Английская Википедия:Amitsur complex
In algebra, the Amitsur complex is a natural complex associated to a ring homomorphism. It was introduced by Шаблон:Harvs. 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.Шаблон:Sfn
Definition
Let <math>\theta: R \to S</math> be a homomorphism of (not-necessary-commutative) rings. First define the cosimplicial set <math>C^\bullet = S^{\otimes \bullet+1}</math> (where <math>\otimes</math> refers to <math>\otimes_R</math>, not <math>\otimes_{\Z}</math>) as follows. Define the face maps <math>d^i : S^{\otimes {n+1}} \to S^{\otimes n+2}</math> by inserting <math>1</math> at the <math>i</math>th spot:Шаблон:Efn
- <math>d^i(x_0 \otimes \cdots \otimes x_n) = x_0 \otimes \cdots \otimes x_{i-1} \otimes 1 \otimes x_i \otimes \cdots \otimes x_n.</math>
Define the degeneracies <math>s^i : S^{\otimes n+1} \to S^{\otimes n}</math> by multiplying out the <math>i</math>th and <math>(i+1)</math>th spots:
- <math>s^i(x_0 \otimes \cdots \otimes x_n) = x_0 \otimes \cdots \otimes x_i x_{i+1} \otimes \cdots \otimes x_n.</math>
They satisfy the "obvious" cosimplicial identities and thus <math>S^{\otimes \bullet + 1}</math> is a cosimplicial set. It then determines the complex with the augumentation <math>\theta</math>, the Amitsur complex:Шаблон:Sfn
- <math>0 \to R \,\overset{\theta}\to\, S \,\overset{\delta^0}\to\, S^{\otimes 2} \,\overset{\delta^1}\to\, S^{\otimes 3} \to \cdots</math>
where <math>\delta^n = \sum_{i=0}^{n+1} (-1)^i d^i.</math>
Exactness of the Amitsur complex
Faithfully flat case
In the above notations, if <math>\theta</math> is right faithfully flat, then a theorem of Alexander Grothendieck states that the (augmented) complex <math>0 \to R \overset{\theta}\to S^{\otimes \bullet + 1}</math> is exact and thus is a resolution. More generally, if <math>\theta</math> is right faithfully flat, then, for each left <math>R</math>-module <math>M</math>,
- <math>0 \to M \to S \otimes_R M \to S^{\otimes 2} \otimes_R M \to S^{\otimes 3} \otimes_R M \to \cdots</math>
is exact.Шаблон:Sfn
Proof:
Step 1: The statement is true if <math>\theta : R \to S</math> splits as a ring homomorphism.
That "<math>\theta</math> splits" is to say <math>\rho \circ \theta = \operatorname{id}_R</math> for some homomorphism <math>\rho : S \to R</math> (<math>\rho</math> is a retraction and <math>\theta</math> a section). Given such a <math>\rho</math>, define
- <math>h : S^{\otimes n+1} \otimes M \to S^{\otimes n} \otimes M</math>
by
- <math>\begin{align}
& h(x_0 \otimes m) = \rho(x_0) \otimes m, \\ & h(x_0 \otimes \cdots \otimes x_n \otimes m) = \theta(\rho(x_0)) x_1 \otimes \cdots \otimes x_n \otimes m. \end{align}</math> An easy computation shows the following identity: with <math>\delta^{-1}=\theta \otimes \operatorname{id}_M : M \to S \otimes_R M</math>,
- <math>h \circ \delta^n + \delta^{n-1} \circ h = \operatorname{id}_{S^{\otimes n+1} \otimes M}</math>.
This is to say that <math>h</math> is a homotopy operator and so <math>\operatorname{id}_{S^{\otimes n+1} \otimes M}</math> determines the zero map on cohomology: i.e., the complex is exact.
Step 2: The statement is true in general.
We remark that <math>S \to T := S \otimes_R S, \, x \mapsto 1 \otimes x</math> is a section of <math>T \to S, \, x \otimes y \mapsto xy</math>. Thus, Step 1 applied to the split ring homomorphism <math>S \to T</math> implies:
- <math>0 \to M_S \to T \otimes_S M_S \to T^{\otimes 2} \otimes_S M_S \to \cdots,</math>
where <math>M_S = S \otimes_R M</math>, is exact. Since <math>T \otimes_S M_S \simeq S^{\otimes 2} \otimes_R M</math>, etc., by "faithfully flat", the original sequence is exact. <math>\square</math>
Arc topology case
Шаблон:Harvs show that the Amitsur complex is exact if <math>R</math> and <math>S</math> 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).
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