Hochschild cohomology ($H{H}^n(R,M)$)

The Hochschild cohomology groups are

The ordinary Hochschild cochain complex is defined using a chosen commutative ground ring $k$. The bar complex is exact via insertion of $1$, but its terms are projective over the enveloping algebra when $R$ is projective over $k$, in particular when $k$ is a field. Without that hypothesis the ordinary cochain groups compute relative Hochschild cohomology and cannot automatically be identified with absolute Ext. Likewise, ordinary second cohomology classifies ground-ring-linearly split square-zero extensions, not arbitrary nonsplit ring extensions.

// Destination: module-theory.bigb

Over a field, the bimodule resolution $0\to k[x]\otimes k[x]\stackrel{x\otimes 1-1\otimes x}{\to }k[x]\otimes k[x]\to k[x]\to 0$ is exact: multiplication identifies the quotient by $x\otimes 1-1\otimes x$, and that element is a non-zero-divisor in the two-variable polynomial ring. Applying the bimodule Hom functor gives zero differential, proving the displayed Hochschild cohomology. Degree-one classes are the derivations $f(x){\partial }_x$.

Fix an $R$-bimodule $M$. A unital ground-ring-linear split extension with square-zero kernel $M$ can be identified with $R\oplus M$, with multiplication $(a,u)(b,v)=(ab,av+ub+f(a,b))$. Associativity is exactly $\delta f=0$. Changing a unital linear section by $g:R\to M$ changes $f$ by $\delta g$. Thus equivalence classes of these split extensions are $H{H}^2(R,M)$. Over a field every such vector-space extension splits; over a general ground ring a splitting hypothesis is essential for this ordinary cochain classification.

For the polynomial ring $A=k[X_1,\dots ,X_r]$ in characteristic zero, antisymmetrization identifies $H{H}^{\ast }(A,A)$ with ${\bigwedge }_A^{\ast }{\mathrm{Der}}_k(A)$. The map on a wedge of $p$ derivations is $\frac{1}{p!}{\sum }_{\sigma \in S_p}\mathrm{sgn}(\sigma ){\prod }_j{D}_{\sigma (j)}(a_j)$. The Hochschild cup product becomes the exterior product, and the left Gerstenhaber bracket becomes the left Schouten-Nijenhuis bracket. This assertion requires a smoothness hypothesis when generalized beyond polynomial rings.

Write $f\circ g={\sum }_{i=0}^{p-1}(-1{)}^{i(q-1)}f{\circ }_{i}g$ for insertion of a degree-$q$ cochain into a degree-$p$ one. With the unsigned Hochschild cup product and the left graded Leibniz rule, take $[f,g]=(-1{)}^{(p-1)(q-1)}f\circ g-g\circ f$. It descends to Hochschild cohomology and gives a Gerstenhaber algebra. The alternative insertion bracket $f\circ g-(-1{)}^{(p-1)(q-1)}g\circ f$ differs by the displayed degree sign and obeys the corresponding right rule. Degree-one brackets are commutators of derivations in either convention.

The Hochschild cohomological dimension of an associative algebra over a field is $\mathrm{Dim}(A)={\mathrm{pd}}_{A\otimes A^{\mathrm{op}}}A$. Equivalently, $H{H}^n(A,M)$ vanishes above that dimension for every bimodule $M$. A bound by one implies $H{H}^2(A,A)=0$ and hence formal rigidity from vanishing second Hochschild cohomology.

A $k$-linear derivation from a $k$-algebra $R$ to an $R$-$R$-bimodule $M$ is a map $d:R\to M$ satisfying

For an associative algebra $A$ over a field $k$, a derivation is a $k$-linear map $D:A\to A$ satisfying $D(ab)=D(a)b+aD(b)$. It is the case $M=A$ of a derivation into a bimodule; the order of factors matters when $A$ is noncommutative. Substituting $a=b=1$ gives $D(1)=0$. Such maps form a Lie algebra under the commutator $[D,E]=D\circ E-E\circ D$.

More generally, for a commutative ring $R$, a derivation from a commutative $R$-algebra $B$ into a $B$-module $M$ is an $R$-linear map $D:B\to M$ with $D(r)=0$ for $r\in R$ and $D(b{b}^{\prime })=bD(b^{\prime })+b^{\prime }D(b)$. The second term has this order because the algebra is commutative. Derivations of smooth functions are vector fields provide a geometric example.

Every real-linear derivation of an algebra $D:C^{\infty }(M)\to C^{\infty }(M)$ on a smooth manifold has the form $Dh=Z(h)$ for a unique smooth vector field $Z$. The product rule makes $D$ local: multiplying a function that vanishes near a point by a smooth cutoff function supported there gives zero derivative at that point. Locally write $h(x)=h(p)+{\sum }_i(x^i-x^i(p))h_i(x)$, where $h_i(p)={\partial }_{i}h(p)$. Applying $D$ at $p$ gives $Dh(p)={\sum }_{i}D(x^i)(p){\partial }_{i}h(p)$; the smooth coefficients $D(x^i)$ define the claimed vector field. No continuity hypothesis is needed. In dimension zero all such derivations vanish.

For $m\in M$, the formula $d_m(r)=rm-mr$ defines an inner derivation $R\to M$.

The degree-one Hochschild cocycles are exactly the maps that are a derivation into a bimodule, while degree-one coboundaries are exactly the inner derivations. Therefore

Let $\mu :R{\otimes }_{k}R\to R$ be multiplication and ${\Omega }_R^{\mathrm{nc}}=\ker \mu$, with the outer $R$-bimodule structure. Thenis a derivation and composition with $D$ gives a natural isomorphismThe inverse sends $d$ toThe condition ${\sum }_i{r}_i{s}_i=0$ makes this map left as well as right $R$-linear. Every element of ${\Omega }_R^{\mathrm{nc}}$ is a sum of terms $D(r)s$, which proves uniqueness.

Under the universal isomorphism, the inner derivation $d_m(r)=rm-mr$ corresponds toThus inner derivations correspond exactly to bimodule maps on ${\Omega }_R^{\mathrm{nc}}$ that extend across the inclusion ${\Omega }_R^{\mathrm{nc}}\subseteq R{\otimes }_{k}R$.