nLab Amitsur complex

Definition

Given a commutative ring $R$ and an $R$-associative algebra $A$, hence a ring homomorphism $R \longrightarrow A$, the Amitsur complex is the Moore complex of the dual Cech nerve of $Spec(A) \to Spec(R)$, hence the chain complex of the form

$$R \to A \to A \otimes_R A \to A \otimes_R A \otimes_R A \to \cdots$$

with differentials given by the alternating sum of the coface-maps.

(See also at Sweedler coring, at commutative Hopf algebroid and at Adams spectral sequence for the same or similar constructions.)

Properties

Descent theorem

Theorem

(descent theorem)

If $A \to B$ is faithfully flat then its Amitsur complex is exact.

This is due to (Grothendieck, FGA1)

The following reproduces the proof in low degree from Milne, prop. 6.8

Proof

We show that

$$0 \to A \stackrel{f}{\longrightarrow} B \stackrel{1 \otimes id - id \otimes 1}{\longrightarrow} B \otimes_A B$$

is an exact sequence if $f \colon A \longrightarrow B$ is faithfully flat.

First observe that the statement follows if $A \to B$ admits a section $s \colon B \to A$. Because then we can define a map

$$k \colon B \otimes_A B \longrightarrow B$$

$$k \;\colon\; b_1 \otimes b_2 \mapsto b_1 \cdot f(s(b_2)) \,.$$

This is such that applied to a coboundary it yields

$$k(1 \otimes b - b \otimes 1) = f(s(b)) - b$$

and hence it exhibits every cocycle $b$ as a coboundary $b = f(s(b))$.

So the statement is true for the special morphism

$$B \to B \otimes_A B$$

$$b \mapsto b \otimes 1$$

because that has a section given by the multiplication map.

But now observe that the morphism $B \to B \otimes_A B$ is the tensor product of the morphism $f$ with $B$ over $A$. That $A \to B$ is faithfully flat by assumption, hence that it exhibits $B$ as a faithfully flat module over $A$ means by definition that the Amitsur complex for $(A \to B)\otimes_A B$ is exact precisely if that for $A \to B$ is exact.

As a bar construction

For $\phi \colon B \longrightarrow A$ a homomorphism of suitable monoids, there is the corresponding pull-push adjunction (extension of scalars $\dashv$ restriction of scalars) on categories of modules

$$((- )\otimes_B A \dashv \phi^\ast ) \;\colon\; Mod_A \stackrel{\overset{(-)\otimes_B A}{\leftarrow}}{\underset{\phi^\ast}{\longrightarrow}} Mod_B \,.$$

The bar construction of the corresponding monad – the higher monadic descent objects – is the corresponding Amitsur complex.

(e.g. Hess 10, section 6)

Related concepts

• Adams-Novikov spectral sequence

References

The Amitsur complex was introduced in

• Shimshon Amitsur, Simple algebras and cohomology groups of arbitrary fields, Transactions of the American Mathematical Society

  Vol. 90, No. 1 (Jan., 1959), pp. 73-112 (JSTOR)

His results were simplified in

• Alex Rosenberg and Daniel Zelinsky, On Amitsur’s complex, Transactions of the American Mathematical Society Vol. 97, No. 2 (Nov., 1960), pp. 327-356

The statement of proof the descent theorem for the Amitsur complex is due to

• Alexander Grothendieck, FGA1

A review of the proof in low degree is in

• James Milne, prop. 6.8 of Lectures on Étale Cohomology

Discussion from the point of view of Sweedler corings and a full proof of the descent theorem is in

• Tomasz Brzezinski, Robert Wisbauer, section 29 of Corings and Comodules, Cambridge University Press, London Math. Soc. LN 309 (2003), (errata pdf)

Disucssion from the point of view of higher monadic descent is in

• Kathryn Hess, section 6 of A general framework for homotopic descent and codescent (arXiv:1001.1556)

Discussion of formal completion of (infinity,1)-modules in terms of totalization of Amitsur complexes is in

• Gunnar Carlsson, Derived completions in stable homotopy theory (arXiv:0707.2585)