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{Tor}_n^{A^e}(A, M) : HH^n(A,M) = \operatorname{Ext}^n_{A^e}(A, M)
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 A^{\otimes n} 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 \otimes A^{\otimes n} with boundary operator d_i defined by :\begin{align} 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{align} where a_i is in
A for all 1\le i\le n and m\in M. If we let : b_n=\sum_{i=0}^n (-1)^i d_i, then b_{n-1} \circ b_{n} =0, so (C_n(A,M),b_n) 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 as simply b.
Remark The maps d_i are
face maps making the family of
modules (C_n(A,M),b) 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 :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. Hochschild homology is the homology of this simplicial module.
Relation with the bar complex There is a similar looking complex B(A/k) called the
bar complex which formally looks very similar to the Hochschild complexpg 4-5. In fact, the Hochschild complex HH(A/k) can be recovered from the bar complex asHH(A/k) \cong A\otimes_{A\otimes A^{op}} B(A/k)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) X over some base scheme S. For example, we can form the derived
fiber productX\times^\mathbf{L}_SXwhich has the sheaf of derived rings \mathcal{O}_X\otimes_{\mathcal{O}_S}^\mathbf{L}\mathcal{O}_X. Then, if embed X with the diagonal map\Delta: X \to X\times^\mathbf{L}_SXthe Hochschild complex is constructed as the pullback of the derived self intersection of the diagonal in the diagonal product schemeHH(X/S) := \Delta^*(\mathcal{O}_X\otimes_{\mathcal{O}_X\otimes_{\mathcal{O}_S}^\mathbf{L}\mathcal{O}_X}^\mathbf{L}\mathcal{O}_X)From this interpretation, it should be clear the Hochschild homology should have some relation to the Kähler differentials \Omega_{X/S} since the
Kähler differentials can be defined using a self-intersection from the diagonal, or more generally, the
cotangent complex \mathbf{L}_{X/S}^\bullet since this is the derived replacement for the Kähler differentials. We can recover the original definition of the Hochschild complex of a commutative k-algebra A by settingS = \text{Spec}(k) and X = \text{Spec}(A)Then, the Hochschild complex is
quasi-isomorphic toHH(A/k) \simeq_{qiso} A\otimes_{A\otimes_{k}^\mathbf{L}A}^\mathbf{L}A If A is a flat k-algebra, then there's the chain of isomorphisms A\otimes_k^\mathbf{L}A \cong A\otimes_kA \cong A\otimes_kA^{op}giving an alternative but equivalent presentation of the Hochschild complex. ==Hochschild homology of functors==