# 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.