at.algebraic topology - Graded commutativity of cup in Hochschild cohomology

The cup product in Hochschild cohomology$H^bullet(A,A)$ is graded commutative for all unitary algebras. If $M$ is an $A$-bimodule, then the cohomology $H^bullet(A,M)$ with values in $M$ is a symmetric graded bimodule over $H^bullet(A,A)$.

(If $M$ itself is also an algebra such that its multiplication map $Motimes Mto M$ is a map of $A$-bimodules, then in general $H^bullet(A,M)$ is not commutative (for example, take $A=k$ to be the ground field, and $M$ to be an arbitrary non-commutative algebra! I do not know of a criterion for commutatitivity in this case)

These results originally appeared in [M. Gerstenhaber, The cohomology structure of an associative ring, Ann. of Math. (2) 78 (1963), 267–288.], and they are discussed at length in [Gerstenhaber, Murray; Schack, Samuel D. Algebraic cohomology and deformation theory. Deformation theory of algebras and structures and applications (Il Ciocco, 1986), 11--264, NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., 247, Kluwer Acad. Publ., Dordrecht, 1988.] Both references give proofs of a rather computational nature.

You can find an element-free proof of the graded commutativity in this paper, which moreover applies to the cup-products of many other cohomologies.

As for what happens with non-unital algebras, I do not know. But they are very much used today as they were before. One particular context in which they show up constantly is in the intersection of K-theory and functional analysis, where people study `algebras' which are really just ideals in rings of operators of functional spaces---one egregius example is the algebra of compact operators in a Hilbert space.

By the way, you say that group cohomology is a special case of Hochschild cohomology: it is only in a sense... There is a close relationship between the group cohomology $H^bullet(G,mathord-)$ of a group $G$, and the Hochschild chomology $H^bullet(kG,mathord-)$ of the group algebra, but they are not the same. You can use the second to compute the first (because more generally if $M$ and $N$ are $G$-modules, then $mathord{Ext}_{kG}^bullet(M,N)=H^bullet(kG, hom(M,N))$, where on the right we have Hochschild cohomology of the group algebra $kG$ with values in the $kG$-bimodule $hom(M,N)$), but the «principal» group cohomology $H^bullet(G,k)$ is only a little part of the «principal» Hochschild cohomology $H^bullet(kG,kG)$.

Finally, in the paper by Sletsjøe the definition for the boundary is given as you say because he only considers commutative algebras and only principal coefficients, that is $H^bullet(A,A)$.

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