representation, ∞-representation?
symmetric monoidal (∞,1)-category of spectra
(also nonabelian homological algebra)
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
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.)
This is due to (Grothendieck, FGA1)
The following reproduces the proof in low degree from Milne, prop. 6.8
We show that
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
This is such that applied to a coboundary it yields
and hence it exhibits every cocycle $b$ as a coboundary $b = f(s(b))$.
So the statement is true for the special morphism
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.
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
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
Vol. 90, No. 1 (Jan., 1959), pp. 73-112 (JSTOR)
His results were simplified in
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.