DESCENT FOR COHERENT SHEAVES ON RIGID-ANALYTIC SPACES
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⊗bAA0→M⊗bAA0⊗bAA0
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
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.
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
to be the difference of the twoA-linear maps just indicated. Finally, define the closed BanachA-submodule M ⊆M0 by the left exact sequence
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 dnM• : M(n) →M(n+1) is given by the usual (n+ 1)-fold “alternating difference”
formula involving the descent data:
dnM•(a01⊗b. . .⊗ba0n⊗bm0) =
(−1)i+1(a01⊗b. . .⊗bδ(a0i⊗bm0)⊗b. . .⊗ba0n)
(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 πm 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
is exact (so M is a finite A-module)and descent is effective forM0. Consider the commutative diagram of complexes ofA-modules
0 //M1 //
//. . .
0 //M //
//. . .
0 //M2 //
. . .
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 //
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) =
(−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. . .⊗bbn+1) =s(b1)b2⊗b. . .⊗bbn+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
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)
. . .
. . . //A0⊗bAM(n)
. . .
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
(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 sectionsq 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◦snq−1(x), withsnq−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 Kq denote the kernel at the far left of the qth row, then Kq 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 Kq 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)
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)⊗bA(n+1) ' //(A0⊗bAA0⊗bAA0)⊗bA0⊗bA A0(n−1) (A0⊗bAA0⊗bAA0)⊗bA0⊗bA A0n
' //(A0⊗bAA0)⊗bA0(n−1) (A0⊗bAA0)⊗bA0n
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. . .⊗banan+1oo (1⊗ba1−a1⊗b1)⊗ba2⊗b. . .⊗ban+1
(−1)n−1a1⊗ba2⊗b. . .⊗banan+1
a1⊗ba2⊗b. . .⊗ban+1
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
for allm. Moreover, since torsion-freeR-modules are flat, we have an isomorphism ToriR(·,N tors)'ToriR(·,N)
for alli >0, whereNtors denotes the torsionR-submodule ofN . Thus, we have an exact sequence (0.0.9) 0→Tor1R(R/Im,N tors)→M0/ImM0 →M00/ImM00
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
is multiplication by someπr. Since theseR-modules are all flat, it follows that the torsion submodule of the cokernel N of ∆ is annihilated by Ir. Thus, the first term in (0.0.9) is annhilated by Ir for all m. If we apply the faithfully flat base changeA/ImA →A0/ImA0 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 byIr (as it is an inverse limit ofR-modules killed by Ir). Hence, upon applying the localization functork⊗R(·), we get aninjectionthat is exactly the map
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
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.
(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
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
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
are exactly the same. Note that the injectivity of these two maps into A0⊗bAA0 follows from the fact that A/ImA →A0/ImA0 is flat for allmandI0→A hasR-flat cokernel (and hence remains injective modulo every power ofI). Passing to quotients by Im, it follows that we have an equality
(I0/ImI0)⊗A/ImA (A0/ImA0) = (A0/ImA0)⊗A/ImA (I0/ImI0)
inside of (A0/ImA0)⊗A/ImA (A0/ImA0) for all m. But by usual faithfully flat descent forA/ImA → A0/ImA0, it follows that there exists a unique finitely generated ideal Im ⊆ A/ImA that induces the finitely generated idealI0/ImI0 ⊆A0/ImA0 under base change. Also, the cokernel of A/ImA by Im
is R/Im-flat (as this can be checked after faithfully flat base change to A0/ImA0), so Im/Im−1Im → A/Im−1A is injective. The uniqueness of descent then implies that this must be the idealIm−1.
To summarize, we have finitely generated idealsIm⊆A/ImA compatible with change inm, so passage to the limit (and a reworking of [EGA, 0I, §7] without noetherian hypotheses, as in [Gu, Appendix]) yields acoherentideal
I def= lim
whose modImreduction 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
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
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.
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Department of Mathematics, University of Michigan, Ann Arbor, MI 48109, USA E-mail address:email@example.com