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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 $T=R/I$.

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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 '12 at 0:28
    
Yes, it looks like something that could be very useful for me. –  Sasha Pavlov Dec 29 '12 at 0:57
    
@Tyler Could you please give a reference for this spectral sequence? –  Sasha Pavlov Dec 29 '12 at 12:24
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@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 '12 at 16:28

1 Answer 1

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.

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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 '12 at 23:49
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Without flatness I have the intuition the result is false. Sorry about that. –  Leo Alonso Dec 28 '12 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 '12 at 0:07
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@Leo: arxiv.org/abs/1002.3636 Proposition 1.26 –  David Ben-Zvi Dec 29 '12 at 1:25
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@David Thank you very much! –  Leo Alonso Dec 29 '12 at 12:21

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