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Let $k$ be a field, $R$ is a commutative algebra over $k$ and $A$ is an associative algebra over $R$. There is a morphism of commutative algebras $R \to T$. Is it possible to reduce calculation of Hochschild homology $HH_*(A\otimes_R T)$ (over basic field $k$) to $HH_*(A)$? I'm mostly interested in a situation when $T$ is separable over $R$, or even more specific $T=R/I$.

flag	asked Dec 28 at 23:21 Sasha Pavlov 276●1●5

	If A is flat over R, or more generally $Tor^R(A,T) = 0$ in positive degrees, then there is a spectral sequence starting with $Tor^{HH_* R}(HH_* A, HH_* T)$ converging to your desired Hochschild term. Does that work in your situation? – Tyler Lawson Dec 29 at 0:28
	Yes, it looks like something that could be very useful for me. – Sasha Pavlov Dec 29 at 0:57
	@Tyler Could you please give a reference for this spectral sequence? – Sasha Pavlov Dec 29 at 12:24
	@Sasha: I'm afraid that I don't have a ready reference. You can prove it by constructing a double complex whose rows are the Hochschild complexes of $A \otimes R^{\otimes q} \otimes T$, with vertical differentials coming from the bar construction. (It's essentially the Hochschild complex of the DGA $A \otimes^{\mathbb L}_R T$.) – Tyler Lawson Dec 29 at 16:28

Leo Alonso · Answer 1 · 2012-12-28 23:41:19Z UTC

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According to the following paper:

Geller, S.; Weibel, C.: Étale descent for Hochschild and cyclic homology, Comment. Math. Helv. 66 (1991), no. 3, 368–388;

—theorem (0.1)— you need that $T$ is étale over $R$. This holds in your case only if the ideal $I$ is generated by an idempotent. Notice that "étale" = "unramified + flat". And the notion of unramifiedness is related to the condition of separable ring extension, but it depends on which of the several used definitions of "separable" you consider.

answered Dec 28 at 23:41

Leo Alonso
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For me, separable means that $T$ is projective over $T \otimes_R T$. I'm aware of this result, but situation when $I$ is generated by an idempotent is not useful for me. – Sasha Pavlov Dec 28 at 23:49

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Without flatness I have the intuition the result is false. Sorry about that. – Leo Alonso Dec 28 at 23:58

Probably won't help, but just a remark in case it's helpful that the etale descent statement extends to smooth descent. – David Ben-Zvi Dec 29 at 0:07

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@Leo: arxiv.org/abs/1002.3636 Proposition 1.26 – David Ben-Zvi Dec 29 at 1:25

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@David Thank you very much! – Leo Alonso Dec 29 at 12:21

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Hochschild homology and change of non-ground ring

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