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Amitsur complex

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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.[1]

Definition

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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

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Faithfully flat case

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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 ,

is exact.[3]

Proof:

Step 1: The statement is true if splits as a ring homomorphism.

That " splits" is to say for some homomorphism ( is a retraction and a section). Given such a , define

by

An easy computation shows the following identity: with ,

.

This is to say that 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.

We remark that is a section of . Thus, Step 1 applied to the split ring homomorphism implies:

where , is exact. Since , etc., by "faithfully flat", the original sequence is exact.

Arc topology case

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Bhargav Bhatt and Peter Scholze (2019, §8) show that the Amitsur complex is exact if and 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).

Notes

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  1. ^ The reference (M. Artin) seems to have a typo, and this should be the correct formula; see the calculation of and in the note.

Citations

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  1. ^ Artin 1999, III.7
  2. ^ Artin 1999, III.6
  3. ^ Artin 1999, Theorem III.6.6

References

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  • Artin, Michael (1999), Noncommutative rings (Berkeley lecture notes) (PDF)
  • Amitsur, Shimshon (1959), "Simple algebras and cohomology groups of arbitrary fields", Transactions of the American Mathematical Society, 90 (1): 73–112
  • Bhatt, Bhargav; Scholze, Peter (2019), Prisms and Prismatic Cohomology, arXiv:1905.08229
  • Amitsur complex at the nLab