18.727, Topics in Algebraic Geometry (rigid analytic geometry) Kiran S. Kedlaya, fall 2004 More on coherent sheaves

References: [FvdP, Chapter 4]. Kiehl’s original papers (in German) are: Der Endlichkeitssatz für eigentliche Abbildungen in der nichtarchimedische Funktionentheorie, Inv. Math. 2 (1967), 191–214; and Theorem A und B in der nichtarchimedische Funktionentheorie, Inv. Math. 2 (1967), 256–273.

Addendum: comment about generic fibres

I forgot to mention in the previous handout that the generic fibre construction loses information in a very precise sense. If P is a formal scheme of finite type over 𝔬_{K}, and P′ is a blowup of 𝔬_{K} within the special fibre, then the induced map from the generic fibre of P′ to that of P is an isomorphism.

Example: if P = Spf 𝔬_{K}⟨x, y⟩, and P′ is the blowup along the ideal (x, y, π) for some π ∈ 𝔪_{K}, then in both cases the generic fibre consists of the closed points of the affine plane over K represented by geometric points (a, b) with |a| ≤ 1 and |b| ≤ 1.

In fact, Raynaud proved an equivalence of categories between a certain category of rigid spaces, and a certain category of formal schemes localized at the blowups in the special fibre. (I don’t know the precise formulation offhand; will go look it up later.) This cuts both ways: sometimes it’s easier to work with rigid spaces than formal schemes because you would just as soon ignore the blowups in the special fibre. But sometimes you can’t make a construction in the rigid setting without keeping track of some choice of an associated formal scheme (as in Laurent Fargues’s seminar talk a couple of weeks ago).

Coherent sheaves on affinoid spaces

Last time I defined a coherent sheaf on a rigid space to be a sheaf which, locally on some admissible affinoid covering, is isomorphic to the sheaf generated by a finitely generated module over the corresponding affinoid algebra. The Gerritzen-Grauert-Tate theorem implies that you actually get a sheaf this way, but it doesn’t imply that this sheaf has “enough” sections. This is fixed by the following theorem of Kiehl, whose proof I’ll sketch here; see [FvdP, Section 4.5] for details.

Theorem 1 (Kiehl). Let F be a coherent sheaf on an affinoid space X = Max(A), and put M = F(X). Then M is finitely generated, and F is isomorphic to the coherent sheaf associated to M.

Sketch of proof. As in the Gerritzen-Grauert-Tate argument, one reduces to the case of a Laurent covering given by a single f. (See [FvdP, Section 4.5] for more on this reduction.) That puts us in the following situation. We are given f ∈ A, finitely generated modules M_{+} and M_{−} over A⟨f⟩ and A⟨f^{−1}⟩, respectively, and an isomorphism between M_{+} ⊗ A⟨f, f^{−1}⟩ and M_{−} ⊗ A⟨f, f^{−1}⟩. For M = F(X), we must show that the maps M ⊗ A⟨f⟩ → M_{+} and M ⊗ A⟨f^{−1}⟩ → M_{−} are surjective.

One can easily reduce this to just checking for M_{−} (as in [FvdP, Lemma 4.5.5], but they seem to get it backwards; see below). For this part, one needs a form of Cartan’s lemma: there exists c > 0 such that any invertible n × n matrix U over A⟨f, f^{−1}⟩ with ∥U − I_{n}∥ < c (where the matrix norm is the maximum over entries) factors as U_{+}U_{−}, with U_{+} invertible over A⟨f⟩ and B invertible over A⟨f^{−1}⟩. See [FvdP, Lemma 4.5.3] for this calculation. (The basic idea is to split U additively as I + V_{+} + V_{−}, where V_{+} has only nonnegative powers of V, V_{−} has only nonnegative powers of V_{−}, and both V_{+} and V_{−} have small norm. Then you replace U by (1 − V_{+})U(1 − V_{−}) and repeat; if c is small enough, this process converges to the desired factorization.)

To check surjectivity of M ⊗ A⟨f⟩ → M_{+} now, you choose a set of n generators of M_{+} and a set of n generators of M_{−} (for some n), and write down change-of-basis matrices U and V between the two sets of generators over A⟨f, f^{−1}⟩. Since the image of A in A⟨f⟩ is dense, we can approximate U closely by a matrix over A, so that ∥(U′ − U)V ∥ < c with c as above. We can then factor I_{n} − (U′ − U)V = U_{+}U_{−} as above, and changing basis from M_{+} via U_{+} gives us a set of generators which are defined on both subspaces, yielding the surjectivity.

Given the surjectivity, we can choose a finitely generated submodule M_{1} of M which generates both M_{+} and M_{−}. Let M_{1} be the associated sheaf: then M_{1} → F is surjective. Let G be the kernel of that map; then G is also coherent and given by finitely generated modules on the two pieces of the cover, so we can find a surjection M_{2} → G, where M_{2} is the coherent sheaf associated to the finitely generated module M_{2}. Hence F is the cokernel of the map M_{2} → M_{1}; by acyclicity, its global sections are precisely M_{2}/M_{1}, and the associated sheaf is precisely F because they match up on the two opens.

This means that we can glue coherent sheaves on any admissible cover of X (I may have said this earlier without justification).

Example: the sheaf of differentials

An important example of a coherent sheaf is the sheaf of continuous differentials.

Theorem 2. Let A be an affinoid algebra. There exists a finitely generated A-module Ω_{A/K} equipped with a K-linear derivation d : A → Ω_{A/K} with the following universal property: given any finitely generated A-module M equipped with a K-linear derivation D : A → M, there exists a unique A-module homomorphism l : Ω_{A/K} → M with D = l ∘ d.

Proof. In case A = T_{n} = K⟨x_{1}, …, x_{n}⟩, this is easy to check: take Ω_{A/K} = Adx_{1}+⋯+Adx_{n} together with the formal total differential. That is,

df = ∑_{i=1}^{n} \frac{∂f}{∂x_{i}} dx_{i},

where the partial derivation is done formally on the series.

For general A, pick a presentation A ≅ T_{n}/(f_{1}, …, f_{m}), and put

Ω_{A/K} = Ω_{T_{n}/K}/(T_{n}df_{1} + ⋯ + T_{n}df_{m}),

with the induced derivation from T_{n}.

The module Ω_{A/K} is called the module of finite differentials, or the universal finite differential module. It is obviously unique up to unique isomorphism, so one gets from it a coherent sheaf of differentials Ω_{X} on any rigid space X.

Note that the module of finite differentials is not always the same as the module of Kähler differentials, because the latter is sometimes badly behaved. In fact, it can be shown (though I don’t remember a reference offhand) that the module of finite differentials is the maximal separated quotient of the completion of the ordinary module of Kähler differentials; it is thus sometimes called the module of continuous differentials, because the latter description makes it universal for continuous derivations into any Banach A-algebra.

Here are some fun facts about modules of differentials. (See [FvdP, Theorem 3.6.3], although they mostly just refer to Springer LNM 38 “Differentialrechnung in der analytische Geometrie”):

Proposition 3. Let A be an affinoid algebra which is an integral domain, and let F be its fraction field.

(a) The dimension of Ω_{A/K} ⊗_{A} F over F equals the Krull dimension d of A.

(b) Let 𝔪 be a maximal ideal of A. Then the following are equivalent.

(i) A is smooth over K at 𝔪. (That means that locally near 𝔪, A can be written as the vanishing locus in some affine space Max T^{n} of some number m of functions whose m × n matrix of partial derivatives has maximal rank. If K is perfect, this is equivalent to A being regular at 𝔪, i.e., the localization A_{𝔪} being a regular local ring.)

(ii) The localization A_{𝔪} ⊗_{A} Ω^{A/K} is free over A_{𝔪} (of rank d).
(iii) The dimension of Ω_{A/K}/𝔪Ω_{A/K} over A/𝔪 is d.

Closed analytic subspaces

Let X be a rigid space over k, and let I be a coherent sheaf of ideals of O. Then we get a subspace of X associated to I whose points are the support of the coherent sheaf O/I (with topology induced from X), and whose structure sheaf is the restriction of O/I. On an affinoid space, this of course corresponds to viewing the vanishing locus of some ideal of functions as the Max of the quotient affinoid algebra. Any such subspace of X is called a closed analytic subspace.

Next time: GAGA

Next time, I’ll talk about separatedness and properness for rigid spaces and sketch the proof of Kiehl’s rigid GAGA theorem.

Exercises

1. Assume that K has characteristic zero. Let X be a smooth rigid space, let F be a coherent sheaf on X, and suppose that there exists a K-linear connection ∇ : F → F ⊗ Ω_{X/K}. That is, given a section s of F and a function f, we have ∇(sf) = f∇(s)+s⊗df. Prove that F must be locally free. (Hint: replace the local ring at a maximal ideal by its completion and check there that the torsion submodule must vanish.)