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| In [[mathematics]], '''Hochschild homology (and cohomology)''' is a [[homology theory]] for [[associative]] [[algebra (ring theory)|algebras]] over [[ring (mathematics)|rings]]. There is also a theory for Hochschild homology of certain [[functor]]s. Hochschild cohomology was introduced by {{harvs|txt|authorlink= Gerhard Hochschild|first=Gerhard |last= Hochschild |year=1945}} for algebras over a [[field (mathematics)|field]], and extended to algebras over more general rings by {{harvtxt|Cartan|Eilenberg|1956}}.
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| ==Definition of Hochschild homology of algebras==
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| Let ''k'' be a ring, ''A'' an [[associative]] ''k''-[[algebra (ring theory)|algebra]], and ''M'' an ''A''-[[bimodule]]. The enveloping algebra of ''A'' is the tensor product ''A<sup>e</sup>''=''A''⊗''A''<sup>o</sup> of ''A'' with its [[opposite ring|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 ''A<sup>e</sup>''-modules. {{harvtxt|Cartan|Eilenberg|1956}} defined the Hochschild homology and cohomology group of ''A'' with coefficients in ''M'' in terms of the [[Tor functor]] and [[Ext functor]] by
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| :<math> HH_n(A,M) = \text{Tor}_n^{A^e}(A, M)</math> | |
| :<math> HH^n(A,M) = \text{Ext}^n_{A^e}(A, M)</math>
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| ===Hochschild complex===
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| Let ''k'' be a ring, ''A'' an [[associative]] ''k''-[[algebra (ring theory)|algebra]] that is a projective ''k''-module, and ''M'' an ''A''-[[bimodule]]. We will write ''A''<sup>⊗''n'' </sup> for the ''n''-fold [[tensor product]] of ''A'' over ''k''. The [[chain complex]] that gives rise to Hochschild homology is given by
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| :<math> C_n(A,M) := M \otimes A^{\otimes n} </math>
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| with boundary operator ''d''<sub>''i''</sub> defined by
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| :<math> d_0(m\otimes a_1 \otimes \cdots \otimes a_n) = ma_1 \otimes a_2 \cdots \otimes a_n </math>
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| :<math> d_i(m\otimes a_1 \otimes \cdots \otimes a_n) = m\otimes a_1 \otimes \cdots \otimes a_i a_{i+1} \otimes \cdots \otimes a_n </math>
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| :<math> d_n(m\otimes a_1 \otimes \cdots \otimes a_n) = a_n m\otimes a_1 \otimes \cdots \otimes a_{n-1} </math>
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| Here ''a''<sub>''i''</sub> is in ''A'' for all 1 ≤ ''i'' ≤ ''n'' and ''m'' ∈ ''M''. If we let
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| :<math> b=\sum_{i=0}^n (-1)^i d_i, </math>
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| then ''b'' ° ''b'' = 0, so (''C''<sub>''n''</sub>(''A'',''M''), ''b'') is a [[chain complex]] called the '''Hochschild complex''', and its homology is the '''Hochschild homology''' of ''A'' with coefficients in ''M''.
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| ===Remark===
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| The maps ''d''<sub>i</sub> are [[face map]]s making the family of [[module (mathematics)|modules]] ''C''<sub>''n''</sub>(''A'',''M'') a [[Face map#Simplicial objects|simplicial object]] in the [[category (mathematics)|category]] of ''k''-modules, i.e. a functor Δ<sup>o</sup> → ''k''-mod, where ''Δ'' is the [[simplicial category]] and ''k''-mod is the category of ''k''-modules. Here Δ<sup>o</sup> is the [[Dual (category theory)|opposite category]] of Δ. The [[degeneracy map]]s are defined by ''s''<sub>''i''</sub>(''a''<sub>0</sub> ⊗ ··· ⊗ ''a''<sub>''n''</sub>) = ''a''<sub>0</sub> ⊗ ··· ''a''<sub>i</sub> ⊗ 1 ⊗ ''a''<sub>''i''+1</sub> ⊗ ··· ⊗ ''a''<sub>''n''</sub>. Hochschild homology is the homology of this simplicial module.
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| ==Hochschild homology of functors==
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| The [[simplicial circle]] ''S''<sup>1</sup> is a simplicial object in the category ''Fin<sub>*</sub>'' of finite pointed sets, i.e. a functor Δ<sup>o</sup> → ''Fin<sub>*</sub>''. Thus, if F is a functor ''F'': ''Fin'' → ''k''-mod, we get a simplicial module by composing F with ''S''<sup>1</sup>
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| :<math> \Delta^o \overset{S^1}{\longrightarrow} \text{Fin}_* \overset{F}{\longrightarrow} k\text{-}\operatorname{mod}.</math>
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| The homology of this simplicial module is the '''Hochschild homology of the functor''' ''F''. The above definition of Hochschild homology of commutative algebras is the special case where ''F'' is the '''Loday functor'''.
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| ===Loday functor===
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| A [[skeleton (category theory)|skeleton]] for the category of finite pointed sets is given by the objects
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| :<math> n_+ = \{0,1,\dots,n\}, \, </math>
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| where 0 is the basepoint, and the [[morphism (category theory)|morphisms]] are the basepoint preserving set maps. Let ''A'' be a commutative k-algebra and ''M'' be a symmetric ''A''-bimodule{{Elucidate|date=March 2012}}. The Loday functor ''L(A,M)'' is given on objects in ''Fin<sub>*</sub>'' by
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| :<math> n_+ \mapsto M \otimes A^{\otimes n}. \, </math>
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| A morphism
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| :<math>f:m_+ \rightarrow n_+</math>
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| is sent to the morphism f<sub>*</sub> given by
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| :<math> f_*(a_0 \otimes \cdots \otimes a_n) = (b_0 \otimes \cdots \otimes b_m) </math>
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| where
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| :<math> b_j = \prod_{f(i)=j} a_i, \,\, j=0,\dots,n, </math>
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| and ''b''<sub>''j''</sub> = 1 if f<sup> −1</sup>(''j'') = ∅. | |
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| ===Another description of Hochschild homology of algebras===
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| The Hochschild homology of a commutative algebra ''A'' with coefficients in a symmetric ''A''-bimodule ''M'' is the homology associated to the composition
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| :<math> \Delta^o \overset{S^1}{\longrightarrow} \text{Fin}_* \overset{\mathcal{L}(A,M)}{\longrightarrow} k\text{-}\operatorname{mod}, </math>
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| and this definition agrees with the one above. | |
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| ==See also==
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| *[[Cyclic homology]]
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| ==References==
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| *{{Citation | last1=Cartan | first1=Henri | last2=Eilenberg | first2=Samuel | author2-link=Samuel Eilenberg | title=Homological algebra | url=http://books.google.com/books?id=0268b52ghcsC | publisher=[[Princeton University Press]] | series=Princeton Mathematical Series | isbn=978-0-691-04991-5 | mr=0077480 | year=1956 | volume=19}}
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| *{{eom|id=C/c023110|title=Cohomology of algebras|first=V.E.|last= Govorov|first2=A.V. |last2=Mikhalev}}
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| *{{Citation | last1=Hochschild | first1=G. | title=On the cohomology groups of an associative algebra | jstor=1969145 | mr=0011076 | year=1945 | journal=[[Annals of Mathematics|Annals of Mathematics. Second Series]] | issn=0003-486X | volume=46 | pages=58–67}}
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| *Jean-Louis Loday, ''Cyclic Homology'', Grundlehren der mathematischen Wissenschaften Vol. 301, Springer (1998) ISBN 3-540-63074-0
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| *Richard S. Pierce, ''Associative Algebras'', Graduate Texts in Mathematics (88), Springer, 1982.
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| *Teimuraz Pirashvili, [http://www.numdam.org/item?id=ASENS_2000_4_33_2_151_0 Hodge decomposition for higher order Hochschild homology]
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| ==External links==
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| * Dylan G.L. Allegretti, [http://www.math.uchicago.edu/~may/VIGRE/VIGRE2009/REUPapers/Allegretti.pdf ''Differential Forms on Noncommutative Spaces'']. An elementary introduction to [[noncommutative geometry]] which uses Hochschild homology to generalize differential forms).
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| *{{nlab|id=Hochschild+cohomology|title=Hochschild cohomology}}
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| [[Category:Ring theory]]
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| [[Category:Homological algebra]]
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Hi there, I am Andrew Berryhill. To play lacross is the factor I love psychic readings most of all. She works as a journey agent but quickly she'll be on her personal. My spouse and I reside in Mississippi and I adore each working day residing right here.