Gabber’s method was extended to the general case by Bosch and G¨ortz [BG]

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DESCENT FOR COHERENT SHEAVES ON RIGID-ANALYTIC SPACES

BRIAN CONRAD

Letkbe a field complete with respect to a non-trivial non-archimedean valuation. Throughout, we work with rigid-analytic spaces overk. The purpose of these notes is to prove:

Theorem. Let f : S⁰ → S be a faithfully flat quasi-compact map of rigid spaces. The functor from coherent sheaves onS to coherent sheaves onS⁰ equipped with descent data is an equivalence of categories.

If one assumeskto have a discrete valuation, then descent theory for coherent sheaves is an old result of Gabber. Gabber’s method was extended to the general case by Bosch and G¨ortz [BG]. Our method is rather different from theirs (though both approaches do use Raynaud’s theory of formal models [BL1], [BL2], we use less of this theory). We think that our approach may be of independent interest, because in contrast with all other known examples of descent (at least to this author) the method of proof is to show non-vanishing of the “descent module”beforeone proves effectivity of descent, and to invoke noetherian induction to obtain the effectivity. Further geometric examples offpqcdescent in the rigid-analytic context are discussed in [C].

Proof. We refer to the lucid exposition in [BLR1,§6.1] for the algebraic version of descent theory, which will play a crucial role in what we do. Faithfulness is obvious (without even requiring quasi-compactness off), and so for the rest we may work locally onS. Thus, we can assumeSis affinoid. Arguing exactly as in the algebraic case, since faithfulness doesn’t require quasi-compactness off we can reduce to the case whereS⁰ is affinoid too. In particular,f is now (quasi-)separated and quasi-compact. Thus, using Raynaud’s theory of formal models [BL1], [BL2] we can run through the localization argument again to reduce to the case in whichS⁰ →S is the “generic fiber” of a map of formal affines Spf(A⁰)→Spf(A) whereA →A⁰ is a faithfully flat map of topologically finitely presented and flatR-algebras (R being the valuation ring of k).

LetA→A⁰ denote the corresponding faithfully flat map ofk-affinoids.

For the full faithfulness, the theory of formal models for coherent sheaves permits us (after running through the affine localization argument a second time on the level of formal models) to use the scheme-theoretic argument, once we establish the analogue of [BLR1, 6.1/2]: ifM is a coherentA-module, then the standard complex ofA-modules

(0.0.1) 0→M →M⊗b_AA⁰→M⊗b_AA⁰⊗b_AA⁰

is exact. But this sequence is the inverse limit of the corresponding sequences of ordinary tensor products moduloIⁿfor alln≥1 (withIan ideal of definition ofR), so by left exactness of inverse limits we reduce to the standard algebraic exactness for the faithfully flat ring extensionA/IⁿA →A⁰/IⁿA⁰ and the module M/IⁿM. Theeffectivity of descent in the coherent sheaf case is much more interesting.

The point of difficulty is that the descent data isomorphism

(M⁰⊗bAA⁰)⊗Rk'(A⁰⊗bA M⁰)⊗Rk

need not respect the “integral structure”, so we cannot trivially reduce to the standard effectivity proof for modules by working modulo powers of an ideal of definition. Instead, we will use a rather curious indirect method to push through the usual commutative algebra argument with completed tensor products on the k-affinoid level.

Date: Jan. 24, 2003.

1991Mathematics Subject Classification. Primary 14G22; Secondary 14H52.

This research was partially supported by a grant from the NSF and the Clay Mathematics Institute.

1

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LetM⁰=M⁰⊗Rkbe the finiteA⁰-module corresponding toF⁰, so we have two continuousA-linear maps p2, ϕ◦p1:M⁰ ⇒A⁰⊗bAM⁰,

withp2the canonical map,p1 the canonical map toM⁰⊗bAA⁰, and ϕ:M⁰⊗bAA⁰'A⁰⊗bAM⁰

the “mysterious” A⁰⊗bAA⁰-linear descent data isomorphism (here and below, we give finite modules over affinoids their unique Banach module topologies). For convenience we will define M⁰⁰ to be the A⁰⊗bAA⁰- moduleA⁰⊗bAM⁰ and we define the continuousA-linear map

(0.0.2) δ=p2−ϕ◦p1:M⁰→M⁰⁰

to be the difference of the twoA-linear maps just indicated. Finally, define the closed BanachA-submodule M ⊆M⁰ by the left exact sequence

0→M →M⁰→M⁰⁰

We wish to prove thatM is a finitely generatedA-module and that the resultingA⁰-linear map A⁰⊗AM 'A⁰⊗bAM →M⁰

(that automatically respects the descent data) is an isomorphism. A priori, M could be 0 or could fail to be A-finite. In any case, the appearance of completed tensor products prevents a trivial application of Grothendieck’s proof from the algebraic case. Surprisingly, we will have to proveM 6= 0 wheneverM⁰ 6= 0 somewhatpriorto being able to prove thatM isA-finite and a descent ofM⁰.

For ease of notation in what follows, we letA⁽ⁿ⁾denote then-fold completed tensor product ofA⁰ overA (withA⁽⁰⁾ =A) and we letM⁽ⁿ⁾denote theA⁽ⁿ⁾-moduleA⁽ⁿ⁻¹⁾⊗bAM⁰. We wish to study that the standard Cech-descent complexˇ

M⁽¹⁾→M⁽²⁾ →M⁽³⁾→. . .

where the A-linear map dⁿ_M• : M⁽ⁿ⁾ →M⁽ⁿ⁺¹⁾ is given by the usual (n+ 1)-fold “alternating difference”

formula involving the descent data:

dⁿ_M•(a⁰₁⊗b. . .⊗ba⁰_n⊗bm⁰) =

n+1

X

i=1

(−1)ⁱ⁺¹(a⁰₁⊗b. . .⊗bδ(a⁰_i⊗bm⁰)⊗b. . .⊗ba⁰_n)

(with the understanding that in theith term, we move the “M⁰-part” back out to the far right). We will writeM⁽^•⁾ as shorthand for this complex. The complexM⁽^•⁾admits the evident “inclusion” augmentation fromM in degree 0. We will present the proof as a series of 8 steps.

Step 1: If the descent is effective, then M must be the finite A-module descent and the A-module ˇCech complex M⁽^•⁾ is a resolution of M via the augmentation.

Since we have already established full faithfulness for descent of coherent sheaves (using the crutch of formal models in (0.0.1)), we just have to show that for anyfiniteA-moduleN, the augmented ˇCech-descent complex attached to this N is exact. By choosing a coherentR-flatA-module N that is a formal model for N, our complex of interest is the result of applying the localization functor k⊗R(·) to the analogous complex ofA-modules

(0.0.3) 0→N →N ⁽^•⁾

It suffices to show that (0.0.3) is exact. Modulo anyπⁿ, (0.0.3) is the ˇCech-descent complex for theA/πⁿA- moduleN /πⁿN relative to the ordinary faithfully flat covering Spec(A⁰/πⁿA⁰)→Spec(A/πⁿA). Thus, (0.0.3) modulo π^m is exact for allm by usual descent theory. Since the transition maps in these inverse systems are surjective and allN ⁽ⁿ⁾’s are R-flat and π-adically separated and complete, the usual Mittag- Leffler argument gives exactness in the limit. This completes Step 1.

Step 2: If 0 → M₁⁰ → M⁰ →M₂⁰ → 0 is an exact sequence of finite A⁰-modules with descent data and if descent is effective forM₁⁰ andM₂⁰, then the sequence ofA-module kernels

0→M1→M →M2→0

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is exact (so M is a finite A-module)and descent is effective forM⁰. Consider the commutative diagram of complexes ofA-modules

(0.0.4) 0

0

0 //M₁ //

M₁⁰ //

M₁⁰⁰

//. . .

0 //M //

M⁰ //

M⁰⁰

//. . .

0 //M₂ //

M₂⁰ //

M₂⁰⁰ //

. . .

0 0 0

The effectivity hypothesis, coupled with Step 1, implies that the top and bottom rows are exact. By hypothesis, the second column is a short exact sequence of finite A⁰-modules. Each successive column in (0.0.4) is obtained from this by applications ofA⁽ⁿ⁺¹⁾⊗bA⁽ⁿ⁾(·) forn= 1,2, . . .. Since this functor takes short exact sequences of finiteA⁽ⁿ⁾-modules to short exact sequences of finiteA⁽ⁿ⁺¹⁾-modules, we get the short exactness of all columns after the first. Now the snake lemma ensures that the first column of kernels in (0.0.4) is short exact. In particular,M is afinite A-module.

It now makes sense to consider the commutative diagram offiniteA⁰-modules 0 //A⁰⊗AM1 //

A⁰⊗AM //

A⁰⊗AM2 //

0

0 //M₁⁰ //M⁰ //M₂⁰ //0

Both rows are short exact and the left and right vertical maps are isomorphisms by hypothesis, so the middle vertical map is an isomorphism, visibly compatible with the descent data (due to howM wasdefined). This completes Step 2.

Now comes a strengthening of Step 1 in which we make no finiteness hypotheses on M or effectivity hypotheses on the descent.

Step 3: The ˇCech-descent complex M⁽^•⁾ is exact.

Before we verify the exactness, we need to review some basic formulas. Consider a faithfully mapB →B⁰ ofk-affinoids. The examples to keep in mind areA⁽ⁿ⁾→A⁽ⁿ⁾⊗bAA⁰ (viax7→xb⊗1) forn≥1. Recall that the ˇCech-descent complex

B⁰→B⁰⊗bBB⁰ →. . .

has continuousB-linear differential dⁿ:B⁽ⁿ⁾→B⁽ⁿ⁺¹⁾forn≥1 determined by dⁿ(b1⊗b. . .⊗bbn) =

n+1

X

i=1

(−1)ⁱ⁺¹b1⊗b. . .⊗b1⊗b. . .⊗bbn

(where theith term has a 1 in theith slot). When we are given aB-linear sections:B⁰B, this complex is exact because we can usesto make an explicit homotopy between 0 and the identity. The standard choice for defining the homotopy mapskⁿ:B⁽ⁿ⁺¹⁾→B⁽ⁿ⁾is

kⁿ(b₁⊗b. . .⊗bb_n+1) =s(b₁)b₂⊗b. . .⊗bb_n+1

forn≥1, but for our purposes it is more convenient to usesⁿ:B⁽ⁿ⁺¹⁾→B⁽ⁿ⁾ defined bys⁰=sand (0.0.5) sⁿ(b1⊗b. . .⊗bbn+1) = (−1)ⁿs(bn+1)b1⊗b. . .⊗bbn

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forn≥1 (the identity dⁿ⁻¹◦sⁿ⁻¹+sⁿ◦dⁿ = id is easily checked for alln≥0). The point is that when n≥1, the homotopy mapsⁿ has no “interaction” with the first factorb₁of the tensor product (in contrast to the more standardkⁿ). This will ensure a certain compatibility later on.

Whenever we are given a ˇCech-descent complex N⁽^•⁾ for Sp(B⁰)→Sp(B) attached to descent data on a finiteB⁰-moduleN⁰ such that the descent is effective, withN⁰ descending to a finiteB-moduleN, we get a B⁰-linear isomorphismB⁰⊗bBN'N⁰ that induces acanonicalB-linear identification ofN⁽^•⁾withB⁽^•⁾⊗bBN as augmented complexes, the latter being exact by Step 1. If we are given the specification of a section s as above, then descent for coherent sheaves is effective (with the same proof as in the algebraic case, via pullback along the section), so in such cases if we let N denote the descended module then the B-linear s^•⊗bidN, with s^•as defined in (0.0.5), induces an explicitB-linear homotopy between 0 and the identity on N⁽^•⁾. This latter formula is hard to “see” in terms ofN⁽^•⁾alone, as it rests on the effectivity of the descent.

Returning to our original situation, consider the first quadrantcommutativediagram of BanachA-modules

(0.0.6) . . . .

. . . //A⁰⊗bAA⁰⊗bAM⁽ⁿ⁾

1b⊗1⊗bdⁿ_M•

//

OO

A⁰⊗bAA⁰⊗bAM⁽ⁿ⁺¹⁾ //

OO

. . .

. . . //A⁰⊗bAM⁽ⁿ⁾

1b⊗dⁿ_M_•

//

p₂−p₁

OO

A⁰⊗bAM⁽ⁿ⁺¹⁾ //

p₂−p₁

OO

. . .

//M⁽ⁿ⁾ _dn

M•

//

OO

M⁽ⁿ⁺¹⁾ //

OO

0

OO

0

OO

withn≥1 and maps defined as follows. The bottom (or 0th) row is theA-linear ˇCech-descent complexM⁽^•⁾ whose exactness we wish to prove. Forq≥1, theA^(q)-linearqth row is thecompleted tensor productextension of scalars byA^(q⁻¹⁾→A⁰⊗bAA^(q⁻¹⁾=A^(q)on theA^(q⁻¹⁾-linear (q−1)th row. Finally, thenth column is the result of applying (·)⊗bAM⁽ⁿ⁾ to the standard augmented A-linear ˇCech complex A → A⁽^•⁾ (associated to thefpqccovering Sp(A⁰)→Sp(A)). The commutativity of (0.0.6) follows fromA-linear functoriality. Using the isomorphism

A^(q)⊗bAM⁽ⁿ⁾'A^(q)⊗bAA⁽ⁿ⁾⊗bA⁽ⁿ⁾M⁽ⁿ⁾'(A⁰⊗bAA⁽ⁿ⁾)^⊗^b^q⊗bA⁽ⁿ⁾M⁽ⁿ⁾

(in which the q-fold tensor product in the right term is taken over A⁽ⁿ⁾), the nth column in (0.0.6) is transformed into the augmented ˇCech-descent complex for the finiteA⁽ⁿ⁾-moduleM⁽ⁿ⁾ relative to thefpqc covering

Sp(A⁽ⁿ⁺¹⁾) = Sp(A⁰⊗bAA⁽ⁿ⁾)→Sp(A⁽ⁿ⁾).

Thus, by Step 1, each column in (0.0.6) is exact.

Forq≥0, base change compatibility of descent theory implies that the qth row in (0.0.6) is exactly the A^(q)-linear ˇCech-descent complex for the finite A^(q+1)-moduleA^(q)⊗bAM⁰ with respect to thefpqccovering Sp(A^(q+1)) = Sp(A^(q)⊗bAA⁰)→Sp(A^(q)). Consider the sections_q that corresponds to thek-affinoid algebra map

(0.0.7) sq(a1⊗b. . .⊗baq+1) =a1⊗b. . .⊗baqaq+1

(geometrically, this is (x1, . . . , xq)7→(x1, . . . , xq, xq)). Following the homotopy convention (0.0.5), we get a homotopys^•_q between 0 and the identity for theqth row of (0.0.6). More specifically, ifx∈A^(q)⊗bAM⁽ⁿ⁾ is killed by the horizontal dⁿ for somen≥2, then x= dⁿ⁻¹◦sⁿ_q⁻¹(x), withsⁿ_q⁻¹(x)∈A^(q)⊗bAM⁽ⁿ⁻¹⁾. This explicates theexactnessof theqth row of (0.0.6).

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By a standard spectral sequence argument, the exactness of the bottom row of (0.0.6) is equivalent to the exactness of the induced sequence of horizontal kernels along the left side in rows above the bottom row. If we let K^q denote the kernel at the far left of the qth row, then K^q is a finite A^(q)-module and A⁰⊗bAK^•'A⁽^•⁾⊗bAM⁰ is an exact sequence (it is just the first column of (0.0.6), in rows≥1). If we could deduce the exactness of K^• from this, then we’d be done. However, since the completed tensor products in A⁰⊗bAK^• cannot be replaced with ordinary tensor products (as K^q is rarely A-finite), we cannot use algebraic faithful flatness to descend exactness toK^•. Thus, we use the following slightly indirect procedure that carefully analyzes the above explication of the exactness of some rows in (0.0.6) via the homotopiess_•. Doing a simple diagram chase through the bottom three rows of (0.0.6) with the help of these homotopies and the exactness of the columns, it follows that M⁽^•⁾ is exact in degree n (with n ≥ 2) as long as the diagram

(0.0.8) A⁰⊗bAA⁰⊗bAM⁽ⁿ⁻¹⁾ A⁰⊗bAA⁰⊗bAM⁽ⁿ⁾

sⁿ⁻¹₂

oo

A⁰⊗bAM⁽ⁿ⁻¹⁾

p₂−p₁

OO

A⁰⊗bAM⁽ⁿ⁾

p₂−p₁

OO

sⁿ⁻¹₁

oo

commutes, where we recall that the horizontal maps are defined in terms of our homotopy conventions (depending on the sectionss₁ ands₂) and the effectivity offpqcdescent in the presence of a section.

Unwinding the effectivity of descent, the descended modules drop out (via functoriality) and we are left with checking the commutativity of the diagram

(A⁰)^⊗^bÂ⁽ⁿ⁺¹⁾ ^' //(A⁰⊗bAA⁰⊗bAA⁰)^⊗^bÂ0^⊗^b^{A A}⁰⁽ⁿ⁻¹⁾ (A⁰⊗bAA⁰⊗bAA⁰)^⊗^bÂ0^⊗^b^{A A}⁰ⁿ

sⁿ⁻¹₂

oo (A⁰)^b^⊗^A⁽ⁿ⁺²⁾^'oo

(A⁰)^⊗^b^Aⁿ

p₂−p₁

OO

' //(A⁰⊗bAA⁰)^⊗^b^A0⁽ⁿ⁻¹⁾ (A⁰⊗bAA⁰)^⊗^b^A0ⁿ

sⁿ⁻¹₁

oo (A⁰)^b^⊗^A⁽ⁿ⁺¹⁾

p₂−p₁

OO

oo '

Recalling (0.0.5) and (0.0.7), checking the commutativity of this diagram is a simple computation that we illustrate as follows:

(−1)ⁿ⁻¹(1⊗ba₁−a₁⊗b1)⊗ba₂⊗b. . .⊗ba_na_n+1oo (1⊗ba₁−a₁⊗b1)⊗ba₂⊗b. . .⊗ba_n+1

(−1)ⁿ⁻¹a1⊗ba2⊗b. . .⊗banan+1

OO

a1⊗ba2⊗b. . .⊗ban+1

oo OO

The point is that sincen+ 1≥3, the products anan+1 do not interfere witha1. Step 4: IfM⁰6= 0 thenM 6= 0.

In other words, ifδin (0.0.2) is injective then we claimM⁰= 0. Beware that it isnotobvious that 1⊗bδis necessarily injective. We will eventually deduce this with the help of Step 3. It seems rather hard to directly prove Step 4. We shall instead prove the contrapositive using formal models and the Banach Open Mapping Theorem.

As we noted already, the A-linear δ in (0.0.2) might not take the R-flat M⁰ ⊆ M⁰ into the R-flat M⁰⁰⊆M⁰⁰. However, if we choose an ideal of definitionI= (π) forRand let ∆ =πⁿδfor suitably largen, then ∆ is injective and takesM⁰ intoM⁰⁰. Consider the exact sequence of R-modules

0→M⁰−→^∆ M⁰⁰→N →0 Thus, byR-flatness of M⁰⁰ we have exact sequences

0→Tor¹_R(R/I^m,N)→M⁰/I^mM⁰→M⁰⁰/I^mM⁰⁰

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for allm. Moreover, since torsion-freeR-modules are flat, we have an isomorphism Torⁱ_R(·,N tors)'Torⁱ_R(·,N)

for alli >0, whereNtors denotes the torsionR-submodule ofN . Thus, we have an exact sequence (0.0.9) 0→Tor¹_R(R/I^m,N tors)→M⁰/I^mM⁰ →M⁰⁰/I^mM⁰⁰

The advantage of usingN tors to compute higher Tor’s is that we can actually produce a single power ofπ that annihilates it! In order to establish this, we note that by Step 3, the map δ:M⁽¹⁾ →M⁽²⁾ hasclosed image (namely, the kernel of a suitable continuous mapM⁽²⁾→M⁽³⁾).

Since δ is a continuous injection between k-Banach spaces, the closedness of its image implies (by the Banach open mapping theorem) that δ must be a closed embedding. But the source and target of δ are countable typek-Banach spaces (i.e., have dense subspaces of countable dimension), so it follows from [BGR, 2.7.1/4] thatδadmits acontinuousk-linear splitting. Multiplying this by a suitable power ofπ, we conclude that there is anR-module mapM⁰⁰→M⁰ such that the composite

M⁰−→^∆ M⁰⁰→M⁰

is multiplication by someπ^r. Since theseR-modules are all flat, it follows that the torsion submodule of the cokernel N of ∆ is annihilated by I^r. Thus, the first term in (0.0.9) is annhilated by I^r for all m. If we apply the faithfully flat base changeA/I^mA →A⁰/I^mA⁰ to (0.0.9) and pass to the inverse limit onm, we arrive at an exact sequence

0→K →A⁰⊗bA M⁰→A⁰⊗bA M⁰⁰

in whichK is killed byI^r (as it is an inverse limit ofR-modules killed by I^r). Hence, upon applying the localization functork⊗R(·), we get aninjectionthat is exactly the map

1⊗bδ:A⁰⊗bAM⁰→A⁰⊗bAM⁰⁰

The map 1⊗bδ is exactly the analogue of δ after we apply the base change Sp(A⁰)→ Sp(A) throughout our original descent data situation. But as we noted in the proof of Step 3, since after such a base change the structure map through which we want to do descent acquires a section, descent for coherent sheaves in such a situation is always effective (with the descended module exactly the kernel of the first map in the Cech-descent complex, by Step 1). We conclude from the vanishing of the kernel of 1ˇ ⊗bδthat the A⁰⊗bAA⁰- moduleA⁰⊗bAM⁰ is the base change (via second projection) of the A⁰-module 0. That is,A⁰⊗bAM⁰= 0. But we have an isomorphism

A⁰⊗bAM⁰'(A⁰⊗bAA⁰)b⊗A⁰M⁰'(A⁰⊗bAA⁰)⊗A⁰M⁰

sinceM⁰is a finiteA⁰-module. SinceA⁰⊗bAA⁰is a faithfully flatA⁰-algebra, it follows thatM⁰= 0, as desired.

In other words, we have proven that whenM⁰ 6= 0 thenM 6= 0.

Step 5: Reduction to the case where M spansM⁰ as an A⁰-module.

We may suppose M⁰ 6= 0, so by Step 4 we have M 6= 0. Let M₁⁰ 6= 0 be the A⁰-span of M inside of the finite A⁰-moduleM⁰. Due to the very definition ofM, it follows that M₁⁰ inherits descent data from M⁰ in an evident sense (sinceM has the sameA-linear map toM⁰⁰under either pullback fromM⁰, due to invariance under the descent data action, andM₁⁰⁰is simply the (A⁰⊗bAA⁰)-span ofM inside ofM⁰⁰). Defining M₂⁰ =M⁰/M₁⁰, Step 2 reduces us to the analysis of M₁⁰ and M₂⁰. Note thatM1 is an A-submodule of M₁⁰ which is nothing other than M, soM₁⁰ is theA⁰-span ofM1. If M₂⁰ = 0 then M⁰ =M₁⁰ is the A⁰-span of M, so we’d be done. If M₂⁰ 6= 0 then M2 6= 0 by Step 4, so we can repeat the process to get a non-zero A⁰-submoduleM₃⁰ ofM₂⁰ (that corresponds to anA⁰-submodule ofM⁰ strictly containingM₁⁰) and the game continues.

As we keep encountering non-zeroA⁰-modules, we build up a strictly increasing chain ofA⁰-submodules ofM⁰ beginning withM₁⁰. SinceM⁰ is a finiteA⁰-module andA⁰is noetherian, this process must eventually stop, at which point we have achieved a descent-data-stable filtration ofM⁰byA⁰-modules whose successive quotients satisfy the condition in our reduction step. Thanks to Step 2, this completes Step 5.

Step 6: Reduction to the case where a single element ofM spansM⁰ as an A⁰-module.

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(Beware that westillhave not yet shown thatM is even afiniteA-module.)

Since M⁰ is spanned by M as an A⁰-module by Step 5, we can let n ≥ 1 denote the minimal positive number of elements of M that suffice to spanM⁰ overA⁰. Letm1, . . . mn∈M be A⁰-module generators of M⁰. Defining M₁⁰ =A⁰m1 and M₂⁰ =M⁰/M₁⁰, it is obvious that M₁⁰, M₂⁰ are compatible with the descent data and that the images of the (n−1) elementsm2, . . . , mn inM2 spanM₂⁰ as anA⁰-module. By induction onnand Step 2, we’re done with Step 6.

Now that a single element ofM spansM⁰ as anA⁰-module, we haveM⁰'A⁰/I⁰as an abstractA⁰-module, whereI⁰= annA⁰(M⁰).

Step 7: The idealI⁰ ofA⁰ descends uniquely to an ideal I ofA.

The uniqueness is clear, so we are only concerned with existence. Since the formation of the annihilator ideal of a coherent sheaf commutes with flat base change, the descent data onM⁰induces compatible descent data onI⁰. More specifically, we have the literal equality

(0.0.10) I⁰⊗bAA⁰=A⁰⊗bAI⁰

inside of A⁰⊗bAA⁰. Let I⁰ = ker(A⁰ →A⁰/I⁰), so A⁰/I⁰ is R-flat. By [BL1, 1.2(c)], I⁰ is automatically a coherent ideal in the coherent ring A⁰, so A⁰/I⁰ is an R-flat formal model of A⁰/I⁰. In particular, the injectionI⁰→A⁰ remains injective modulo all powers ofπ.

We conclude that the two closed immersion

(0.0.11) Spf((A⁰/I⁰)b⊗AA⁰),Spf(A⁰⊗bA(A⁰/I⁰)),→Spf(A⁰⊗bAA⁰)

are Spf(R)-flat closed formal subschemes that are formal models for the closed rigid analytic subspaces Sp((A⁰/I⁰)⊗bAA⁰),Sp(A⁰⊗bA(A⁰/I⁰)),→Sp(A⁰⊗bAA⁰)

But (0.0.10) implies that these latter closed subspaces literally coincide, whence byR-flatness we conclude that the formal closed subschemes (0.0.11) literally coincide. In other words, the defining ideals

I⁰⊗bAA⁰,A⁰⊗bAI⁰ ,→A⁰⊗bAA⁰

are exactly the same. Note that the injectivity of these two maps into A⁰⊗b_AA⁰ follows from the fact that A/I^mA →A⁰/I^mA⁰ is flat for allmandI⁰→A hasR-flat cokernel (and hence remains injective modulo every power ofI). Passing to quotients by I^m, it follows that we have an equality

(I⁰/I^mI⁰)⊗_A/I^mA (A⁰/I^mA⁰) = (A⁰/I^mA⁰)⊗_A/I^mA (I⁰/I^mI⁰)

inside of (A⁰/I^mA⁰)⊗_A/I^mA (A⁰/I^mA⁰) for all m. But by usual faithfully flat descent forA/I^mA → A⁰/I^mA⁰, it follows that there exists a unique finitely generated ideal Im ⊆ A/I^mA that induces the finitely generated idealI⁰/I^mI⁰ ⊆A⁰/I^mA⁰ under base change. Also, the cokernel of A/I^mA by Im

is R/I^m-flat (as this can be checked after faithfully flat base change to A⁰/I^mA⁰), so Im/I^m⁻¹Im → A/I^m⁻¹A is injective. The uniqueness of descent then implies that this must be the idealIm−1.

To summarize, we have finitely generated idealsIm⊆A/I^mA compatible with change inm, so passage to the limit (and a reworking of [EGA, 0_I, §7] without noetherian hypotheses, as in [Gu, Appendix]) yields acoherentideal

I ^def= lim

←−Im⊆A

whose modI^mreduction isImfor allm≥1. Since theI-adic topology onA induces that on I by [BL1, 1.2(a)], the topologically faithfully flat base changeA →A⁰ consequently induces an identification

A⁰⊗bAI =I⁰

inside ofA⁰. Now passing to the affinoid world by localization, the idealI=k⊗RI ⊆A inducesI⁰ under base change toA⁰. This finishes Step 7.

Step 8: Completion of the proof.

It is clear that we can now replace A with A/I so as to reduce to the case I⁰ = 0. That is, we have M⁰ ' A⁰ as A⁰-modules where 1 ∈ A⁰ corresponds to an element of M, which is to say it is an element of M⁰ that is invariant under the descent data. It is then automatic that carrying the descent data across

(8)

the isomorphismM⁰ 'A⁰ must induce on A⁰⊗bAA⁰ 'M⁰⁰ exactly theusualtrivially effective descent data structure for the finiteA-moduleArelative to the faithfully flat covering Sp(A⁰)→Sp(A). Translating back

toM⁰, the original descent is effective.

References

[BGR] S. Bosch, U. G¨unzter, R. Remmert,Non-Archimedean analysis, Springer-Verlag, 1984.

[BL1] S. Bosch, W. L¨utkebohmert,Formal and rigid geometryI, Math. Annalen,295(1993), pp. 291–317.

[BL2] S. Bosch, W. L¨utkebohmert,Formal and rigid geometryII, Math. Annalen,296(1993), pp. 403–429.

[BLR1] S. Bosch, W. L¨utkebohmert, M. Raynaud,N´eron Models, Springer-Verlag, 1984.

[BG] S. Bosch, U. G¨ortz,Coherent modules and their descent on relative rigid spaces, J. Reine Angew. Math.495(1998), pp.

119–134.

[C] B. Conrad,Ampleness and descent on rigid-analytic spaces, in preparation.

[EGA] J. Dieudonné, A. Grothendieck,El´éments de géométrie algébrique, Publ. Math. IHES,4, 8, 11, 17, 20, 24, 28, 32, (1960-7).

[Gu] W. Gubler,Local heights on subvarieties over non-Archimedean fields, J. Reine Angew. Math.498(1998), pp. 61–113.

Department of Mathematics, University of Michigan, Ann Arbor, MI 48109, USA E-mail address:bdconrad@math.lsa.umich.edu