Let k be a field, A an associativek-algebra, and M an A-bimodule. The enveloping algebra of A is the tensor product 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. 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
Hochschild complexedit
Let k be a ring, A an associativek-algebra that is a projective k-module, and M an A-bimodule. We will write for the n-fold tensor product of A over k. The chain complex that gives rise to Hochschild homology is given by
with boundary operator defined by
where is in A for all and . If we let
then , so is a chain complex called the Hochschild complex, and its homology is the Hochschild homology of A with coefficients in M. Henceforth, we will write as simply .
Hochschild homology is the homology of this simplicial module.
Relation with the Bar complexedit
There is a similar looking complex called the Bar complex which formally looks very similar to the Hochschild complex[1]pg 4-5. In fact, the Hochschild complex can be recovered from the Bar complex as
giving an explicit isomorphism.
As a derived self-intersectionedit
There's another useful interpretation of the Hochschild complex in the case of commutative rings, and more generally, for sheaves of commutative rings: it is constructed from the derived self-intersection of a scheme (or even derived scheme) over some base scheme . For example, we can form the derived fiber product
which has the sheaf of derived rings . Then, if embed with the diagonal map
the Hochschild complex is constructed as the pullback of the derived self intersection of the diagonal in the diagonal product scheme
From this interpretation, it should be clear the Hochschild homology should have some relation to the Kähler differentials since the Kähler differentials can be defined using a self-intersection from the diagonal, or more generally, the cotangent complex since this is the derived replacement for the Kähler differentials. We can recover the original definition of the Hochschild complex of a commutative -algebra by setting
If is a flat -algebra, then there's the chain of isomorphism
giving an alternative but equivalent presentation of the Hochschild complex.
Hochschild homology of functorsedit
The simplicial circle is a simplicial object in the category of finite pointed sets, i.e., a functor Thus, if F is a functor , we get a simplicial module by composing F with .
The homology of this simplicial module is the Hochschild homology of the functorF. The above definition of Hochschild homology of commutative algebras is the special case where F is the Loday functor.
Loday functoredit
A skeleton for the category of finite pointed sets is given by the objects
where 0 is the basepoint, and the morphisms are the basepoint preserving set maps. Let A be a commutative k-algebra and M be a symmetric A-bimodule[further explanation needed]. The Loday functor is given on objects in by
A morphism
is sent to the morphism given by
where
Another description of Hochschild homology of algebrasedit
The Hochschild homology of a commutative algebra A with coefficients in a symmetric A-bimodule M is the homology associated to the composition
and this definition agrees with the one above.
Examplesedit
The examples of Hochschild homology computations can be stratified into a number of distinct cases with fairly general theorems describing the structure of the homology groups and the homology ring for an associative algebra . For the case of commutative algebras, there are a number of theorems describing the computations over characteristic 0 yielding a straightforward understanding of what the homology and cohomology compute.
Commutative characteristic 0 caseedit
In the case of commutative algebras where , the Hochschild homology has two main theorems concerning smooth algebras, and more general non-flat algebras ; but, the second is a direct generalization of the first. In the smooth case, i.e. for a smooth algebra , the Hochschild-Kostant-Rosenberg theorem[2]pg 43-44 states there is an isomorphism
for every . This isomorphism can be described explicitly using the anti-symmetrization map. That is, a differential -form has the map
If the algebra isn't smooth, or even flat, then there is an analogous theorem using the cotangent complex. For a simplicial resolution , we set . Then, there exists a descending -filtration on whose graded pieces are isomorphic to
Note this theorem makes it accessible to compute the Hochschild homology not just for smooth algebras, but also for local complete intersection algebras. In this case, given a presentation for , the cotangent complex is the two-term complex .
Polynomial rings over the rationalsedit
One simple example is to compute the Hochschild homology of a polynomial ring of with -generators. The HKR theorem gives the isomorphism
where the algebra is the free antisymmetric algebra over in -generators. Its product structure is given by the wedge product of vectors, so
for .
Commutative characteristic p caseedit
In the characteristic p case, there is a userful counter-example to the Hochschild-Kostant-Rosenberg theorem which elucidates for the need of a theory beyond simplicial algebras for defining Hochschild homology. Consider the -algebra . We can compute a resolution of as the free differential graded algebras
giving the derived intersection where and the differential is the zero map. This is because we just tensor the complex above by , giving a formal complex with a generator in degree which squares to . Then, the Hochschild complex is given by
In order to compute this, we must resolve as an -algebra. Observe that the algebra structure
forces . This gives the degree zero term of the complex. Then, because we have to resolve the kernel , we can take a copy of shifted in degree and have it map to , with kernel in degree We can perform this recursively to get the underlying module of the divided power algebra
with and the degree of is , namely . Tensoring this algebra with over gives
since multiplied with any element in is zero. The algebra structure comes from general theory on divided power algebras and differential graded algebras.[3] Note this computation is seen as a technical artifact because the ring is not well behaved. For instance, . One technical response to this problem is through Topological Hochschild homology, where the base ring is replaced by the sphere spectrum.
The above construction of the Hochschild complex can be adapted to more general situations, namely by replacing the category of (complexes of) -modules by an ∞-category (equipped with a tensor product) , and by an associative algebra in this category. Applying this to the category of spectra, and being the Eilenberg–MacLane spectrum associated to an ordinary ring yields topological Hochschild homology, denoted . The (non-topological) Hochschild homology introduced above can be reinterpreted along these lines, by taking for the derived category of -modules (as an ∞-category).
Replacing tensor products over the sphere spectrum by tensor products over (or the Eilenberg–MacLane-spectrum ) leads to a natural comparison map . It induces an isomorphism on homotopy groups in degrees 0, 1, and 2. In general, however, they are different, and tends to yield simpler groups than HH. For example,
is the polynomial ring (with x in degree 2), compared to the ring of divided powers in one variable.
Dylan G.L. Allegretti, Differential Forms on Noncommutative Spaces. An elementary introduction to noncommutative geometry which uses Hochschild homology to generalize differential forms).
Ginzburg, Victor (2005). "Lectures on Noncommutative Geometry". arXiv:math/0506603.
Topological Hochschild homology in arithmetic geometry
Antieau, Benjamin; Bhatt, Bhargav; Mathew, Akhil (2019). "Counterexamples to Hochschild–Kostant–Rosenberg in characteristic p". arXiv:1909.11437 [math.AG].
Noncommutative caseedit
Richard, Lionel (2004). "Hochschild homology and cohomology of some classical and quantum noncommutative polynomial algebras". Journal of Pure and Applied Algebra. 187 (1–3): 255–294. arXiv:math/0207073. doi:10.1016/S0022-4049(03)00146-4.
Yashinski, Allan (2012). "The Gauss-Manin connection and noncommutative tori". arXiv:1210.4531 [math.KT].
January 01, 1970
hochschild, homology, mathematics, cohomology, homology, theory, associative, algebras, over, rings, there, also, theory, certain, functors, hochschild, cohomology, introduced, gerhard, hochschild, 1945, algebras, over, field, extended, algebras, over, more, g. 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 Gerhard Hochschild 1945 for algebras over a field and extended to algebras over more general rings by Henri Cartan and Samuel Eilenberg 1956 Contents 1 Definition of Hochschild homology of algebras 1 1 Hochschild complex 1 2 Remark 1 3 Relation with the Bar complex 1 4 As a derived self intersection 2 Hochschild homology of functors 2 1 Loday functor 2 2 Another description of Hochschild homology of algebras 3 Examples 3 1 Commutative characteristic 0 case 3 1 1 Polynomial rings over the rationals 3 2 Commutative characteristic p case 4 Topological Hochschild homology 5 See also 6 References 7 External links 7 1 Introductory articles 7 2 Commutative case 7 3 Noncommutative caseDefinition of Hochschild homology of algebras editLet 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 A o displaystyle A e A otimes A o nbsp 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 Cartan amp 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 H H n A M Tor n A e A M displaystyle HH n A M operatorname Tor n A e A M nbsp H H n A M Ext A e n A M displaystyle HH n A M operatorname Ext A e n A M nbsp Hochschild complex edit Let k be a ring A an associative k algebra that is a projective k module and M an A bimodule We will write A n displaystyle A otimes n nbsp for the n fold tensor product of A over k The chain complex that gives rise to Hochschild homology is given by C n A M M A n displaystyle C n A M M otimes A otimes n nbsp with boundary operator d i displaystyle d i nbsp defined by d 0 m a 1 a n m a 1 a 2 a n d i m a 1 a n m a 1 a i a i 1 a n d n m a 1 a n a n m a 1 a n 1 displaystyle begin aligned d 0 m otimes a 1 otimes cdots otimes a n amp ma 1 otimes a 2 cdots otimes a n d i m otimes a 1 otimes cdots otimes a n amp m otimes a 1 otimes cdots otimes a i a i 1 otimes cdots otimes a n d n m otimes a 1 otimes cdots otimes a n amp a n m otimes a 1 otimes cdots otimes a n 1 end aligned nbsp where a i displaystyle a i nbsp is in A for all 1 i n displaystyle 1 leq i leq n nbsp and m M displaystyle m in M nbsp If we let b n i 0 n 1 i d i displaystyle b n sum i 0 n 1 i d i nbsp then b n 1 b n 0 displaystyle b n 1 circ b n 0 nbsp so C n A M b n displaystyle C n A M b n nbsp is a chain complex called the Hochschild complex and its homology is the Hochschild homology of A with coefficients in M Henceforth we will write b n displaystyle b n nbsp as simply b displaystyle b nbsp Remark edit The maps d i displaystyle d i nbsp are face maps making the family of modules C n A M b displaystyle C n A M b nbsp a simplicial object in the category of k modules i e a functor Do k mod where D is the simplex category and k mod is the category of k modules Here Do is the opposite category of D The degeneracy maps are defined by s i a 0 a n a 0 a i 1 a i 1 a n displaystyle s i a 0 otimes cdots otimes a n a 0 otimes cdots otimes a i otimes 1 otimes a i 1 otimes cdots otimes a n nbsp Hochschild homology is the homology of this simplicial module Relation with the Bar complex edit There is a similar looking complex B A k displaystyle B A k nbsp called the Bar complex which formally looks very similar to the Hochschild complex 1 pg 4 5 In fact the Hochschild complex H H A k displaystyle HH A k nbsp can be recovered from the Bar complex asH H A k A A A o p B A k displaystyle HH A k cong A otimes A otimes A op B A k nbsp giving an explicit isomorphism As a derived self intersection edit There s another useful interpretation of the Hochschild complex in the case of commutative rings and more generally for sheaves of commutative rings it is constructed from the derived self intersection of a scheme or even derived scheme X displaystyle X nbsp over some base scheme S displaystyle S nbsp For example we can form the derived fiber productX S L X displaystyle X times S mathbf L X nbsp which has the sheaf of derived rings O X O S L O X displaystyle mathcal O X otimes mathcal O S mathbf L mathcal O X nbsp Then if embed X displaystyle X nbsp with the diagonal mapD X X S L X displaystyle Delta X to X times S mathbf L X nbsp the Hochschild complex is constructed as the pullback of the derived self intersection of the diagonal in the diagonal product schemeH H X S D O X O X O S L O X L O X displaystyle HH X S Delta mathcal O X otimes mathcal O X otimes mathcal O S mathbf L mathcal O X mathbf L mathcal O X nbsp From this interpretation it should be clear the Hochschild homology should have some relation to the Kahler differentials W X S displaystyle Omega X S nbsp since the Kahler differentials can be defined using a self intersection from the diagonal or more generally the cotangent complex L X S displaystyle mathbf L X S bullet nbsp since this is the derived replacement for the Kahler differentials We can recover the original definition of the Hochschild complex of a commutative k displaystyle k nbsp algebra A displaystyle A nbsp by settingS Spec k displaystyle S text Spec k nbsp and X Spec A displaystyle X text Spec A nbsp Then the Hochschild complex is quasi isomorphic toH H A k q i s o A A k L A L A displaystyle HH A k simeq qiso A otimes A otimes k mathbf L A mathbf L A nbsp If A displaystyle A nbsp is a flat k displaystyle k nbsp algebra then there s the chain of isomorphismA k L A A k A A k A o p displaystyle A otimes k mathbf L A cong A otimes k A cong A otimes k A op nbsp giving an alternative but equivalent presentation of the Hochschild complex Hochschild homology of functors editThe simplicial circle S 1 displaystyle S 1 nbsp is a simplicial object in the category Fin displaystyle operatorname Fin nbsp of finite pointed sets i e a functor D o Fin displaystyle Delta o to operatorname Fin nbsp Thus if F is a functor F Fin k m o d displaystyle F colon operatorname Fin to k mathrm mod nbsp we get a simplicial module by composing F with S 1 displaystyle S 1 nbsp D o S 1 Fin F k mod displaystyle Delta o overset S 1 longrightarrow operatorname Fin overset F longrightarrow k text mod nbsp 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 Loday functor edit A skeleton for the category of finite pointed sets is given by the objects n 0 1 n displaystyle n 0 1 ldots n nbsp where 0 is the basepoint and the morphisms are the basepoint preserving set maps Let A be a commutative k algebra and M be a symmetric A bimodule further explanation needed The Loday functor L A M displaystyle L A M nbsp is given on objects in Fin displaystyle operatorname Fin nbsp by n M A n displaystyle n mapsto M otimes A otimes n nbsp A morphism f m n displaystyle f m to n nbsp is sent to the morphism f displaystyle f nbsp given by f a 0 a m b 0 b n displaystyle f a 0 otimes cdots otimes a m b 0 otimes cdots otimes b n nbsp where j 0 n b j i f 1 j a i f 1 j 1 f 1 j displaystyle forall j in 0 ldots n qquad b j begin cases prod i in f 1 j a i amp f 1 j neq emptyset 1 amp f 1 j emptyset end cases nbsp Another description of Hochschild homology of algebras edit The Hochschild homology of a commutative algebra A with coefficients in a symmetric A bimodule M is the homology associated to the composition D o S 1 Fin L A M k mod displaystyle Delta o overset S 1 longrightarrow operatorname Fin overset mathcal L A M longrightarrow k text mod nbsp and this definition agrees with the one above Examples editThe examples of Hochschild homology computations can be stratified into a number of distinct cases with fairly general theorems describing the structure of the homology groups and the homology ring H H A displaystyle HH A nbsp for an associative algebra A displaystyle A nbsp For the case of commutative algebras there are a number of theorems describing the computations over characteristic 0 yielding a straightforward understanding of what the homology and cohomology compute Commutative characteristic 0 case edit In the case of commutative algebras A k displaystyle A k nbsp where Q k displaystyle mathbb Q subseteq k nbsp the Hochschild homology has two main theorems concerning smooth algebras and more general non flat algebras A displaystyle A nbsp but the second is a direct generalization of the first In the smooth case i e for a smooth algebra A displaystyle A nbsp the Hochschild Kostant Rosenberg theorem 2 pg 43 44 states there is an isomorphismW A k n H H n A k displaystyle Omega A k n cong HH n A k nbsp for every n 0 displaystyle n geq 0 nbsp This isomorphism can be described explicitly using the anti symmetrization map That is a differential n displaystyle n nbsp form has the mapa d b 1 d b n s S n sign s a b s 1 b s n displaystyle a db 1 wedge cdots wedge db n mapsto sum sigma in S n operatorname sign sigma a otimes b sigma 1 otimes cdots otimes b sigma n nbsp If the algebra A k displaystyle A k nbsp isn t smooth or even flat then there is an analogous theorem using the cotangent complex For a simplicial resolution P A displaystyle P bullet to A nbsp we set L A k i W P k i P A displaystyle mathbb L A k i Omega P bullet k i otimes P bullet A nbsp Then there exists a descending N displaystyle mathbb N nbsp filtration F displaystyle F bullet nbsp on H H n A k displaystyle HH n A k nbsp whose graded pieces are isomorphic to F i F i 1 L A k i i displaystyle frac F i F i 1 cong mathbb L A k i i nbsp Note this theorem makes it accessible to compute the Hochschild homology not just for smooth algebras but also for local complete intersection algebras In this case given a presentation A R I displaystyle A R I nbsp for R k x 1 x n displaystyle R k x 1 dotsc x n nbsp the cotangent complex is the two term complex I I 2 W R k 1 k A displaystyle I I 2 to Omega R k 1 otimes k A nbsp Polynomial rings over the rationals edit One simple example is to compute the Hochschild homology of a polynomial ring of Q displaystyle mathbb Q nbsp with n displaystyle n nbsp generators The HKR theorem gives the isomorphismH H Q x 1 x n Q x 1 x n L d x 1 d x n displaystyle HH mathbb Q x 1 ldots x n mathbb Q x 1 ldots x n otimes Lambda dx 1 dotsc dx n nbsp where the algebra d x 1 d x n displaystyle bigwedge dx 1 ldots dx n nbsp is the free antisymmetric algebra over Q displaystyle mathbb Q nbsp in n displaystyle n nbsp generators Its product structure is given by the wedge product of vectors so d x i d x j d x j d x i d x i d x i 0 displaystyle begin aligned dx i cdot dx j amp dx j cdot dx i dx i cdot dx i amp 0 end aligned nbsp for i j displaystyle i neq j nbsp Commutative characteristic p case edit In the characteristic p case there is a userful counter example to the Hochschild Kostant Rosenberg theorem which elucidates for the need of a theory beyond simplicial algebras for defining Hochschild homology Consider the Z displaystyle mathbb Z nbsp algebra F p displaystyle mathbb F p nbsp We can compute a resolution of F p displaystyle mathbb F p nbsp as the free differential graded algebrasZ p Z displaystyle mathbb Z xrightarrow cdot p mathbb Z nbsp giving the derived intersection F p Z L F p F p e e 2 displaystyle mathbb F p otimes mathbb Z mathbf L mathbb F p cong mathbb F p varepsilon varepsilon 2 nbsp where deg e 1 displaystyle text deg varepsilon 1 nbsp and the differential is the zero map This is because we just tensor the complex above by F p displaystyle mathbb F p nbsp giving a formal complex with a generator in degree 1 displaystyle 1 nbsp which squares to 0 displaystyle 0 nbsp Then the Hochschild complex is given byF p F p Z L F p L F p displaystyle mathbb F p otimes mathbb F p otimes mathbb Z mathbb L mathbb F p mathbb L mathbb F p nbsp In order to compute this we must resolve F p displaystyle mathbb F p nbsp as an F p Z L F p displaystyle mathbb F p otimes mathbb Z mathbf L mathbb F p nbsp algebra Observe that the algebra structure F p e e 2 F p displaystyle mathbb F p varepsilon varepsilon 2 to mathbb F p nbsp forces e 0 displaystyle varepsilon mapsto 0 nbsp This gives the degree zero term of the complex Then because we have to resolve the kernel e F p Z L F p displaystyle varepsilon cdot mathbb F p otimes mathbb Z mathbf L mathbb F p nbsp we can take a copy of F p Z L F p displaystyle mathbb F p otimes mathbb Z mathbf L mathbb F p nbsp shifted in degree 2 displaystyle 2 nbsp and have it map to e F p Z L F p displaystyle varepsilon cdot mathbb F p otimes mathbb Z mathbf L mathbb F p nbsp with kernel in degree 3 displaystyle 3 nbsp e F p Z L F p Ker F p Z L F p e F p Z L F p displaystyle varepsilon cdot mathbb F p otimes mathbb Z mathbf L mathbb F p text Ker displaystyle mathbb F p otimes mathbb Z mathbf L mathbb F p to displaystyle varepsilon cdot mathbb F p otimes mathbb Z mathbf L mathbb F p nbsp We can perform this recursively to get the underlying module of the divided power algebra F p Z L F p x F p Z L F p x 1 x 2 x i x j i j i x i j displaystyle mathbb F p otimes mathbb Z mathbf L mathbb F p langle x rangle frac mathbb F p otimes mathbb Z mathbf L mathbb F p x 1 x 2 ldots x i x j binom i j i x i j nbsp with d x i e x i 1 displaystyle dx i varepsilon cdot x i 1 nbsp and the degree of x i displaystyle x i nbsp is 2 i displaystyle 2i nbsp namely x i 2 i displaystyle x i 2i nbsp Tensoring this algebra with F p displaystyle mathbb F p nbsp over F p Z L F p displaystyle mathbb F p otimes mathbb Z mathbf L mathbb F p nbsp givesH H F p F p x displaystyle HH mathbb F p mathbb F p langle x rangle nbsp since e displaystyle varepsilon nbsp multiplied with any element in F p displaystyle mathbb F p nbsp is zero The algebra structure comes from general theory on divided power algebras and differential graded algebras 3 Note this computation is seen as a technical artifact because the ring F p x displaystyle mathbb F p langle x rangle nbsp is not well behaved For instance x p 0 displaystyle x p 0 nbsp One technical response to this problem is through Topological Hochschild homology where the base ring Z displaystyle mathbb Z nbsp is replaced by the sphere spectrum S displaystyle mathbb S nbsp Topological Hochschild homology editMain article Topological Hochschild homology The above construction of the Hochschild complex can be adapted to more general situations namely by replacing the category of complexes of k displaystyle k nbsp modules by an category equipped with a tensor product C displaystyle mathcal C nbsp and A displaystyle A nbsp by an associative algebra in this category Applying this to the category C Spectra displaystyle mathcal C textbf Spectra nbsp of spectra and A displaystyle A nbsp being the Eilenberg MacLane spectrum associated to an ordinary ring R displaystyle R nbsp yields topological Hochschild homology denoted T H H R displaystyle THH R nbsp The non topological Hochschild homology introduced above can be reinterpreted along these lines by taking for C D Z displaystyle mathcal C D mathbb Z nbsp the derived category of Z displaystyle mathbb Z nbsp modules as an category Replacing tensor products over the sphere spectrum by tensor products over Z displaystyle mathbb Z nbsp or the Eilenberg MacLane spectrum H Z displaystyle H mathbb Z nbsp leads to a natural comparison map T H H R H H R displaystyle THH R to HH R nbsp It induces an isomorphism on homotopy groups in degrees 0 1 and 2 In general however they are different and T H H displaystyle THH nbsp tends to yield simpler groups than HH For example T H H F p F p x displaystyle THH mathbb F p mathbb F p x nbsp H H F p F p x displaystyle HH mathbb F p mathbb F p langle x rangle nbsp is the polynomial ring with x in degree 2 compared to the ring of divided powers in one variable Lars Hesselholt 2016 showed that the Hasse Weil zeta function of a smooth proper variety over F p displaystyle mathbb F p nbsp can be expressed using regularized determinants involving topological Hochschild homology See also editCyclic homologyReferences edit Morrow Matthew Topological Hochschild homology in arithmetic geometry PDF Archived PDF from the original on 24 Dec 2020 Ginzburg Victor 2005 06 29 Lectures on Noncommutative Geometry arXiv math 0506603 Section 23 6 09PF Tate resolutions The Stacks project stacks math columbia edu Retrieved 2020 12 31 Cartan Henri Eilenberg Samuel 1956 Homological algebra Princeton Mathematical Series vol 19 Princeton University Press ISBN 978 0 691 04991 5 MR 0077480 Govorov V E Mikhalev A V 2001 1994 Cohomology of algebras Encyclopedia of Mathematics EMS Press Hesselholt Lars 2016 Topological Hochschild homology and the Hasse Weil zeta function Contemporary Mathematics vol 708 pp 157 180 arXiv 1602 01980 doi 10 1090 conm 708 14264 ISBN 9781470429119 S2CID 119145574 Hochschild Gerhard 1945 On the cohomology groups of an associative algebra Annals of Mathematics Second Series 46 1 58 67 doi 10 2307 1969145 ISSN 0003 486X JSTOR 1969145 MR 0011076 Jean Louis Loday Cyclic Homology Grundlehren der mathematischen Wissenschaften Vol 301 Springer 1998 ISBN 3 540 63074 0 Richard S Pierce Associative Algebras Graduate Texts in Mathematics 88 Springer 1982 Pirashvili Teimuraz 2000 Hodge decomposition for higher order Hochschild homology Annales Scientifiques de l Ecole Normale Superieure 33 2 151 179 doi 10 1016 S0012 9593 00 00107 5 External links editIntroductory articles edit Dylan G L Allegretti Differential Forms on Noncommutative Spaces An elementary introduction to noncommutative geometry which uses Hochschild homology to generalize differential forms Ginzburg Victor 2005 Lectures on Noncommutative Geometry arXiv math 0506603 Topological Hochschild homology in arithmetic geometry Hochschild cohomology at the nLab Commutative case edit Antieau Benjamin Bhatt Bhargav Mathew Akhil 2019 Counterexamples to Hochschild Kostant Rosenberg in characteristic p arXiv 1909 11437 math AG Noncommutative case edit Richard Lionel 2004 Hochschild homology and cohomology of some classical and quantum noncommutative polynomial algebras Journal of Pure and Applied Algebra 187 1 3 255 294 arXiv math 0207073 doi 10 1016 S0022 4049 03 00146 4 Quddus Safdar 2020 Non commutative Poisson Structures on quantum torus orbifolds arXiv 2006 00495 math KT Yashinski Allan 2012 The Gauss Manin connection and noncommutative tori arXiv 1210 4531 math KT Retrieved from https en wikipedia org w index php title Hochschild homology amp oldid 1184103076, wikipedia, wiki, book, books, library,