I.10: Almost mathematics (II)

I continue by introducing standard notions from commutative algebra to the context of almost mathematics. For the references to the proof of the following definition/proposition, see Proposition 4.7 in Scholze’s “`Perfectoid spaces”‘. I follow closely the exposition in the Scholze’s paper in this blog post (consult it for all references).

Definition 42 Let {A} be a {K^{\circ a}}-algebra

  1. An {A}-module {M} is flat if the functor {X \mapsto M \otimes _A X} on {A} is exact. If {R} is a {K^{\circ}}-algebra and {N} is an {R}-module, then {R^a}-module {N^a} is flat if and only if for all {R}-modules {X} and all {i >0}, the module {\mathrm{Tor} ^R _i(N,X)} is almost zero.
  2. An {A}-module {M} is almost projective if the functor {X \mapsto al\mathrm{Hom} _A(M,X)} on {A}-modules is exat. If {R} is a {K^{\circ}}-algebra and {N} is an {R}-module, then {N^a} is almost projective over {R^a} if and only if for all {R}-modules {X} and all {i >0}, the module {\mathrm{Ext} ^R _i(N,X)} is almost zero.
  3. If {R} is a {K^{\circ}}-algebra and {N} is an {R}-module then we say {M=N^a} is almost finitely generated (resp. almost finitely presented) {R^a}-module if and only if for all {\epsilon \in \mathfrak{m}}, there is some finitely generated (resp. finitely presented) {R}-module {N_{\epsilon}} with a map {f _{\epsilon}: N_{\epsilon} \rightarrow N_{\epsilon}} such that the kernel and cokernel of {f_{\epsilon}} are annigilated by {\epsilon}. We say {M} is uniformly almost finitely generated if there is some integer {n} such that {N_{\epsilon}} can be chosen to be generated by {n} elements for all {\epsilon}.

 

Proposition 43 Let {A} be a {K^{\circ a}}-algebra. Then an {A}-module {M} is flat and almost finitely presented if and only if it is almost projective and almost finitely generated.

We shall call such {A}-modules {M} finite projective. If moreover {M} is uniformly almost finitely generated, we say {M} is uniformly finite projective. For such modules we have a good notion of rank.

Theorem 44 Let {A} be a {K^{\circ a}}-algebra and let {M} be a uniformly finite projective {A}-module. Then there is a unique decomposition {A = A_0 \times A_1 \times ... \times A_k} such that for each {i=0,...,k} the {A_i}-module {M_i = M \otimes _A A_i} has the property that {\wedge ^i M_i} is invertible, and {\wedge ^{i+1} M_i = 0}. Here, we call {A}-module {L} invertible if {L \otimes _A al\mathrm{Hom} _A(L,A)= A}.

We introduce the notion of étale morphisms

Definition 45 Let {A} be a {K^{\circ a}}-algebra and let {B} be an {A}-algebra. Let {\mu : B\otimes _A B \rightarrow B} denote the multiplication morphism.

(1) The morphism {A \rightarrow B} is unramified if there is an element {e \in (B\otimes _A B)_{*}} such that {e^2 = e}, {\mu(e) = 1} and {xe =0} for all {x\in \ker (\mu)_{*}}.

(2) The morphism {A \rightarrow B} is étale if it is unramified and {B} is a flat {A}-module.

 

Definition 46 A morphism {A \rightarrow B} of {K^{\circ a}}-algebras is finite étale if it is étale and {B} is an almost finitely presented {A}-module. We write {A_{fet}} for the category of finite étale {A}-algebras.

Observe that any finite étale {A}-algebra is also finite projective {A}-module. There is an equivalent characterization of finite étale morphisms in terms of trace morphisms. For a {K^{\circ a}}-algebra {A} and a finite projective {A}-module {P} we define {P^{*} = al\mathrm{Hom}(P,A)} which is also a finite projective {A}-module. Moreover {P{**} \simeq P} canonically and there is an isomorphism

\displaystyle \mathrm{End}(P)^a = P \otimes _A P^{*}

In particular, we get a trace morphism {\mathrm{tr} _{P/A} : \mathrm{End}(P)^a \rightarrow A}

Definition 47 Let {A} be a {K^{\circ a}}-algebra and let {B} be an {A}-algebra such that {B} is a finite projective {A}-module. We define the trace form as the bilinear form

\displaystyle t _{B/A} : B\otimes _A B \rightarrow A

given by the composition of {\mu : B \otimes _A B \rightarrow B} and the map {B \rightarrow A} sending any {b\in B} to the trace of the endomorphism {b' \mapsto bb'} of {B}.

We remark that the last map is defined on {(.)_{*}} as we cannot talk about elements {b} of an almost object {B}. That is, we construct a map {B _{*} \rightarrow \mathrm{End} _{A_{*}}(B_{*})^a} in the same way and then pass to the almost setting by taking {(.)^a}.

Theorem 48 If {A,B} are as in the definition above, then {A \rightarrow B} is finite étale if and only if the trace map is a perfect pairing, that is it induces an isomorphism {B \simeq B^{*}}.

We finish by saying that finite étale covers lift uniquely over nilpotents (recall {\varpi} was a uniformizer of our base field):

Theorem 49 Let {A} be a {K^{\circ a}}-algebra. Assume that {A} is flat over {K^{\circ a}} and {\varpi}-adically complete, that is

\displaystyle A \simeq \varprojlim _n A/\varpi ^n

Then the functor {B \mapsto B \otimes _A A/\varpi} induces an equivalence of categories {A _{fet} \simeq (A/\varpi)_{fet}}. Any {B \in A_{fet}} is again flat over {K^{\circ a}} and {\varpi}-adically complete. Moreover {B} is a uniformly finite projective {A}-module if and only if {B \otimes _A A/\varpi} is a uniformly finite projective {A/\varpi}-module.

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