Algebraization of bundles on non-proper schemes

We consider the algebraization problem for principal bundles with reductive structure group, defined on the complement of a closed subset Z in a proper formal scheme. We show that, when Z is of codimension at least 3, an algebraization always exists. For codimension 2 we show that an algebraization exists precisely when a certain additional condition is satisfied.


Introduction
This work is a contribution toward an algebraic understanding of the Uhlenbeck compactification. Recall, cf. [DK] that for a complex projective surface S the moduli space M n of semistable vector bundles with fixed rank, determinant and c 2 = n is non-compact, but the union U hl n = s≥0 M n−s × Sym s S can be given a topology of a compact space (since one deals with semistable bundles for s ≫ 0 the space M n−s will be empty). We will call U hl n the Uhlenbeck moduli space although sometimes this name is reserved for the closure of M n in U hl n .
Some time ago, see e.g. [Li], [BFG], [FGK], the Uhlenbeck moduli space started to appear in algebraic geometry and higher dimensional Langlands Program. For instance, it is a convenient tool for the study of higher versions of Hecke correspondences which modify a vector bundle on S (more generally, a principal bundle) along a divisor, obtaining a new bundle. For several reasons, we would like to have a definition of U hl n as a "functor", i.e. we want to be able to describe in geometric terms the set of maps F (T ) (actually, a category of maps) from any test scheme T = Spec(A) to U hl n . Firstly, that would allow to define U hl n over any field k and not to require stability. Secondly, in the study of the cohomology of U hl n and the action of Hecke correspondences on it, one needs to deal with the phenomenon of unexpected dimension of U hl n . A possible approach involves defining a "derived moduli space" DU hl n in the sense of [Lu] which would amount to considering more general "spaces" T . Thus, defining U hl n as a functor is a necessary preliminary step to constructing DU hl n .
Very roughly, it is expected that a map T → U hl n should be described by a vector bundle F on an open subset U ⊂ T × S such that its complement Z is finite over T , a family ξ of effective zero cycles on X parametrized by T plus an agreement condition between ξ and F . Such a definition gives a "reasonable space" U hl n if it satisfies a criterion due to Artin, cf. [Ar], or its "derived" generalization proved in [Lu]. The most difficult part of Artin's criterion is the effectiveness condition: if A is a complete noetherian local k-algebra with maximal ideal m and A p = A/m p+1 one needs to show that F (Spec(A)) = lim ← − F (Spec(A p )). Ignoring the family of zero cycles ξ (as will be done in this paper), if X = Spec(A) × k S and X is its formal completion along the fiber over the closed point of Spec(A), we are trying to find whether a bundle F on an open subset U ⊂ X comes from a bundle F on an open subset U ⊂ X. Such F is called an algebraization of F.
In this paper we prove that, when S has arbitrary dimension and U has complement of codimension ≥ 3, algebraization always exists (for vector bundles and also for principal bundles over reductive groups). If U has complement of codimension ≥ 2 then algebraization exists only under an additional condition (which, in the Uhlenbeck functor case, a guaranteed due to the presence of the relatize zero cycle ξ).
Earlier similar questions were studied for coherent sheaves on proper schemes by Grothendieck, see [EGAIII], and in the case of Lefschetz type theorems by Grothendieck and Raynaud in [SGA2], and [R]. Although these results do not apply in our case directly, our proof is based on the tools developed in [EGAIII], [SGA2].
In Section 2 we fix the notation, give examples illustrating some problems to be encountered, and prove algebraization results for vector bundles, summarized in Corollary 8. In Section 3 we formulate an algebraization criterion for principal bundles over reductive groups, see Theorem 9. Finally, Section 4 provides a categorical restatement of our results, see Theorem 13.
Acknowledgements. The author thanks V. Ginzburg who first formulated the problem of defining the Uhlenbeck functor and whose unpublished preprint on it (written jointly with the present author) served as a principal motivation for this work. Many thanks are also due to V. Drinfeld who conjectured the statement of Theorem 6(i), brought the author's attention to the references [SGA2] and [Ha1], and also suggested Example 3 in Section 2.2 below. This work was supported by the Sloan Research Fellowship.

Setup
We refer the reader to Expose III in [SGA2] regarding basic properties of depth and its relation to local cohomology. Let S be an irreducible noetherian scheme of finite type over a field k. We will assume that S is proper and satisfies Serre's S 2 condition: for any s ∈ S, depth s O S ≥ min(dim O S,s , 2). Let V ⊂ S be an open subset with closed complement of codimension ≥ 2 in S and A a complete noetherian local k-algebra with residue field K = A/m and associated graded K-algebra gr(A) = ⊕ p≥0 gr p (A) = ⊕ p≥0 m p /m p+1 . Define X = S × k Spec(A) and p ≥ 0 Let i p : U p → X p be the natural open embeddings. Denote by X the completion of X along X 0 , which may be viewed at the limit of {X p } p≥0 , cf. Section 10.6 in [EGAI]. The open subset U 0 ⊂ X 0 defines an open formal subscheme i : U → X, given by the limit of {U p } p≥0 . The ideal sheaf of X 0 in X will be denoted by by J X and the closed subset X 0 \ U 0 by Z 0 . Finally, f : X → Spec(A) is the natural proper projection and, for any s ∈ Spec(A), X s stands for the fiber f −1 (s).
Observe that X may no longer satisfy the S 2 conditon (since we made no depth assumptions on A).
However, for f (x) = s we can lift a regular sequence from O Xs,x to O X,x which gives Consider a vector bundle F on U , i.e. a sequence of vector bundles F p on U p with isomorphisms Definition. We will say that a vector bundle F on U admits an algebraization if there exists an open subset U ⊂ X with U ∩ X 0 = U 0 and a vector bundle F on U such that F is isomorphic to the completion of F , i.e. for J U = J X | U there exist isomorphisms F p ≃ F/J p+1 U F compatible with (1). In Section 3 we apply similar terminology to principal bundles.
Let Z be the closed subset X \ U and i : U ֒→ X the open embedding.
Lemma 2 Let codim X 0 Z 0 ≥ 2 and suppose that U ⊂ X is an open subset such that U ∩ X 0 = U 0 . For any s ∈ Spec(A) define Z s = Z ∩X s . Then codim Xs Z s ≥ 2 for all s ∈ SpecA and codim X Z ≥ 2.
Proof. Since f is proper, the image f (Z s ) contains the unique closed point s 0 ∈ Spec(A). Therefore Z s ∩ X 0 ⊂ Z 0 is not empty. By semicontinuity of dimensions in the fibers we have codim Xs Z s ≥ codim X 0 (Z s ∩ X 0 ) ≥ codim X 0 Z 0 = 2. The second assertion of the lemma follows from the first.
In our discussion, we repeatedly use the following results Proposition 3 In the notation introduced above (i). Completion along X 0 induces an equivalence between the category of coherent sheaves on X and the category of coherent sheaves on the formal scheme X.

(ii). For any locally free sheaf
. Let E be a coherent sheaf on X and ψ : E → i * i * E the canonical morphism. Then ψ is an isomorphism if and only if depth x E ≥ 2 for any point x ∈ Z = X \ U .
Proof. Part (i) follows from Corollary 5.1.6 in [EGAIII]. To check the coherence of i * F , by Corollary VIII.2.3 in [SGA2] it suffices to check that depth x F ≥ 1 for any point x ∈ U such that {x} ∩ Z has codimension 1 in {x}. But Lemma 1 and local freeness of F imply that any x with depth x F = 0 must be generic in its fiber, and the Lemma 2 implies that {x} ∩ Z would in fact have codimension 2 in {x}. The same proof applies to (i 0 ) * F 0 . If codim X 0 Z 0 ≥ 3 then the above argument can also by applied to R 1 (i 0 ) * F 0 once we show that depth x F 0 ≥ 2 for any x ∈ U 0 such that {x} ∩ Z 0 has codimension 1 in {x}. But by S 2 condition depth x F 0 ≤ 1 can only hold for points x of codimension ≤ 1 in U 0 , which would imply that {x} ∩ Z 0 has codimension ≥ 2 in {x}. This proves (ii). Part (iii) is a particular case of Corollary II.3.5 in loc.cit.

Examples.
The first example with codim X 0 Z 0 = 3 and K = k shows that one may not be able to take U = U 0 × k Spec(A).

Lemma 4 The bundle F admits no algebraization
Proof. Set F to be the kernel of morphism ϕ : O ⊕3 → O(1)of vector bundles on U , given by the same formula as for ϕ p . By definition, ϕ is not surjective only at P = [0 : 0 : t : 1] ∈ U which projects to the generic point ξ = Spec(k[t −1 , t]]) ∈ Spec(A). The specialization at t = 0 is not in U 0 , hence P is closed in U and U \ P is an open subset containing U 0 . Since on U \ P we have the short exact sequence of locally free sheaves the restriction of F to each U p is given by F p , i.e. F is indeed the completion of F . On the other hand, F is not locally free at P : Suppose that E is a locally free sheaf on U with completion isomorphic to F. We will see later in Proposition 7(ii) that in such situation we must have: The second example illustrates that for codim X 0 Z 0 = 2, a pair (U, F ) may not exist at all.
given by non-vanishing of x, resp. y, form a covering of U p and we can glue trivial rank 2 bundles on these open sets, using the transition function in a natural way, and we obtain a vector bundle F on U .

Lemma 5 There exists no vector bundle
Proof. Suppose otherwise and take the direct image of F with respect to the open embedding i : U → X. By Proposition 3, i * F is coherent and has depth ≥ 2 at all codimension 2 points of X. Since modules of depth 2 over two-dimensional regular local rings are free by Auslander-Buchsbaum formula, i * F will be locally free in codimension two. Therefore shrinking Z we can assume that Z has codimension 3 in X which in our case means that Z is a finite set of points in X 0 . Then the short exact sequence of sheaves on X \ Z leads to a long exact sequence on X: where R 1 i * F is coherent for the same reason as in Proposition 3(ii). Since R 1 i * F is supported at the finite set Z of closed points, it has finite length at each of them and the last arrow is zero for To that end, replace X 0 with the affine open subset X 0 ≃ A 2 given by non-vanishing of z, with affine coordinates Xp is the sheaf associated to H 0 (W p , F p | Wp ) viewed as a module over A( X p ) = k[u, v, t]/t p+1 . By its definition, F p is an extension of O Up with O Up which leads to long exact sequence where the last arrow sends the constant function 1 to the class of the extension. Let M p be the kernel of the last arrow. It suffices to show that where ψ p is multiplication by p l=0 t uv ) l (i.e. the upper right corner of the transition matrix in the definition of F p ), and π p is the natural projection Example 3. (Suggested to the author by V. Drinfeld.) The bundle in the previous example has trivial determinant, but if we don't insist on this condition, then there is a rank one example: glue two trivial bundles on U

Algebraization of vector bundles.
Theorem 6 In the notation of section 2.1, (i). If codim X 0 Z 0 ≥ 3 then F admits an algebraization.
(ii). If codim X 0 Z 0 ≥ 2 and the cokernel of the natural morphism (i p ) * F p | X p−1 → (i p−1 ) * F p−1 is supported in codimension ≥ 3 for all p large enough, then F admits an algebraization.
(iii). In either of the two situations (codimension ≥ 3 or codimension ≥ 2 with the additional support assumption) the projective system {(i p ) * F p } p≥0 satisfies the Mittag-Leffler condition, the direct image i * F is coherent and isomorphic to lim ← − (i p ) * F p .
Proof. We split the proof of (i) and (ii) in a number of steps. Part (iii) will follow from Step 2.
Step 2. Therefore (i) and (ii) are reduced to showing that, under the conditions stated, i * F is coherent. To that end we modify the argument of 0.13.7.7 in [EGAIII] which will also prove (iii). First, as in 0.13.7.2 of loc. cit., we choose injective resolutions F k → L • k such that L • k+1 /J k+1 U L • k+1 ≃ L • k and the natural filtrations by J n U (. . .) agree with those on F k . Each i * (L • k ) is a filtered complex and has a spectral sequence with E 1 term given by As in 0.13.7.3 of loc.cit. we pass to the limit as k → ∞ and get a spectral sequence with We are interested in the components We would like to show that the spectral sequence converges at the E 0 = ⊕E p,−p terms. Note that each E 1 k+1 = ⊕E p,1−p k+1 is a quotient of a subsheaf in E 1 k while each E 0 k+1 is a subsheaf E 0 k (since E p,−1−p terms are zero). Taking successive preimages of the boundaries in E r−1 , E r−2 , . . . , E 1 we get a sequence of boundary subsheaves B 1 ⊂ B 2 ⊂ B 3 ⊂ . . . ⊂ E 1 1 , and taking preimages of cycles in E k we get a sequence of cycle subsheaves E 0 1 ⊃ Z 1 ⊃ Z 2 ⊃ Z 3 ⊃ . . . . By 0.13.7.6 in loc.cit. these are actually O X 0 ⊗ K gr(A)-submodules.
Suppose that sequence of cycles stabilizes, i.e. for some r 0 one has Z r = Z r 0 whenever r ≥ r 0 , then by 0.13.7.4 in [EGAIII], the projective system { i * (F k )} k≥0 satisfies the Mittag-Leffler condition and the associated graded of i by the noetherian property of X 0 and A. By loc.cit. 13.7.7.2, i * F is itself coherent on X. Also, i * F ≃ lim ← − (i p ) * F p by 0.13.7.5.1 in loc.cit..
Step 3. Now the assertion of the theorem is reduced to showing that the sequence of cycles Z 1 ⊃ Z 2 ⊃ . . . stabilizes. By definition of Z i this is equivalent to saying that the higher differentials of the spectral seqence d r : E 0 r → E 1 r become zero for r ≥ r 0 . That in turn is equivalent to saying that the sequence of boundaries B 1 ⊂ B 2 ⊂ B 3 ⊂ . . ., also stabilizes.
If codim X 0 Z 0 ≥ 3 by Proposition 3(ii), R 1 (i 0 ) * F 0 is also coherent and {B r } r≥1 stabilizes by the noetherian property of R 1 (i 0 ) * F 0 ⊗ K gr(A), which proves (i). If codim X 0 Z 0 ≥ 2 we need to find a coherent subsheaf of R 1 (i 0 ) * F 0 ⊗ K gr(A) containing B r for all r ≥ 1.
Step 4. At this point we reduced (ii) to showing that, under the assumptions stated, there exists a coherent subsheaf G ⊂ R 1 (i 0 ) * F 0 such that B r ⊂ G ⊗ K gr(A) for all r. By 0.11.2.2 in [EGAIII] for r ≥ p the term B p,1−p r is the image of the connecting homomorphism in the long exact sequence obtained by applying R i * to the short exact sequence on U : Observe that by our assumptions each Im(ρ p ) is coherent, and supported in codimension ≥ 3 for p ≫ 0. Thefore we are done once we show that the subsheaf of R 1 (i 0 ) * F 0 formed by all sections with support in codimension ≥ 3, is coherent whenever codim X 0 Z 0 ≥ 2 and F 0 is locally free on U 0 .
Step 5. Set Q = (i 0 ) * F 0 , a coherent sheaf on X 0 by Step 2. By the standard exact sequence we have is the functor of sections supported in codimension ≥ 3. Let H i ≥3 be the higher derived functors.
First, the standard spectral sequence for the composition of functors RH 0 where H 2 Φ is the local cohomology with the family of supports Φ formed by all codimension ≥ 3 closed subsets in Z 0 .
Step 6. To show that H 2 Φ Q is coherent let ω be the dualizing complex of X 0 , cf. [Ha2]. More precisely, by loc.cit ω is quasi-isomorphic to a complex of injective sheaves is an injective envelope of the residue field k(x) as a module over O X 0 ,x . An easy but important observation which we use below, is that K p has no sections supported in codimension ≥ p + 1.
By definition of a dualizing complex, the double complex K p,q = Hom(Hom(Q, K q ), K p ) has total complex quasi-isomorphic to Q. Moreover, by Proposition IV.2.1 and the remark on page 123 in [Ha2], this total complex is a flasque resolution of Q and hence can be used to compute H • Φ (Q). This leads to a spectral sequence: where Ext p Φ = R p (Γ Φ • Hom) and the Ext sheaves are understood in the sense of hypercohomology. Since E p,q 2 = 0 only for when p and (−q) are between 0 and dim K X 0 only finitely many terms with fixed p + q will be non-trivial and to show that H 2 Φ (Q) is coherent it suffices to show that Ext p Φ (Ext p−2 (Q, ω), ω) is coherent for p ≥ 2.
Step 7. First observe that Ext 2 Φ (G, ω) = 0 for any quasi-coherent sheaf G since K 2 has no sections supported in codimension ≥ 3 and hence no sections with support in Φ. Hence we can assume that p ≥ 3. Denote R p = Ext p−2 (Q, ω). We first claim that codim X 0 Supp(R p ) = d ≥ p ≥ 3. In fact, let x ∈ Supp(R p ) be a point with dim O X 0 ,x = d. Since the stalk of R p x is non zero, and by local duality, cf. V.6 in [Ha2] its completion is dual to H d+2−p x (Q) we conclude that H d+2−p x (Q) = 0. Then d + 2 − p ≥ 0 and d ≥ p − 2 ≥ 1. If d = 1 then p = 3 and also x / ∈ Z 0 hence the stalk Q x is free. Thus H 0 x (O) = 0, contradicting the S 2 assumption. If d ≥ 2 then applying the S 2 condition when x / ∈ Z 0 and Proposition 3(iii) when x ∈ Z 0 we actually have d + 2 − p ≥ 2 so d ≥ p as required. By primary decomposition, the coherent sheaf R p admits a finite filtration by coherent subsheaves such that all successive quotients have irreducible supports of codimension ≥ p. By the standard long exact sequence for Ext • Φ (·, ω) is suffices to show that Ext p Φ (G, ω) is coherent whenever p ≥ 3 and G is a coherent sheaf with irredicuble support Y of codimension ≥ p.
If Y Z 0 for any W in the family Φ, the intersection Y ∩ W is not equal to Y and therefore has codimension ≥ p + 1. But then Ext p Φ (G, ω) = 0 because any section ρ of Hom(G, K p ) representing a class in Ext p Φ (G, ω) has zero values since K p has no sections supported in codimension ≥ p + 1. If Y ⊆ Z 0 then Y is an element of Φ and Ext p Φ (G, ω) ≃ Ext p (G, ω) since all sections of Hom(G, K t ) have support in Φ. But Ext p (G, ω) is coherent which finishes the proof.
The converse to Theorem 6 can be formulated as follows.
Proposition 7 In the setting of Section 2.1, assume that F admits an algebraization (U, F ) and view each F p as a sheaf on U . Then Proof. To prove (i) observe that the cokernel of i * F p → i * F p−1 is is annihilated by J X , being a subsheaf of R 1 i * F 0 ⊗ K gr p (A), and is therefore isomorphic to the cokernel of We will first show that the natural map i * F p | X 0 → i * F 0 is an embedding of sheaves for all p. Considering the exact sequence and its map to the first terms of the sequence Using Lemma 1 and the Cohen-Macaulay assumption on X 0 we see that Similarly, i * F | X 0 → i * F 0 is an embedding. So for any p ≥ 1 we have embeddings Consequently, the coherent sheaf K = Coker(i * (F )| X 0 → i * F 0 ) has a decreasing filtration by images of i * F p | X 0 and each Coker(i * F p | X 0 → i * F p−1 | X 0 ) is its successive quotient. But K is a coherent sheaf with Supp(K) ⊂ Z 0 and Z 0 has at most finitely many points of codimension 2. Since for each point x ∈ X 0 of codimension 2, the localization K x is a module of finite length, only finitely many successive quotients of the filtration of K can be non-trivial in codimension 2, which proves (i).
To prove (ii) first observe that i * F and E = i * F are coherent by Theorem 6(iii) and Proposition 3(ii), respectively. By Proposition 3(i) we can find a sheaf E ′ such that E ′ ≃ i * F. The isomorphism E| b U ≃ F = i * i * F extends uniquely to a morphism of sheaves φ : E → i * F = E ′ . By Proposition 3(i), φ is the completion of a unique morphism φ : E → E ′ which by Corollary 10.8.14 in [EGAI] should be an isomorphism on an open subset W containing U 0 . Shrinking W if necessary we can assume W ⊂ U . By Lemma 2, each point x ∈ U \ W has codimension ≥ 2 in its fiber, hence depth x E ≥ 2 by Lemma 1. For x ∈ X \ U we still have depth x E ≥ 2 by Proposition 3(iii). Applying the same result to j : W ֒→ X instead of U we see that E = j * j * E. By adjunction of j * and j * the isomorphism (φ| W ) −1 : j * E ′ → j * E extends uniquely to a morphism ψ : E ′ → j * j * E = E.
By construction, the composition ψφ : E → E restricts to identity on W hence ψφ = Id E , by the same adjunction. Similarly, the composition φ ψ = E ′ → E ′ restricts to identity on U and since E ′ ≃ i * F, we must have φ ψ = Id c E ′ , so φψ = Id E ′ by Proposition 3(i). We have proved that

Corollary 8 The following conditions are equivalent:
(i). The cokernel of (i p ) * F p → (i p−1 ) * F p−1 is supported in codimension ≥ 3 for p ≫ 0.
(iii). The direct image i * F is coherent.
Proof. The implications (i) ⇒ (ii) and (iii) ⇒ (iv) are established in the proof of Theorem 6. The implication (iv) ⇒ (i) is proved in Proposition 7. If the projective system { i * F p } p≥1 satisfies the Mittag-Leffler condition, by 0.13.3.1 in [EGAIII] the natural map i * F → lim ← − i * F p is an isomorphism.
By the Mittag-Leffler condition we can replace i * F p by a system of subsheaves G p ⊂ i * F p so that the property i * F ≃ lim ← − G p still holds and G p | X p−1 → G p−1 is surjective. Since each G p is coherent by the noetherian property of X p , Proposition 10.11.3 in [EGAI] tells that lim ← − G p is also coherent. Therefore, (ii) ⇒ (iii).
Remark. Suppose that X 0 is a smooth projective surface over K, ξ = k 1 P 1 + . . . + k l P l an effective zero cycle and F 0 a rank n vector bundle on U 0 = X 0 \ {P 1 , . . . , P l }. The pair (F 0 , ξ 0 ) should define a point Spec(K) → U hl n of the Uhlenbeck functor. Assume that (F, ξ) : Spec(A) → U hl n extends (F 0 , ξ 0 ). Then it is expected that Coker(i * F → i * F 0 ) can be supported only at the points P 1 , . . . , P l , with multiplicities bounded by k 1 , . . . , k l , respectively (in the differential geometry picture, cf. [DK], ξ 0 represents the singular part of a connection which may be smoothed out by F but may not acquire any negative coefficients; since the multiplicities of Coker(i * F → i * F 0 ) measure the local change of c 2 one obtains the bound mentioned). But the proof of Proposition 7 shows that the multulplicities of Coker(i * F → i * F 0 ) give an upper bound for the total sum, over all p, of similar multiplicites for Coker((i p ) * F p → (i p−1 ) * F p−1 ). Hence the condition of Corollary 8(i) is rather natural from the point of view of Uhlenbeck spaces.

Algebraization of principal bundles.
Let G be an affine algebraic group over k. We keep the notation of Section 2.1. and consider left principal G-bundles which are locally trivial in fppf topology. For such a G-bundle P (over U or an open subset U ⊂ X) and any scheme Y over k with left G-action, denote by P Y = G \ (Y × k P ) the associated fiber bundle, i.e. the quotient by the left diagonal action of G. For instance, when ρ : G → H is a homomorphism of linear algebraic groups over k, we can consider a left G-action on H given by g · h = hρ(g) −1 and then P H is simply the principal H-bundle induced via ρ.
Theorem 9 Assume that the identity component G • is reductive. Then a principal G-bundle P over the formal scheme U admits an algebraization if and only if for a fixed exact representation G ֒→ GL(V ) the associated vector bundle P V admits an algebraization, i.e. satisfies the conditions of Corollary 8.
The "only if" part is obvious. Since by a result of Haboush, cf. Theorem 3.3 in [Ha1], the quotient GL(V )/η(G) is affine, the "if" part follows from the following general statement.
Proposition 10 Let H an affine algebraic group over k and G its closed subgroup such that H/G is affine. Suppose that P is a principal G-bundle over U such that the associated principal H-bundle Q = P H admits an algebraization. Then P admits an algebraization.
First we establish a preparatory result. As before, U ⊂ X is an open subset satisfying U ∩ X 0 = U 0 .
Therefore s becomes a section of the vector bundle Q V . By Proposition 7(ii) the completion of the coherent sheaf i * Q V is isomorphic to i * Q V and therefore by Proposition 3(i) there exists a unique sections of i * Q V with completion given by i * s. Set s =s| U .
It remains to show that s(W ) ⊂ Q V on some W as above. Let A = Sym * (Q ∨ V ) be the sheaf of symmetric algebras on U corresponding to Q V and I ⊂ A the ideal sheaf of Q Y . The section s gives the evaluation morphism ρ : A → O U . The sheaf G = ρ(I) is coherent, being a subsheaf of O U . Since s takes values in Q Y , the completion G is zero. By Corollary 10.8.12 in [EGAI] this implies Supp(G) ∩ U 0 = ∅ hence W = U \ Supp(G) satisfies the conditions of the lemma. The uniqueness of s follows from the uniqueness of s.
Proof of Proposition 10. Let (U, Q) be an algebraization of Q. In general, giving a principal Gbundle is equivalent to giving a principal H-bundle R together with a reduction to G, i.e. a section of the associated bundle R H/G with the fiber H/G. Since Q is induced from P, we get a section s : U → Q H/G and by the above lemma there exists s : W → Q H/G such that s is equal to its completion. Then P admits an algebraization (W, P ) where P is the pullback of the principal G-bundle Q → Q H/G via s : W → Q H/G .

Categorical formulations.
Proposition 12 The functor F → F | b U induces and eqivalence between the full subcategory of all coherent sheaves E on X which are locally free at the points of U 0 ⊂ X and have depth x E ≥ 2 at the points where E is not locally free, and the full subcategory of locally free sheaves on U admitting algebraization.
Proof. Let (U, F ) be an algebraization of F. Then the sheaf E = i * F satisfies E ≃ i * i * E hence by Proposition 3(iii) depth x E ≥ 2 for all x ∈ Z = X \ U . We also observe that E is uniquely determined by F, since by Propositions 3(i) and 7(ii) it is the unique coherent sheaf on X such that E ≃ i * F. Thus the functor described is essentially surjective on objects. For the morphisms, let F 1 , F 2 be a pair of vector bundles on U with algebraizations (U, F 1 ) and (U, F 2 ), respectively, which we may assume to be defined on the same U . Denote by E 1 = i * F 1 , E 2 = i * F 2 the corresponding coherent sheaves on X. Then Hom b U (F 1 , F 2 ) = Hom b X ( i * F 1 , i * F 2 ) = Hom X (E 1 , E 2 ) where the first equality is by adjunction of i * and i * and the second by Propositions 3(i) and 7(ii).
To formulate a result for principal bundles, let B(G, U 0 ) be the groupoid category in which the objects are given by pairs (U, P ) where U ⊂ X is an open subset with U ∩ X 0 = U 0 , and P is a principal G-bundle on U . Morphisms from (U, P ) to (U ′ , P ′ ) are given by the set of equivalence classes of pairs (W, ψ) where W ⊂ U ∩ U ′ is an open subset with W ∩ X 0 = U 0 and ψ : P | W → P ′ | W an isomorphism of G-bundles. Two such pairs (W, ψ) and (W, ψ ′ ) are equivalent if ψ = ψ ′ on W ∩W ′ . Also denote by Bun(G, U ) the groupoid category of G-bundles on the formal scheme U . Completion along U 0 defines a functor Ψ : B(G, U 0 ) → Bun(G, U ). The following statement summarizes our results on algebraization of principal bundles Theorem 13 With the notation of Section 2.1, (i). For any affine algebraic group G over k, Ψ : B(G, U 0 ) → Bun(G, U ) is full and strict.
(ii). For G = GL n (k) the essential image of Ψ is the full subcategory of rank n vector bundles F = lim ← − F p on U which satisfy the equivalent conditions (i)-(iii) of Corollary 8.
(iii). Let G ֒→ H be a closed embedding of affine algebraic groups over k such that H/G is affine. Then the natural functor from G-bundles to H-bundles induces an equivalence of categories Proof. To prove (i) suppose that P, P ′ are two principal bundles on U admitting algebraizations P, P ′ , respectively, which we may assume to be defined on the same U ⊂ X. Let ψ : P → P ′ be an isomorphism. We need to prove that there exists (perhaps after shrinking U ) a unique isomorphism ψ : P → P ′ with completion given by ψ. Let Isom(P, P ′ ) be the bundle of isomorphisms P → P ′ . Considering graphs of isomorphisms, we can identify Isom(P, P ′ ) ≃ G \ (P × U P ′ ). On the other hand, P × U P ′ is a principal bundle over G × k G. Define a left action of G × k G on G by (g, h) · f = gf h −1 , then G \ (P × U P ′ ) ≃ (P × U P ′ ) G . Since ψ gives a section s of Isom(P, P ′ ), applying Lemma 11 to H = G× k G and Y = G, we get a unique algebraization s : W → (P × U P ′ ) G ≃ Isom(P, P ′ )| W , which corresponds to the required isomorphism ψ. This proves (i). The statement of (ii) for objects holds by Corollary 8 and for morphisms by (i). On objects, this functor is an equivalence if for a G-bundle P on U , an H-bundle Q on U ⊂ X and an isomorphism φ : P H ≃ Q, there exists an open subset W ⊂ U with W ∩ X 0 = U 0 , a G-bundle P on W and isomorphisms P ≃ P and P H ≃ Q| W which induce φ in a natural way. This is equivalent to finding an algebraization of the section s : U → Q H/G induced by φ, which was done in the proof of Proposition 10. On morphisms, without loss of generality it suffices to consider two G-bundles P, P ′ defined on the same open set U , and isomorphisms ψ : P H ≃ P ′ H , φ : P → P ′ which have the same image in Bun(H, U ). We need to show that there exists a unique isomorphism φ : P → P ′ inducing φ and ψ in the natural sense. But by (i) there exists a unique φ with completion equal to φ. Since by assumption the isomorphisms ψ ′ = φ H and ψ are equal after completion, ψ ′ = ψ by part (i). This finishes the proof.