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Let $A$ be a finite dimensional algebra over an algebraically closed field (I think this can be generalized to Artinian algebras over commutative rings, but lets work with this hypothesis for now).

Then the cup product on $HH^*(A)=\oplus_{n\geq 0} HH^n(A)$, gives an algebra structure.

The vector space $B= \oplus_{n \geq 0}Hom_{D^b(A^e)}(A,A[n])$ with product given by composition and the shift of $D^b(A)$ gives also an algebra.

I have found a lot of references where they claim that $HH^*(A) \cong B$ as algebras, but I have never seen a proof for this.

Can anyone help me out with this proof? or a reference for it?

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    Your $B$ is in fact isomorphic to $A$. You should instead define $B=\bigoplus_{n\geq 0}\operatorname{Hom}_{D^b(A^e)}(A,A[n])$, that is in the derived category of $A^e$ modules where $A^e=A\otimes_k A^{op}$. – 2017-02-27
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    With this change, it should be more or less obvious depending on your definition of Hochschild cohomology. You might need the fact that $\operatorname{Hom}_{D^b(R)}(M,N[n])=\operatorname{Ext}_R^n(M,N)$ if $M,N$ are $R$-modules. – 2017-02-27
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    You're right, that was a little typo. Thanks. – 2017-02-27
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    Yes, the thing is that I also have not found a proof for the isomorphism between the Ext with the Yoneda product and the Hochschild cohomology algebra. Neither for the Hom's in derived category and Ext's with Yoneda product. But I now they are isomorphic as vector spaces. – 2017-02-27

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