structure on \begin HH_*(\mathbb_p/\mathbb) &= \mathbb_p\langle u \rangle \\ &= \mathbb_p ,u^2/2!, u^3/3!,\ldots \end (as a divided power algebra structure) then there is a significant technical issue: if we set u \in HH_2(\mathbb_p/\mathbb), so u^2 \in HH_4(\mathbb_p/\mathbb), and so on, we have u^p = 0 from the resolution of \mathbb_p as an algebra over \mathbb_p\otimes^\mathbf\mathbb_p, i.e. HH_k(\mathbb_p/\mathbb) = H_k(\mathbb_p\otimes_\mathbb_p) This calculation is further elaborated on the Hochschild homology page, but the key point is the pathological behavior of the ring structure on the Hochschild homology of \mathbb_p. In contrast, th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hochschild Homology
In mathematics, Hochschild homology (and cohomology) is a homology theory for associative algebras over rings. There is also a theory for Hochschild homology of certain functors. Hochschild cohomology was introduced by for algebras over a field, and extended to algebras over more general rings by . Definition of Hochschild homology of algebras Let ''k'' be a field, ''A'' an associative ''k''-algebra, and ''M'' an ''A''-bimodule. The enveloping algebra of ''A'' is the tensor product A^e=A\otimes A^o of ''A'' with its opposite algebra. 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 ''Ae''-modules. defined the Hochschild homology and cohomology group of ''A'' with coefficients in ''M'' in terms of the Tor functor and Ext functor by : HH_n(A,M) = \operatorname_n^(A, M) : HH^n(A,M) = \operatorname^n_(A, M) Hochschild complex Let ''k'' be a ring, ''A'' an associative ''k''-algebra th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Divided Power Structure
In mathematics, specifically commutative algebra, a divided power structure is a way of making expressions of the form x^n / n! meaningful even when it is not possible to actually divide by n!. Definition Let ''A'' be a commutative ring with an ideal ''I''. A divided power structure (or PD-structure, after the French ''puissances divisées'') on ''I'' is a collection of maps \gamma_n : I \to A for ''n'' = 0, 1, 2, ... such that: #\gamma_0(x) = 1 and \gamma_1(x) = x for x \in I, while \gamma_n(x) \in I for ''n'' > 0. #\gamma_n(x + y) = \sum_^n \gamma_(x) \gamma_i(y) for x, y \in I. #\gamma_n(\lambda x) = \lambda^n \gamma_n(x) for \lambda \in A, x \in I. #\gamma_m(x) \gamma_n(x) = ((m, n)) \gamma_(x) for x \in I, where ((m, n)) = \frac is an integer. #\gamma_n(\gamma_m(x)) = C_ \gamma_(x) for x \in I and m > 0, where C_ = \frac is an integer. For convenience of notation, \gamma_n(x) is often written as x^ when it is clear what divided power structure is meant. The term ''divide ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Eilenberg–MacLane Spectrum
In mathematics, specifically algebraic topology, there is a distinguished class of spectra called Eilenberg–Maclane spectra HA for any Abelian group Apg 134. Note, this construction can be generalized to commutative rings R as well from its underlying Abelian group. These are an important class of spectra because they model ordinary integral cohomology and cohomology with coefficients in an abelian group. In addition, they are a lift of the homological structure in the derived category D(\mathbb) of abelian groups in the homotopy category of spectra. In addition, these spectra can be used to construct resolutions of spectra, called Adams resolutions, which are used in the construction of the Adams spectral sequence. Definition For a fixed abelian group A let HA denote the set of Eilenberg–MacLane spaces \with the adjunction map coming from the property of loop spaces of Eilenberg–Maclane spaces: namely, because there is a homotopy equivalenceK(A,n-1)\simeq \Omega K(A,n)we c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ring Spectrum
In stable homotopy theory, a ring spectrum is a spectrum ''E'' together with a multiplication map :''μ'': ''E'' ∧ ''E'' → ''E'' and a unit map : ''η'': ''S'' → ''E'', where ''S'' is the sphere spectrum. These maps have to satisfy associativity and unitality conditions up to homotopy, much in the same way as the multiplication of a ring is associative and unital. That is, : ''μ'' (id ∧ ''μ'') ∼ ''μ'' (''μ'' ∧ id) and : ''μ'' (id ∧ ''η'') ∼ id ∼ ''μ''(''η'' ∧ id). Examples of ring spectra include singular homology with coefficients in a ring, complex cobordism, K-theory, and Morava K-theory. See also *Highly structured ring spectrum In mathematics, a highly structured ring spectrum or A_\infty-ring is an object in homotopy theory encoding a refinement of a multiplicative structure on a cohomology theory. A commutative version of an A_\infty-ring is called an E_\infty-ring. W ... References * Algebraic topology Homotopy theo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Stable Homotopy Group Of Spheres
In the mathematical field of algebraic topology, the homotopy groups of spheres describe how spheres of various dimensions can wrap around each other. They are examples of topological invariants, which reflect, in algebraic terms, the structure of spheres viewed as topological spaces, forgetting about their precise geometry. Unlike homology groups, which are also topological invariants, the homotopy groups are surprisingly complex and difficult to compute. The -dimensional unit sphere — called the -sphere for brevity, and denoted as — generalizes the familiar circle () and the ordinary sphere (). The -sphere may be defined geometrically as the set of points in a Euclidean space of dimension located at a unit distance from the origin. The -th ''homotopy group'' summarizes the different ways in which the -dimensional sphere can be mapped continuously into the sphere . This summary does not distinguish between two mappings if one can be continuously deformed to the oth ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Differential Graded Algebra
In mathematics, in particular abstract algebra and topology, a differential graded algebra is a graded associative algebra with an added chain complex structure that respects the algebra structure. __TOC__ Definition A differential graded algebra (or DG-algebra for short) ''A'' is a graded algebra equipped with a map d\colon A \to A which has either degree 1 (cochain complex convention) or degree −1 (chain complex convention) that satisfies two conditions: A more succinct way to state the same definition is to say that a DG-algebra is a monoid object in the monoidal category of chain complexes. A DG morphism between DG-algebras is a graded algebra homomorphism which respects the differential ''d''. A differential graded augmented algebra (also called a DGA-algebra, an augmented DG-algebra or simply a DGA) is a DG-algebra equipped with a DG morphism to the ground ring (the terminology is due to Henri Cartan). ''Warning:'' some sources use the term ''DGA'' for a DG-alge ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bar Complex
In mathematics, the standard complex, also called standard resolution, bar resolution, bar complex, bar construction, is a way of constructing resolutions in homological algebra. It was first introduced for the special case of algebras over a commutative ring by and and has since been generalized in many ways. The name "bar complex" comes from the fact that used a vertical bar , as a shortened form of the tensor product \otimes in their notation for the complex. Definition If ''A'' is an associative algebra over a field ''K'', the standard complex is :\cdots\rightarrow A\otimes A\otimes A\rightarrow A\otimes A\rightarrow A \rightarrow 0\,, with the differential given by :d(a_0\otimes \cdots\otimes a_)=\sum_^n (-1)^i a_0\otimes\cdots\otimes a_ia_\otimes\cdots\otimes a_\,. If ''A'' is a unital ''K''-algebra, the standard complex is exact. Moreover, cdots\rightarrow A\otimes A\otimes A\rightarrow A\otimes A/math> is a free ''A''-bimodule resolution of the ''A''-bimodule ''A ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Homological Algebra
Homological algebra is the branch of mathematics that studies homology (mathematics), homology in a general algebraic setting. It is a relatively young discipline, whose origins can be traced to investigations in combinatorial topology (a precursor to algebraic topology) and abstract algebra (theory of module (mathematics), modules and Syzygy (mathematics), syzygies) at the end of the 19th century, chiefly by Henri Poincaré and David Hilbert. Homological algebra is the study of homological functors and the intricate algebraic structures that they entail; its development was closely intertwined with the emergence of category theory. A central concept is that of chain complexes, which can be studied through both their homology and cohomology. Homological algebra affords the means to extract information contained in these complexes and present it in the form of homological invariant (mathematics), invariants of ring (mathematics), rings, modules, topological spaces, and other 'tan ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
