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Transcription
Definition
Let be a homomorphism of (not-necessary-commutative) rings. First define the cosimplicial set (where refers to , not ) as follows. Define the face maps by inserting at the th spot:[a]
Define the degeneracies by multiplying out the th and 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
Exactness of the Amitsur complex
Faithfully flat case
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. More generally, if is right faithfully flat, then, for each left -module ,