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 : S^{0} → S be a faithfully flat quasi-compact map of rigid spaces. The functor from
coherent sheaves onS to coherent sheaves onS^{0} 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 whereS^{0}
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 whichS^{0} →S is the “generic fiber” of a map of formal affines Spf(A^{0})→Spf(A) whereA →A^{0} is a
faithfully flat map of topologically finitely presented and flatR-algebras (R being the valuation ring of k).

LetA→A^{0} 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_{A}A^{0}→M⊗b_{A}A^{0}⊗b_{A}A^{0}

is exact. But this sequence is the inverse limit of the corresponding sequences of ordinary tensor products
moduloI^{n}for 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/I^{n}A →A^{0}/I^{n}A^{0} and the module
M/I^{n}M. Theeffectivity of descent in the coherent sheaf case is much more interesting.

The point of difficulty is that the descent data isomorphism

(M^{0}⊗bAA^{0})⊗Rk'(A^{0}⊗bA M^{0})⊗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

LetM^{0}=M^{0}⊗Rkbe the finiteA^{0}-module corresponding toF^{0}, so we have two continuousA-linear maps
p2, ϕ◦p1:M^{0} ⇒A^{0}⊗bAM^{0},

withp2the canonical map,p1 the canonical map toM^{0}⊗bAA^{0}, and
ϕ:M^{0}⊗bAA^{0}'A^{0}⊗bAM^{0}

the “mysterious” A^{0}⊗bAA^{0}-linear descent data isomorphism (here and below, we give finite modules over
affinoids their unique Banach module topologies). For convenience we will define M^{00} to be the A^{0}⊗bAA^{0}-
moduleA^{0}⊗bAM^{0} and we define the continuousA-linear map

(0.0.2) δ=p2−ϕ◦p1:M^{0}→M^{00}

to be the difference of the twoA-linear maps just indicated. Finally, define the closed BanachA-submodule
M ⊆M^{0} by the left exact sequence

0→M →M^{0}→M^{00}

We wish to prove thatM is a finitely generatedA-module and that the resultingA^{0}-linear map
A^{0}⊗AM 'A^{0}⊗bAM →M^{0}

(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 wheneverM^{0} 6= 0
somewhatpriorto being able to prove thatM isA-finite and a descent ofM^{0}.

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

M^{(1)}→M^{(2)} →M^{(3)}→. . .

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

formula involving the descent data:

d^{n}_{M}•(a^{0}_{1}⊗b. . .⊗ba^{0}_{n}⊗bm^{0}) =

n+1

X

i=1

(−1)^{i+1}(a^{0}_{1}⊗b. . .⊗bδ(a^{0}_{i}⊗bm^{0})⊗b. . .⊗ba^{0}_{n})

(with the understanding that in theith term, we move the “M^{0}-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 anyπ^{n}, (0.0.3) is the ˇCech-descent complex for theA/π^{n}A-
moduleN /π^{n}N relative to the ordinary faithfully flat covering Spec(A^{0}/π^{n}A^{0})→Spec(A/π^{n}A). 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 → M_{1}^{0} → M^{0} →M_{2}^{0} → 0 is an exact sequence of finite A^{0}-modules with descent data and if
descent is effective forM_{1}^{0} andM_{2}^{0}, then the sequence ofA-module kernels

0→M1→M →M2→0

is exact (so M is a finite A-module)and descent is effective forM^{0}.
Consider the commutative diagram of complexes ofA-modules

(0.0.4) 0

0

0

0 //M_{1} //

M_{1}^{0} //

M_{1}^{00}

//. . .

0 //M //

M^{0} //

M^{00}

//. . .

0 //M_{2} //

M_{2}^{0} //

M_{2}^{00} //

. . .

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 A^{0}-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 offiniteA^{0}-modules
0 //A^{0}⊗AM1 //

A^{0}⊗AM //

A^{0}⊗AM2 //

0

0 //M_{1}^{0} //M^{0} //M_{2}^{0} //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 →B^{0}
ofk-affinoids. The examples to keep in mind areA^{(n)}→A^{(n)}⊗bAA^{0} (viax7→xb⊗1) forn≥1. Recall that
the ˇCech-descent complex

B^{0}→B^{0}⊗bBB^{0} →. . .

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

n+1

X

i=1

(−1)^{i+1}b1⊗b. . .⊗b1⊗b. . .⊗bbn

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

k^{n}(b_{1}⊗b. . .⊗bb_{n+1}) =s(b_{1})b_{2}⊗b. . .⊗bb_{n+1}

forn≥1, but for our purposes it is more convenient to uses^{n}:B^{(n+1)}→B^{(n)} defined bys^{0}=sand
(0.0.5) s^{n}(b1⊗b. . .⊗bbn+1) = (−1)^{n}s(bn+1)b1⊗b. . .⊗bbn

forn≥1 (the identity d^{n}^{−}^{1}◦s^{n}^{−}^{1}+s^{n}◦d^{n} = id is easily checked for alln≥0). The point is that when
n≥1, the homotopy maps^{n} has no “interaction” with the first factorb_{1}of the tensor product (in contrast
to the more standardk^{n}). This will ensure a certain compatibility later on.

Whenever we are given a ˇCech-descent complex N^{(}^{•}^{)} for Sp(B^{0})→Sp(B) attached to descent data on a
finiteB^{0}-moduleN^{0} such that the descent is effective, withN^{0} descending to a finiteB-moduleN, we get a
B^{0}-linear isomorphismB^{0}⊗bBN'N^{0} 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) . . . .

. . . //A^{0}⊗bAA^{0}⊗bAM^{(n)}

1b⊗1⊗bd^{n}_{M•}

//

OO

A^{0}⊗bAA^{0}⊗bAM^{(n+1)} //

OO

. . .

. . . //A^{0}⊗bAM^{(n)}

1b⊗d^{n}_{M}_{•}

//

p_{2}−p_{1}

OO

A^{0}⊗bAM^{(n+1)} //

p_{2}−p_{1}

OO

. . .

//M^{(n)} _{d}n

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)}→A^{0}⊗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(A^{0})→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)}'(A^{0}⊗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(A^{0}⊗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)}⊗bAM^{0} with respect to thefpqccovering
Sp(A^{(q+1)}) = Sp(A^{(q)}⊗bAA^{0})→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 d^{n} for somen≥2, then x= d^{n}^{−}^{1}◦s^{n}_{q}^{−}^{1}(x), withs^{n}_{q}^{−}^{1}(x)∈A^{(q)}⊗bAM^{(n}^{−}^{1)}. This
explicates theexactnessof theqth row of (0.0.6).

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
A^{0}⊗bAK^{•}'A^{(}^{•}^{)}⊗bAM^{0} 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 A^{0}⊗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) A^{0}⊗bAA^{0}⊗bAM^{(n}^{−}^{1)} A^{0}⊗bAA^{0}⊗bAM^{(n)}

s^{n−1}_{2}

oo

A^{0}⊗bAM^{(n}^{−}^{1)}

p_{2}−p_{1}

OO

A^{0}⊗bAM^{(n)}

p_{2}−p_{1}

OO

s^{n−1}_{1}

oo

commutes, where we recall that the horizontal maps are defined in terms of our homotopy conventions
(depending on the sectionss_{1} ands_{2}) 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

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

s^{n−1}_{2}

oo (A^{0})^{b}^{⊗}^{A}^{(n+2)}^{'}oo

(A^{0})^{⊗}^{b}^{A}^{n}

p_{2}−p_{1}

OO

' //(A^{0}⊗bAA^{0})^{⊗}^{b}^{A0}^{(n}^{−}^{1)} (A^{0}⊗bAA^{0})^{⊗}^{b}^{A0}^{n}

s^{n−1}_{1}

oo (A^{0})^{b}^{⊗}^{A}^{(n+1)}

p_{2}−p_{1}

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⊗ba_{1}−a_{1}⊗b1)⊗ba_{2}⊗b. . .⊗ba_{n}a_{n+1}oo (1⊗ba_{1}−a_{1}⊗b1)⊗ba_{2}⊗b. . .⊗ba_{n+1}

(−1)^{n}^{−}^{1}a1⊗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: IfM^{0}6= 0 thenM 6= 0.

In other words, ifδin (0.0.2) is injective then we claimM^{0}= 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 M^{0} ⊆ M^{0} into the R-flat
M^{00}⊆M^{00}. However, if we choose an ideal of definitionI= (π) forRand let ∆ =π^{n}δfor suitably largen,
then ∆ is injective and takesM^{0} intoM^{00}. Consider the exact sequence of R-modules

0→M^{0}−→^{∆} M^{00}→N →0
Thus, byR-flatness of M^{00} we have exact sequences

0→Tor^{1}_{R}(R/I^{m},N)→M^{0}/I^{m}M^{0}→M^{00}/I^{m}M^{00}

for allm. Moreover, since torsion-freeR-modules are flat, we have an isomorphism
Tor^{i}_{R}(·,N tors)'Tor^{i}_{R}(·,N)

for alli >0, whereNtors denotes the torsionR-submodule ofN . Thus, we have an exact sequence
(0.0.9) 0→Tor^{1}_{R}(R/I^{m},N tors)→M^{0}/I^{m}M^{0} →M^{00}/I^{m}M^{00}

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 mapM^{00}→M^{0} such that the composite

M^{0}−→^{∆} M^{00}→M^{0}

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^{m}A →A^{0}/I^{m}A^{0} to (0.0.9) and pass to the inverse limit onm, we
arrive at an exact sequence

0→K →A^{0}⊗bA M^{0}→A^{0}⊗bA M^{00}

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:A^{0}⊗bAM^{0}→A^{0}⊗bAM^{00}

The map 1⊗bδ is exactly the analogue of δ after we apply the base change Sp(A^{0})→ 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 A^{0}⊗bAA^{0}-
moduleA^{0}⊗bAM^{0} is the base change (via second projection) of the A^{0}-module 0. That is,A^{0}⊗bAM^{0}= 0. But
we have an isomorphism

A^{0}⊗bAM^{0}'(A^{0}⊗bAA^{0})b⊗A^{0}M^{0}'(A^{0}⊗bAA^{0})⊗A^{0}M^{0}

sinceM^{0}is a finiteA^{0}-module. SinceA^{0}⊗bAA^{0}is a faithfully flatA^{0}-algebra, it follows thatM^{0}= 0, as desired.

In other words, we have proven that whenM^{0} 6= 0 thenM 6= 0.

Step 5: Reduction to the case where M spansM^{0} as an A^{0}-module.

We may suppose M^{0} 6= 0, so by Step 4 we have M 6= 0. Let M_{1}^{0} 6= 0 be the A^{0}-span of M inside of
the finite A^{0}-moduleM^{0}. Due to the very definition ofM, it follows that M_{1}^{0} inherits descent data from
M^{0} in an evident sense (sinceM has the sameA-linear map toM^{00}under either pullback fromM^{0}, due to
invariance under the descent data action, andM_{1}^{00}is simply the (A^{0}⊗bAA^{0})-span ofM inside ofM^{00}). Defining
M_{2}^{0} =M^{0}/M_{1}^{0}, Step 2 reduces us to the analysis of M_{1}^{0} and M_{2}^{0}. Note thatM1 is an A-submodule of M_{1}^{0}
which is nothing other than M, soM_{1}^{0} is theA^{0}-span ofM1. If M_{2}^{0} = 0 then M^{0} =M_{1}^{0} is the A^{0}-span of
M, so we’d be done. If M_{2}^{0} 6= 0 then M2 6= 0 by Step 4, so we can repeat the process to get a non-zero
A^{0}-submoduleM_{3}^{0} ofM_{2}^{0} (that corresponds to anA^{0}-submodule ofM^{0} strictly containingM_{1}^{0}) and the game
continues.

As we keep encountering non-zeroA^{0}-modules, we build up a strictly increasing chain ofA^{0}-submodules
ofM^{0} beginning withM_{1}^{0}. SinceM^{0} is a finiteA^{0}-module andA^{0}is noetherian, this process must eventually
stop, at which point we have achieved a descent-data-stable filtration ofM^{0}byA^{0}-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 spansM^{0} as an A^{0}-module.

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

Since M^{0} is spanned by M as an A^{0}-module by Step 5, we can let n ≥ 1 denote the minimal positive
number of elements of M that suffice to spanM^{0} overA^{0}. Letm1, . . . mn∈M be A^{0}-module generators of
M^{0}. Defining M_{1}^{0} =A^{0}m1 and M_{2}^{0} =M^{0}/M_{1}^{0}, it is obvious that M_{1}^{0}, M_{2}^{0} are compatible with the descent
data and that the images of the (n−1) elementsm2, . . . , mn inM2 spanM_{2}^{0} as anA^{0}-module. By induction
onnand Step 2, we’re done with Step 6.

Now that a single element ofM spansM^{0} as anA^{0}-module, we haveM^{0}'A^{0}/I^{0}as an abstractA^{0}-module,
whereI^{0}= annA^{0}(M^{0}).

Step 7: The idealI^{0} ofA^{0} 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 onM^{0}induces compatible descent
data onI^{0}. More specifically, we have the literal equality

(0.0.10) I^{0}⊗bAA^{0}=A^{0}⊗bAI^{0}

inside of A^{0}⊗bAA^{0}. Let I^{0} = ker(A^{0} →A^{0}/I^{0}), so A^{0}/I^{0} is R-flat. By [BL1, 1.2(c)], I^{0} is automatically
a coherent ideal in the coherent ring A^{0}, so A^{0}/I^{0} is an R-flat formal model of A^{0}/I^{0}. In particular, the
injectionI^{0}→A^{0} remains injective modulo all powers ofπ.

We conclude that the two closed immersion

(0.0.11) Spf((A^{0}/I^{0})b⊗AA^{0}),Spf(A^{0}⊗bA(A^{0}/I^{0})),→Spf(A^{0}⊗bAA^{0})

are Spf(R)-flat closed formal subschemes that are formal models for the closed rigid analytic subspaces
Sp((A^{0}/I^{0})⊗bAA^{0}),Sp(A^{0}⊗bA(A^{0}/I^{0})),→Sp(A^{0}⊗bAA^{0})

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

I^{0}⊗bAA^{0},A^{0}⊗bAI^{0} ,→A^{0}⊗bAA^{0}

are exactly the same. Note that the injectivity of these two maps into A^{0}⊗b_{A}A^{0} follows from the fact that
A/I^{m}A →A^{0}/I^{m}A^{0} is flat for allmandI^{0}→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

(I^{0}/I^{m}I^{0})⊗_{A}/I^{m}A (A^{0}/I^{m}A^{0}) = (A^{0}/I^{m}A^{0})⊗_{A}/I^{m}A (I^{0}/I^{m}I^{0})

inside of (A^{0}/I^{m}A^{0})⊗_{A}/I^{m}A (A^{0}/I^{m}A^{0}) for all m. But by usual faithfully flat descent forA/I^{m}A →
A^{0}/I^{m}A^{0}, it follows that there exists a unique finitely generated ideal Im ⊆ A/I^{m}A that induces the
finitely generated idealI^{0}/I^{m}I^{0} ⊆A^{0}/I^{m}A^{0} under base change. Also, the cokernel of A/I^{m}A by Im

is R/I^{m}-flat (as this can be checked after faithfully flat base change to A^{0}/I^{m}A^{0}), so Im/I^{m}^{−}^{1}Im →
A/I^{m}^{−}^{1}A is injective. The uniqueness of descent then implies that this must be the idealIm−1.

To summarize, we have finitely generated idealsIm⊆A/I^{m}A 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^{m}reduction isImfor allm≥1. Since theI-adic topology onA induces that on I by [BL1,
1.2(a)], the topologically faithfully flat base changeA →A^{0} consequently induces an identification

A^{0}⊗bAI =I^{0}

inside ofA^{0}. Now passing to the affinoid world by localization, the idealI=k⊗RI ⊆A inducesI^{0} under
base change toA^{0}. 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 I^{0} = 0. That is, we have
M^{0} ' A^{0} as A^{0}-modules where 1 ∈ A^{0} corresponds to an element of M, which is to say it is an element
of M^{0} that is invariant under the descent data. It is then automatic that carrying the descent data across

the isomorphismM^{0} 'A^{0} must induce on A^{0}⊗bAA^{0} 'M^{00} exactly theusualtrivially effective descent data
structure for the finiteA-moduleArelative to the faithfully flat covering Sp(A^{0})→Sp(A). Translating back

toM^{0}, 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. Dieudonn´e, A. Grothendieck,El´´ements de g´eom´etrie alg´ebrique, 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