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Let $A$ be an associative algebra. Its zeroth Hochschild homology $\mathrm{HH}_0(A)$ is the cokernel of the linear map $A^{\wedge 2} \to A$, $a \wedge b \mapsto ab - ba$. I.e. you quotient the vector space $A$ by its subvector space of commutators (and not the ideal that subvector space generates).

I am interested in a situation where $A = \prod_{i\in \mathbb N} A_i$ is an infinite product of algebras. Then there is a canonical map $$ \mathrm{HH}_0\left( \prod_i A_i\right) \to \prod_i \mathrm{HH}_0\left(A_i\right) $$ which in general has no reason to be an isomorphism. (It would be an isomorphism if the product were finite.)

The abstract reason for such a map is that $\mathrm{HH}_0$ is built from colimits whereas $\prod$ is a type of limit, and there is always an arrow $\mathrm{colim} \circ \mathrm{lim} \to \mathrm{lim} \circ \mathrm{colim}$, which in general is not an isomorphism. The specific reason for this particular map is that the LHS is the quotient of $\prod_i A_i$ by the subspace of finite sums of sequences $(a_ib_i - b_ia_i)$. The RHS is the quotient by the subspace of possibly infinite sums of such sequences which are finite for each $i$, but as $i\to \infty$ the number of summands might grow without bound. So in fact the canonical map is a surjection.

Are there criteria on the $A_i$ that guarantee that in fact this canonical map is an isomorphism?

This is the type of thing that properties like "Noetherian" are supposed to solve. Indeed, this question is formally similar to asking when the natural transformation commuting infinite products past tensor products is injective, for which the answer has very much to do with Noetherian-ity: Goodearl, Distributing tensor product over direct product, Pacific Journal of Mathematics, Vol. 43 (1972), No. 1, 107–110 (DOI: 10.2140/pjm.1972.43.107).

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