In algebra, the Amitsur complex is a natural complex associated to a ring homomorphism.
It was introduced by Shimshon 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.
be a homomorphism of (not-necessary-commutative) rings.
First define the cosimplicial set
refers to
Define the face maps
th spot:[a] Define the degeneracies
th spots: They satisfy the "obvious" cosimplicial identities and thus
is a cosimplicial set.
It then determines the complex with the augumentation
, the Amitsur complex:[2] where
In the above notations, if
is right faithfully flat, then a theorem of Alexander Grothendieck states that the (augmented) complex
is exact and thus is a resolution.
is right faithfully flat, then, for each left
, is exact.
[3] Proof: Step 1: The statement is true if
splits as a ring homomorphism.
ρ ∘ θ =
, define by An easy computation shows the following identity: with
is a homotopy operator and so
determines the zero map on cohomology: i.e., the complex is exact.
Step 2: The statement is true in general.
Thus, Step 1 applied to the split ring homomorphism
, is exact.
, etc., by "faithfully flat", the original sequence is exact.
Bhargav Bhatt and Peter Scholze (2019, §8) show that the Amitsur complex is exact if
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).