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Amitsur complex

Context

Algebra

Homological algebra

homological algebra

(also nonabelian homological algebra)

Introduction

Context

Basic definitions

Stable homotopy theory notions

Constructions

Lemmas

diagram chasing

Homology theories

Theorems

Locality and descent

Contents

Definition

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

RAA RAA RA RA 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 ABA \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

0AfB1idid1B AB 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:ABf \colon A \longrightarrow B is faithfully flat.

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

k:B ABB k \colon B \otimes_A B \longrightarrow B
k:b 1b 2b 1f(s(b 2)). 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(1bb1)=f(s(b))b k(1 \otimes b - b \otimes 1) = f(s(b)) - b

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

So the statement is true for the special morphism

BB AB B \to B \otimes_A B
bb1 b \mapsto b \otimes 1

because that has a section given by the multiplication map.

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

As a bar construction

For ϕ:BA\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

(() BAϕ *):Mod Aϕ *() BAMod B. ((- )\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)

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

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

Last revised on March 7, 2016 at 11:19:32. See the history of this page for a list of all contributions to it.