Amitsur complex

In algebra, the Amitsur complex is a natural complex associated to a ring homomorphism. It was introduced in (Amitsur 1959). When the homomorphism is faithfully flat, the Amitsur complex is exact (thus determining a resolution), which is the basis of the theory of faithfully flat descent.

The notion should be thought of as a mechanism to go beyond the conventional localization of rings and modules.[1]

Definition

Let $\theta :R\to S$ be a homomorphism of (not-necessary-commutative) rings. First define the cosimplicial set $C^{\bullet }=S^{\otimes \bullet +1}$ (where $\otimes$ refers to $\otimes _{R}$ , not $\otimes _{\mathbb {Z} }$ ) as follows. Define the face maps $d^{i}:S^{\otimes {n+1}}\to S^{\otimes n+2}$ by inserting 1 at the i-th spot:[note 1]

d^{i}(x_{0}\otimes \cdots \otimes x_{n})=x_{0}\otimes \cdots \otimes x_{i-1}\otimes 1\otimes x_{i}\otimes \cdots \otimes x_{n}.

Define the degeneracies $s^{i}:S^{\otimes n+1}\to S^{\otimes n}$ by multiplying out the i-th and (i + 1)-th spots:

s^{i}(x_{0}\otimes \cdots \otimes x_{n})=x_{0}\otimes \cdots \otimes x_{i}x_{i+1}\otimes \cdots \otimes x_{n}.

They satisfy the "obvious" cosimplicial identities and thus $S^{\otimes \bullet +1}$ is a cosimplicial set. It then determines the complex with the augumentation $\theta$ , the Amitsur complex:[2]

0\to R\,{\overset {\theta }{\to }}\,S\,{\overset {\delta ^{0}}{\to }}\,S^{\otimes 2}\,{\overset {\delta ^{1}}{\to }}\,S^{\otimes 3}\to \cdots

where $\delta ^{n}=\sum _{i=0}^{n+1}(-1)^{i}d^{i}.$

Exactness of the Amitsur complex

Faithfully flat case

In the above notations, if $\theta$ is right faithfully flat, then a theorem of Grothendieck states that the (augmented) complex $0\to R{\overset {\theta }{\to }}S^{\otimes \bullet +1}$ is exact and thus is a resolution. More generally, if $\theta$ is right faithfully flat, then, for each left R-module M,

0\to M\to S\otimes _{R}M\to S^{\otimes 2}\otimes _{R}M\to S^{\otimes 3}\otimes _{R}M\to \cdots

is exact.[3]

Proof:

Step 1: The statement is true if $\theta :R\to S$ splits as a ring homomorphism.

That " $\theta$ splits" is to say $\rho \circ \theta =\operatorname {id} _{R}$ for some homomorphism $\rho :S\to R$ ( $\rho$ is a retraction and $\theta$ a section). Given such a $\rho$ , define

h:S^{\otimes n+1}\otimes M\to S^{\otimes n}\otimes M

by

{\begin{aligned}&h(x_{0}\otimes m)=\rho (x_{0})\otimes m,\\&h(x_{0}\otimes \cdots \otimes x_{n}\otimes m)=\theta (\rho (x_{0}))x_{1}\otimes \cdots \otimes x_{n}\otimes m.\end{aligned}}

An easy computation shows the following identity: with $\delta ^{-1}:M{\overset {\theta \otimes \operatorname {id} _{M}}{\to }}S\otimes _{R}M$ ,

h\circ \delta ^{n}+\delta ^{n-1}\circ h=\operatorname {id} _{S^{\otimes n+1}\otimes M}

.

This is to say that h is a homotopy operator and so $\operatorname {id} _{S^{\otimes n+1}\otimes M}$ determines the zero map on cohomology: i.e., the complex is exact.

Step 2: The statement is true in general.

We remark that $S\to T:=S\otimes _{R}S,\,x\mapsto 1\otimes x$ is a section of $T\to S,\,x\otimes y\mapsto xy$ . Thus, Step 1 applied to the split ring homomorphism $S\to T$ implies:

0\to M_{S}\to T\otimes _{S}M_{S}\to T^{\otimes 2}\otimes _{S}M_{S}\to \cdots ,

where $M_{S}=S\otimes _{R}M$ , is exact. Since $T\otimes _{S}M_{S}\simeq S^{\otimes 2}\otimes _{R}M$ , etc., by "faithfully flat", the original sequence is exact. $\square$

The case of the arc topology

Bhatt & Scholze (2019, §8) show that the Amitsur complex is exact if R and S are (commutative) perfect rings, and the map is required to be a covering in the arc topology (which is a weaker condition than being a cover in the flat topology).

References

Note the reference (M. Artin) seems to have a typo, and this should be the correct formula; see the calculation of s⁰ and d² in the note.

(Artin 1999, III.7.)
Artin 1999, III.6.
Artin, Theorem III.6.6.

Artin, Michael (1999), Noncommutative rings (Berkeley lecture notes) (PDF)
Amitsur, Shimshon (1959), "Simple algebras and cohomology groups of arbitrary fields", Transactions of the American Mathematical Society, 90 (1): 73–112
Bhatt, Bhargav; Scholze, Peter (2019), Prisms and Prismatic Cohomology, arXiv:1905.08229
Amitsur complex in nLab

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[2] Note the reference (M. Artin) seems to have a typo, and this should be the correct formula; see the calculation of s⁰ and d² in the note.

[1] (Artin 1999, III.7.)

[3] Artin 1999, III.6.

[4] Artin, Theorem III.6.6.