Theory for associative algebras over rings
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).
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
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
![{\displaystyle HH_{n}(A,M)=\operatorname {Tor} _{n}^{A^{e}}(A,M)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e0d685811e2cec4a9c875e90ca288c5ad912f2cc)
![{\displaystyle HH^{n}(A,M)=\operatorname {Ext} _{A^{e}}^{n}(A,M)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6a3b57fe86491d3acb4668fab0072d74512b3ac6)
Hochschild complex
Let k be a ring, A an associative k-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
![{\displaystyle C_{n}(A,M):=M\otimes A^{\otimes n}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f11215c23c5d3128a3aa5c48cfce9b5b7de19afc)
with boundary operator
defined by
![{\displaystyle {\begin{aligned}d_{0}(m\otimes a_{1}\otimes \cdots \otimes a_{n})&=ma_{1}\otimes a_{2}\cdots \otimes a_{n}\\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}\\d_{n}(m\otimes a_{1}\otimes \cdots \otimes a_{n})&=a_{n}m\otimes a_{1}\otimes \cdots \otimes a_{n-1}\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/aa335bdc7f9dbe409d26158e7f07271e1ac014e1)
where
is in A for all
and
. If we let
![{\displaystyle b_{n}=\sum _{i=0}^{n}(-1)^{i}d_{i},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7e7da62a86b7036155865ddc606915bfb62836e3)
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
.
The maps
are face maps making the family of modules
a simplicial object in the category of k-modules, i.e., a functor Δo → k-mod, where Δ is the simplex category and k-mod is the category of k-modules. Here Δo is the opposite category of Δ. The degeneracy maps are defined by
![{\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}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/961273d3edcfc084d36e87ad5a8e1e66f44ea1f4)
Hochschild homology is the homology of this simplicial module.
Relation with the Bar complex
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
![{\displaystyle HH(A/k)\cong A\otimes _{A\otimes A^{op}}B(A/k)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/82bbde9333892cff86b4f2066cd56bcc5eddf64f)
giving an explicit isomorphism.
As a derived self-intersection
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
![{\displaystyle X\times _{S}^{\mathbf {L} }X}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f236ea966b2bf2a02d203372bb3307bf1b0e8c9e)
which has the sheaf of derived rings
![{\displaystyle {\mathcal {O}}_{X}\otimes _{{\mathcal {O}}_{S}}^{\mathbf {L} }{\mathcal {O}}_{X}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4ed2e5a7b6361ccf8620edc81d29050723cc1230)
. Then, if embed
![{\displaystyle X}](https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab)
with the diagonal map
![{\displaystyle \Delta :X\to X\times _{S}^{\mathbf {L} }X}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c95ecf00b43675e9b328e45757dcddbf25946e2d)
the Hochschild complex is constructed as the pullback of the derived self intersection of the diagonal in the diagonal product scheme
![{\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})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/913fff5c9d40133491b813aa55a7994966c5b518)
From this interpretation, it should be clear the Hochschild homology should have some relation to the Kähler differentials
![{\displaystyle \Omega _{X/S}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b403d4143c562a13c19a6f57a9870d8f9f12b8f9)
since the
Kähler differentials can be defined using a self-intersection from the diagonal, or more generally, the
cotangent complex ![{\displaystyle \mathbf {L} _{X/S}^{\bullet }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d54156e468d57e14f8a16d524bb794b355a518c7)
since this is the derived replacement for the Kähler differentials. We can recover the original definition of the Hochschild complex of a commutative
![{\displaystyle k}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40)
-algebra
![{\displaystyle A}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3)
by setting
![{\displaystyle S={\text{Spec}}(k)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/efe4d3c3e4fad420bae18dada75fe1cc58ab9d65)
and
![{\displaystyle X={\text{Spec}}(A)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/66ca672b2931d5610ec73627859cc89c726b5e57)
Then, the Hochschild complex is
quasi-isomorphic to
![{\displaystyle HH(A/k)\simeq _{qiso}A\otimes _{A\otimes _{k}^{\mathbf {L} }A}^{\mathbf {L} }A}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6fa0de3d92410a6d5f70f2cc9751d49fc0b91547)
If
![{\displaystyle A}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3)
is a flat
![{\displaystyle k}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40)
-algebra, then there's the chain of isomorphism
![{\displaystyle A\otimes _{k}^{\mathbf {L} }A\cong A\otimes _{k}A\cong A\otimes _{k}A^{op}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/65384484fd7cd253d05a4aba22fdd66a6252ab80)
giving an alternative but equivalent presentation of the Hochschild complex.
Hochschild homology of functors
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
.
![{\displaystyle \Delta ^{o}{\overset {S^{1}}{\longrightarrow }}\operatorname {Fin} _{*}{\overset {F}{\longrightarrow }}k{\text{-mod}}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0e51913f29b9e35269d7c2295588a8504f1ae21d)
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
A skeleton for the category of finite pointed sets is given by the objects
![{\displaystyle n_{+}=\{0,1,\ldots ,n\},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b0f222982837beb98f77fbc94b106eff8e01d0de)
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
![{\displaystyle n_{+}\mapsto M\otimes A^{\otimes n}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/269ce07a4ca1ccae5819bd0cad0784088e77164b)
A morphism
![{\displaystyle f:m_{+}\to n_{+}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/862ec76f60dfa4cd123cc7f7b7644b06c709a1f7)
is sent to the morphism
given by
![{\displaystyle f_{*}(a_{0}\otimes \cdots \otimes a_{m})=b_{0}\otimes \cdots \otimes b_{n}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ec4b4ab850d6c386bdc38821f067beded1380d2c)
where
![{\displaystyle \forall j\in \{0,\ldots ,n\}:\qquad b_{j}={\begin{cases}\prod _{i\in f^{-1}(j)}a_{i}&f^{-1}(j)\neq \emptyset \\1&f^{-1}(j)=\emptyset \end{cases}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/77a31182689c3d294f7bd9f044c9ee8203880fc6)
Another description of Hochschild homology of algebras
The Hochschild homology of a commutative algebra A with coefficients in a symmetric A-bimodule M is the homology associated to the composition
![{\displaystyle \Delta ^{o}{\overset {S^{1}}{\longrightarrow }}\operatorname {Fin} _{*}{\overset {{\mathcal {L}}(A,M)}{\longrightarrow }}k{\text{-mod}},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a0b28ef3487aa7ac46d971f445d6c73c95bb8e44)
and this definition agrees with the one above.
Examples
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 case
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
![{\displaystyle \Omega _{A/k}^{n}\cong HH_{n}(A/k)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6f0ca7dcbdec0cdc527cf9ce7f4860b0e99e3156)
for every
![{\displaystyle n\geq 0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ce8a1b7b3bc3c790054d93629fc3b08cd1da1fd0)
. This isomorphism can be described explicitly using the anti-symmetrization map. That is, a differential
![{\displaystyle n}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b)
-form has the map
![{\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)}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/773daddba60ac3fd328cd78ab057c3db8c264963)
If the algebra
![{\displaystyle A/k}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0f88648bf6836b4507748dc2dec3a15ad2e660de)
isn't smooth, or even flat, then there is an analogous theorem using the
cotangent complex. For a simplicial resolution
![{\displaystyle P_{\bullet }\to A}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c0756a3392c0f7ccd12cb39b7662af8ed03ce200)
, we set
![{\displaystyle \mathbb {L} _{A/k}^{i}=\Omega _{P_{\bullet }/k}^{i}\otimes _{P_{\bullet }}A}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e910135cb4377638cae345b3a13e88958e7b5cc5)
. Then, there exists a descending
![{\displaystyle \mathbb {N} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/fdf9a96b565ea202d0f4322e9195613fb26a9bed)
-filtration
![{\displaystyle F_{\bullet }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b5f2063b38d24eca838e16288057850e953c26d1)
on
![{\displaystyle HH_{n}(A/k)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3075836748770b4289795a109aad1be8d15bb3a8)
whose graded pieces are isomorphic to
![{\displaystyle {\frac {F_{i}}{F_{i+1}}}\cong \mathbb {L} _{A/k}^{i}[+i].}](https://wikimedia.org/api/rest_v1/media/math/render/svg/db3cf3a3b21350e12af7ebd41db2fb635f10575b)
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
![{\displaystyle A=R/I}](https://wikimedia.org/api/rest_v1/media/math/render/svg/45aaf590840d376315df53d1d4ce77b04f3d6e0b)
for
![{\displaystyle R=k[x_{1},\dotsc ,x_{n}]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d1f762caf2a5417370bc67f31d6b1882cc197822)
, the cotangent complex is the two-term complex
![{\displaystyle I/I^{2}\to \Omega _{R/k}^{1}\otimes _{k}A}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ef663a1f830d046d870fbd035c9dfb05f9dc022c)
.
Polynomial rings over the rationals
One simple example is to compute the Hochschild homology of a polynomial ring of
with
-generators. The HKR theorem gives the isomorphism
![{\displaystyle HH_{*}(\mathbb {Q} [x_{1},\ldots ,x_{n}])=\mathbb {Q} [x_{1},\ldots ,x_{n}]\otimes \Lambda (dx_{1},\dotsc ,dx_{n})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2169c3e27f491a4c43c9c0ba1066c303c88607a5)
where the algebra
![{\displaystyle \bigwedge (dx_{1},\ldots ,dx_{n})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f7bcd6e0eb94ab17ca919ab16129117ea0eab12d)
is the free antisymmetric algebra over
![{\displaystyle \mathbb {Q} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/c5909f0b54e4718fa24d5fd34d54189d24a66e9a)
in
![{\displaystyle n}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b)
-generators. Its product structure is given by the
wedge product of vectors, so
![{\displaystyle {\begin{aligned}dx_{i}\cdot dx_{j}&=-dx_{j}\cdot dx_{i}\\dx_{i}\cdot dx_{i}&=0\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/52325fcad69f94ab92f9dd02fd0d87f780d6d41e)
for
![{\displaystyle i\neq j}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d95aeb406bb427ac96806bc00c30c91d31b858be)
.
Commutative characteristic p case
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
![{\displaystyle \mathbb {Z} \xrightarrow {\cdot p} \mathbb {Z} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/18df829c7bc6515ddbbe31f5c976bad7ca68e4e0)
giving the derived intersection
![{\displaystyle \mathbb {F} _{p}\otimes _{\mathbb {Z} }^{\mathbf {L} }\mathbb {F} _{p}\cong \mathbb {F} _{p}[\varepsilon ]/(\varepsilon ^{2})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/46d11956d00ffef2469011a8e19c25e0452aa90e)
where
![{\displaystyle {\text{deg}}(\varepsilon )=1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f4bec5a1717797bffa8364adf4e7f674eee20ad5)
and the differential is the zero map. This is because we just tensor the complex above by
![{\displaystyle \mathbb {F} _{p}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2d35035371db7bee93733c68c1802114c17d8bb4)
, giving a formal complex with a generator in degree
![{\displaystyle 1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/92d98b82a3778f043108d4e20960a9193df57cbf)
which squares to
![{\displaystyle 0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2aae8864a3c1fec9585261791a809ddec1489950)
. Then, the Hochschild complex is given by
![{\displaystyle \mathbb {F} _{p}\otimes _{\mathbb {F} _{p}\otimes _{\mathbb {Z} }^{\mathbb {L} }\mathbb {F} _{p}}^{\mathbb {L} }\mathbb {F} _{p}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fef14469171b7dd6c6c1528e0ccc0e7d92e9d69d)
In order to compute this, we must resolve
![{\displaystyle \mathbb {F} _{p}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2d35035371db7bee93733c68c1802114c17d8bb4)
as an
![{\displaystyle \mathbb {F} _{p}\otimes _{\mathbb {Z} }^{\mathbf {L} }\mathbb {F} _{p}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0edc4d814008d8afbd4f97cfb7660d0f307b2a08)
-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 ![{\displaystyle 3}](https://wikimedia.org/api/rest_v1/media/math/render/svg/991e33c6e207b12546f15bdfee8b5726eafbbb2f)
We can perform this recursively to get the underlying module of the divided power algebra
![{\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}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/30643364231311ef5143166fde26a4e83db07f6b)
with
![{\displaystyle dx_{i}=\varepsilon \cdot x_{i-1}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cad5d2993e916dbc79659a86c1b8b448190bcd9f)
and the degree of
![{\displaystyle x_{i}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e87000dd6142b81d041896a30fe58f0c3acb2158)
is
![{\displaystyle 2i}](https://wikimedia.org/api/rest_v1/media/math/render/svg/717fa22d62566808b398d03504bc04a9599da936)
, namely
![{\displaystyle |x_{i}|=2i}](https://wikimedia.org/api/rest_v1/media/math/render/svg/87a1ca3940302475fa179569794e6a61254c68e1)
. Tensoring this algebra with
![{\displaystyle \mathbb {F} _{p}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2d35035371db7bee93733c68c1802114c17d8bb4)
over
![{\displaystyle \mathbb {F} _{p}\otimes _{\mathbb {Z} }^{\mathbf {L} }\mathbb {F} _{p}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0edc4d814008d8afbd4f97cfb7660d0f307b2a08)
gives
![{\displaystyle HH_{*}(\mathbb {F} _{p})=\mathbb {F} _{p}\langle x\rangle }](https://wikimedia.org/api/rest_v1/media/math/render/svg/04dd140b3532c9ede98d42692076c194795c1860)
since
![{\displaystyle \varepsilon }](https://wikimedia.org/api/rest_v1/media/math/render/svg/a30c89172e5b88edbd45d3e2772c7f5e562e5173)
multiplied with any element in
![{\displaystyle \mathbb {F} _{p}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2d35035371db7bee93733c68c1802114c17d8bb4)
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
![{\displaystyle \mathbb {F} _{p}\langle x\rangle }](https://wikimedia.org/api/rest_v1/media/math/render/svg/31697b450191b9dfa25e19b85c90bf4f62e8948e)
is not well behaved. For instance,
![{\displaystyle x^{p}=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3f406cbaadc521291b87b8b2d948e31e3417f65f)
. One technical response to this problem is through Topological Hochschild homology, where the base ring
![{\displaystyle \mathbb {Z} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/449494a083e0a1fda2b61c62b2f09b6bee4633dc)
is replaced by the
sphere spectrum ![{\displaystyle \mathbb {S} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/9f9d5874c5d7f68eba1cec9da9ccbe53903303bb)
.
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)
-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,
![{\displaystyle THH(\mathbb {F} _{p})=\mathbb {F} _{p}[x],}](https://wikimedia.org/api/rest_v1/media/math/render/svg/49bc242b404d1768fa27b21dcc0f06c4b83e234b)
![{\displaystyle HH(\mathbb {F} _{p})=\mathbb {F} _{p}\langle x\rangle }](https://wikimedia.org/api/rest_v1/media/math/render/svg/d1a87b97fbfc1b551508c0429fa256a84ffea7d4)
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
can be expressed using regularized determinants involving topological Hochschild homology.
See also
References
- ^ 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'École Normale Supérieure. 33 (2): 151–179. doi:10.1016/S0012-9593(00)00107-5.
External links
Introductory articles
- 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
- Antieau, Benjamin; Bhatt, Bhargav; Mathew, Akhil (2019). "Counterexamples to Hochschild–Kostant–Rosenberg in characteristic p". arXiv:1909.11437 [math.AG].
Noncommutative case
- 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].