# nLab Amitsur complex

Contents

### Context

#### Algebra

higher algebra

universal algebra

## Theorems

#### Homological algebra

homological algebra

Introduction

diagram chasing

# Contents

## 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)

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

A review of the proof in low degree is in

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

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

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