aa r X i v : . [ m a t h . AG ] F e b SHEAFINESS OF STRONGY RIGID-NOETHERIAN HUBER PAIRS
BOGDAN ZAVYALOV
Abstract.
We show that any strongly rigid-noetherian Huber ring A is sheafy. In particular, wepositively answer Problem 31 in the Nonarchimedean Scottish Book. Introduction
In the paper [Hub94], Huber defined the notion of an adic spectrum Spa(
A, A + ) for a Huberpair ( A, A + ). One of the main nuisances of this theory is that the structure presheaf O ( A,A + ) is not always a sheaf on Spa( A, A + ) (see [Hub94, Example after Proposition 1.6]). He showedthat O ( A,A + ) is a sheaf in two important cases: if A is a strongly noetherian Tate ring; and if A has a noetherian ring of definition. The former case was later generalized in [Ked17] to thestrongly noetherian analytic case. Huber gave different arguments for the two cases. In the formercase Huber’s argument is very close in the spirit to Raynaud’s theory of admissible blow-ups and“generic fibers”; in the latter case he was able to adapt the Tate’s proof of sheafiness of O A is therigid geometry, this argument is more analytic.These two examples cover adic spaces that come the theory rigid spaces and noetherian formalschemes. But one disadvantage of these results is that they do not cover formal schemes thatare (locally) topologically finitely presented over O C p as the ring O C p is not noetherian. In con-trast, there is a good theory of formal schemes over O C p developed, for example, in [Bos14], andsignificantly generalized in [FK18].David Hansen proposed a question in the Nonarchimedean Scottish Book if any complete, uni-versally topologically rigid-noetherian ring A (see Definition 2.8) is sheafy. The main reason whyHuber’s proof in the case of a noetherian ring of definition does not work in this more general setupis that Huber needs to refer to some finiteness results from [EGAIII] that require the noetheriannhypotheses. The main new idea is to use results from the recent book [FK18] in place of [EGAIII]to make Huber’s argument work in a bigger generality.Based on this approach, we are able to show that any Huber ring A with a topologically univer-sally rigid-noetherian ring of definition is sheafy. Theorem 1.1.
Let (
A, A + ) be a Huber pair with a strongly rigid-noetherian A (see Definition 2.8).Then the structure presheaf O X is a sheaf of topological rings on X = Spa( A, A + ). Furthermore,H i ( X, O X ) = 0 for any rational subdomain U ⊂ X and i ≥ proofs will appear only in their upcoming work. The second problem is that even if we try towork in a less general situation (i.e. topologically universally adhesive rings) where the finitenessresults are known, there is a problem due to the issue that certain morphisms/sheaves are onlyof finite type and not of finite presentation. The finiteness results (probably) hold only under thefinite presentation assumption. This issue can also be elegantly resolved by using the theory ofFP-approximated sheaves.There are other sheafiness results that are somehow orthogonal to the result of this paper. Forinstance, Scholze showed sheafiness of perfectoid algebras in [Sch12], Buzzard and Verberkmoesgeneralized it to any stably uniform Tate ring in [BV18], and recently Hansen and Kedlaya [KH17]gave new examples of sheafy rings by verifying the stable uniformity of a certain class of rings. Acknowledgements.
The author expresses gratitude to B. Conrad for reading the first draft ofthis paper and making lots of suggestions on how to improve the exposition of the paper.2.
Rational Localizations
We review the theory of rational localizations of Huber pairs. We spell out the main definitionsfrom [Hub94]. One reason for doing this is that the construction of (uncompleted) rational local-izations does not show up much once the foundational aspects of the theory are developed, but wewill really need it in our proof.
Definition 2.1.
Let A be a Huber ring with a pair of definition ( A , I ) and elements f , . . . , f n , s ∈ A such that f A + f A + · · · + f n A is an open ideal in A . • The rational localization A (cid:16) f s , . . . , f n s (cid:17) is a Huber ring such that:(1) The ring structure is given by A (cid:16) f s , . . . , f n s (cid:17) = A (cid:2) s (cid:3) .(2) A ring of definition is given by A (cid:20) f s , . . . , f n s (cid:21) ⊂ A (cid:20) s (cid:21) (3) An ideal of definition is given by IA h f s , . . . , f n s i ⊂ A (cid:2) s (cid:3) . • The completed rational localization A D f s , . . . , f n s E is defined as the completion of the Huberring A (cid:16) f s , . . . , f n s (cid:17) . Remark 2.2.
One can check that A (cid:16) f s , . . . , f n s (cid:17) is well-defined, i.e. it is indeed a Huber ring andit is independent of a choice of a couple of definition ( A , I ). See [Hub94, Lemma and definitionon p.516 and the universal property (1.2) on p.517]. Remark 2.3. [Hub93, Lemma 1.6(ii)] implies that A D f s , . . . , f n s E is a Huber ring with a ring ofdefinition equal to A (cid:28) f s , . . . , f n s (cid:29) := A (cid:20) f s , . . . , f n s (cid:21) ∧ , and an ideal of definition IA D f s , . . . , f n s E . We point out that this definition depends on the ambient ring A . HEAFINESS OF STRONGY RIGID-NOETHERIAN HUBER PAIRS 3
Remark 2.4.
The main importance of this construction is that it gives values of the structurepresheaf on rational subdomains. More precisely, suppose that X = Spa( A, A + ) for a completeHuber pair ( A, A + ). Then we have a topological isomorphism O X (cid:18) X ( f s , . . . , f n s ) (cid:19) ≃ A (cid:28) f s , . . . , f n s (cid:29) for any f , . . . , f n , s ∈ A such that the ideal f A + · · · + f n A is open in A .We also review a slightly more general version of this construction that will be convenient forour later purposes. Definition 2.5.
Let A be a Huber ring, s , s , . . . , s n elements of A , and finite sets F i of elementsof A such that the ideal generated by F i is open. Let ( A , I ) be a pair of definition. • The rational localization A (cid:16) F s ; . . . ; F n s n (cid:17) is a Huber ring such that:(1) The ring structure is given by A (cid:16) F s ; . . . ; F n s n (cid:17) = A h s , . . . , s n i .(2) A ring of definition is given by A (cid:20) F s ; . . . ; F n s n (cid:21) := A (cid:20) fs i | i = 1 , . . . , n, f ∈ F i (cid:21) ⊂ A (cid:20) s , . . . , s n (cid:21) . (3) An ideal of definition is given by IA h F s ; . . . ; F n s n i ⊂ A (cid:16) F s ; . . . ; F n s n (cid:17) . • The completed rational localization A D F s ; . . . ; F n s n E is defined as the completion of the Huberring A (cid:16) F s ; . . . ; F n s n (cid:17) . Remark 2.6.
If we set F = { f , . . . , f n } , it is clear that A (cid:18) Fs (cid:19) = A (cid:18) f s , . . . , f n s (cid:19) , A (cid:28) Fs (cid:29) = A (cid:28) f s , . . . , f n s (cid:29) A (cid:20) Fs (cid:21) = A (cid:20) f s , . . . , f n s (cid:21) , A (cid:28) Fs (cid:29) = A (cid:28) f s , . . . , f n s (cid:29) . Remark 2.7.
Similarly to Remark 2.4, we have a canonical topological isomorphism O X (cid:18) X ( F s ) ∩ · · · ∩ X ( F n s n ) (cid:19) ≃ A (cid:28) F s ; · · · : F n s n (cid:29) for any Huber pair ( A, A + ) with complete A , elements s , . . . , s n ∈ A , and finite sets F , . . . , F n ⊂ A such that the ideal generated by F i is open for any i . Definition 2.8.
Let ( A , I ) be a pair of a ring A and a finitely generated ideal I . We saythat A is topologically universally rigid-noetherian if Spec c A h X , . . . , X d i is noetherian outside I c A h X , . . . , X d i for every d ≥ .A Huber ring A is strongly rigid-noetherian if A admits a pair of definition ( A , I ) that istopologically universally rigid-noetherian. Remark 2.9. If A is complete and Tate, Definition 2.8 recovers the usual definition of a stronglynoetherian Huber-Tate pair. This definition differs from [FK18, Definition 0.8.4.3] as we do not require A to be noetherian outside I . BOGDAN ZAVYALOV
Remark 2.10.
We want to emphasize that strong rigid-noetherianness of A does not imply that A is noetherian. For instance, the ring O C p is strongly rigid-noetherian, but O C p is not noetherian. Remark 2.11.
The definition of a strongly rigid-noetherian Huber pair does not depend on achoice of a pair of definition ( A , I ). Indeed, it clearly does not depend on a choice of an ideal ofdefinition I inside a fixed ring of definition A .Now we may and do assume that A is complete. Suppose A and A are two rings of definition,so [Hub93, Corollary 1.3] implies that A · A is again a ring of definition, so it suffices to show thatclaim under the additional assumption that A ⊂ A . If I is an ideal of definition in A , then IA is an ideal of definition in A . So it is enough to show that Spec A is noetherian outside f ∈ I ifand only if so is Spec A . Now [Hub93, Lemma 3.7] ensures that ( A ) f → A f and ( A ) f → A f are isomorphisms. This finishes the proof. Lemma 2.12.
Let A be a strongly rigid-noetherian Huber ring, and f , . . . , f n , s ∈ A elementssuch that f A + f A + · · · + f n A is an open ideal in A . Then the completed rational localization A (cid:28) f s , . . . , f n s (cid:29) is a strongly rigid-noetherian Huber ring. Proof.
Without loss of generality, we can assume that A is complete, equivalently, any ring ofdefinition A is complete. Now, it suffices to show that A D f s , . . . , f n s E admits a topologicallyuniversally rigid-noetherian pair of definition. Remark 2.3 implies that this ring admits a pair ofdefinition (cid:16) A D f s , . . . , f n s E , IA D f s , . . . , f n s E(cid:17) . Clearly, there a surjection A h X , . . . , X n i → A (cid:28) f s , . . . , f n s (cid:29) . Thus, A D f s , . . . , f n s E is a topologically finitely generated A -algebra. Therefore, the pair (cid:18) A (cid:28) f s , . . . , f n s (cid:29) , IA (cid:28) f s , . . . , f n s (cid:29)(cid:19) is topologically universally rigid-noetherian as ( A , I ) is so. (cid:3) Definition 2.13.
A pair (
A, I ) of finite type is pseudo-adhesive (or A is I -adically pseudo-adhesive )if Spec A is noetherian outside V( I ) and any finite A -module M has bounded I -power torsion (i.e. M [ I ∞ ] = M [ I n ] for some n ).A pair ( A, I ) of finite type is universally pseudo-adhesive (or A is I -adically universally pseudo-adhesive ) if ( A [ X , . . . , X d ] , IA [ X , . . . , X d ]) is pseudo-adhesive for any d ≥ Remark 2.14.
It is easy to see that any finite type A -algebra over a universally pseudo-adhesivepair ( A, I ) is I -adically universally pseudo-adhesive.The following theorem of Fujiwara, Gabber, and Kato will play a crucial role in what follows. Itwill give us a way to apply results from Appendix A in our context. Theorem 2.15. [FK18, Theorem 0.8.4.8] Let ( A , I ) be a complete topologically universally rigid-noetherian pair. Then it is universally pseudo-adhesive. Then apply the same reasoning to A h T i and A h T i . This means that I is finitely generated HEAFINESS OF STRONGY RIGID-NOETHERIAN HUBER PAIRS 5 Sheafiness Of Strognly Noetherian Huber Pairs
We show that any strongly rigid-noetherian Huber ring A is sheafy. Our proof follows Huber’sproof of the same result for Huber pairs with a noetherian ring definition very closely (see [Hub94,Theorem 2.2]). The main obstacle why his proof does not work in this more general situationis that he needs to use certain finiteness of cohomology groups from [EGAIII] that require thenoetherian hypothesis. Instead, we use the results from Section A in place of the results from[EGAIII]. However, we want to point out one complication is that Theorem A.5 does not give anhonest finiteness result.The following lemma plays a crucial role in our argument: Lemma 3.1. [Hub94, Lemma 2.6] Let (
A, A + ) be a complete Huber pair, and { V j } j ∈ J be an opencovering of X = Spa( A, A + ). Then there exist f , . . . , f n ∈ A such that A = f A + f A + · · · + f n A and, for every i ∈ { , . . . , n } , the rational subset X (cid:16) f f i , . . . , f n f i (cid:17) is contained in some V j . Definition 3.2. A standard covering of X = Spa( A, A + ) is a covering of the form X = n [ i =0 X (cid:18) f f i , . . . , f n f i (cid:19) for some f , . . . , f n ∈ A such that A = f A + f A + · · · + f n A . Definition 3.3.
A morphism of topological groups ϕ : A → B is called strict of it is continuousand A → ϕ ( A ) is open if the target has the subspace topology. Definition 3.4.
A presheaf of topological rings F on a topological space X is a sheaf if F is asheaf of sets and the natural map F ( U ) → Q i ∈ I F ( U i ) is a topological embedding for any covering U = ∪ i ∈ I U i . Theorem 3.5.
Let (
A, A + ) be a Huber pair with a strongly rigid-noetherian A . Then the structurepresheaf O X is a sheaf of topological rings on X = Spa( A, A + ). Furthermore, H i ( X, O X ) = 0 forany rational subdomain U ⊂ X and i ≥ M ⊗ O X for any finite A -module M . Theproof should be slightly modified as done in [Hub94, Theorem 2.5]. We prefer to write the argumentonly in the case of the structure presheaf as it simplifies the exposition significantly. Proof. Step 0. We may assume that A is complete : This follows from the fact that there is acanonical isomorphism (cid:0) Spa(
A, A + ) , O A,A + (cid:1) ≃ (cid:16) Spa( b A, b A + ) , O b A, b A + (cid:17) . Step 1. We reduce theorem to showing that ˇ C • aug ( U , O X ) is exact with strict differentials for astandard covering U = { U , . . . , U n } of X : The sheaf condition means that the sequence0 → O X ( U ) d −→ Y i O X ( U i ) → Y i Likewise, the ˇCech-to-derived spectral sequence and Lemma 3.1 implies that it is sufficient toshow that ˇH i ( U , O X ) = 0 for any i > 0, rational U , and its standard covering U . Lemma 2.12ensures that we can replace X with U to assume that U is a covering of X .Therefore, we reduced the original question to show that the augmented (alternating) ˇ C echcomplex ˇ C • aug ( U , O X ) := ( A [1] → ˇ C • ( U , O X ))is exact with strict differentials for any standard covering U of X . Step 2. We show that the “decompleted” augmented ˇ C ech complex is exact : Now suppose thatthe standard covering is given by elements f , . . . , f n ∈ A with f A + · · · + f n A = A . Then wechoose a pair of definition ( A , I ) with a topologically universally rigid-noetherian A . We considerthe A -module J := f A + f A + · · · + f n A inside A . We denote S := Spec A , U := Spec A , P := Proj ⊕ J m , and P ′ := Proj ⊕ ( J A ) m . Then we have a commutative square P ′ UP S. ps jg Clearly, p is an isomorphism as J A = A , so s induces a morphism s : U → P such that the diagram UP S s jg is commutative. We note that s is an affine morphism as j = g ◦ s is affine and g is separated.Therefore, R i s ∗ O U vanish for i > 0. This implies that H i ( P, s ∗ O U ) = H i ( U, O U ) = 0 for i > 0, andH ( P, s ∗ O U ) = A .Now we compute the same cohomology groups in a different way using the ˇCech complex. Wechoose an affine covering P := { D + ( f i ) } of P . Since s is quasi-compact and quasi-separated, the O P -module s ∗ O U is quasi-coherent. So we can compute its cohomology via the ˇCech complex.Consider C • := ˇ C • ( P , s ∗ O U ) . The above computation of the cohomology groups H i ( P, s ∗ O U ) implies that the augmented ˇCechcomplex C • aug = ( A [1] → C • ) The notation J m means the A -submodule of A generated by all m -fold products of elements in J . In particular, J = A . HEAFINESS OF STRONGY RIGID-NOETHERIAN HUBER PAIRS 7 is exact. For brevity, write F = { f , . . . , f n } . Now we note that C iaug = Y j 0, we conclude thatˇ C iaug ( P , O P ) → C iaug is injective and identifies ˇ C iaug ( P , O P ) with a ring of definition of C iaug .Now we deal with the case of i = − 1, separately. We note thatˇ C − aug ( P , O P ) ≃ H ( P, O P ) ⊂ ˇ C aug ( P , O P )and C − aug ≃ H ( U, O U ) ⊂ C aug . Therefore, injectivity of ˇ C − aug ( P , O P ) → C − aug follows from injectivity in degree 0. So we only needto topologize ˇ C − aug ( P , O P ) in a way that ˇ C − aug ( P , O P ) is a ring of definition of C − aug = A . BOGDAN ZAVYALOV We topologize it using the subspace topology from ˇ C ( P , O P ). This topology coincides with thenatural I -topology (see Definition A.7) by Remark A.8. Now Theorem A.13 ensures that thistopology is the I -adic topology. So we need to show that B := H ( P, O P ) ⊂ A is an open subring and the subspace topology is the I -adic topology.Now we note that the morphism g ∗ O S → O P gives the morphism A → B such that whencomposed with the inclusion B → A it is equal to the embedding A → A . This implies that I m ⊂ I m B , so it suffices to show that, for any k , there is an m such that I m B ⊂ I k .Theorem A.5 guarantees that B is FP-approximated as an A -module, i.e. there is a finite A -submodule M ⊂ B such that the module quotient is annihilated by I d for some d . Since I is anideal of definition in A , and M is finitely generated, we can find c such that I c M ⊂ I k . Therefore, I c + d B ⊂ I c M ⊂ I k . This finishes the argument.Overall, we see that the ˇ C iaug ( P , O P ) → C iaug is injective and identifies ˇ C iaug ( P , O P ) with a ringof definition in C iaug for every i ≥ − d iC : C iaug → ker d i +1 C are open to conclude that d iC : ˇ C iaug → ˇ C i +1 aug are strict for every i ≥ − 1. We claim that it is actually sufficient to show thatthe differentials δ i : K i → ker δ i +1 is open, where K • := ˇ C iaug ( P , O P ) and δ is the differential of thiscomplex.Grant this opennness. We just need to deduce that d iC ( I m K i ) is open for any m ≥ { I m K i } for a fundamental system of neighborhoods of 0 in C iaug ( K i is a ring of definition in C iaug ). Weknow that d iC ( I m K i ) = δ i ( I m K i )is open in ker δ i +1 = ker d i +1 C ∩ K i +1 . So as K i +1 is open in C i +1 aug , we conclude that ker δ i +1 is openin ker d i +1 C . As a result, we get that d iC ( I m K i ) is open in C i +1 aug for every m ≥ i ≥ − Step 4. We show that the differentials of δ i : K i → ker δ i +1 are open : The claim for δ i is trivialif i < − 1. If i = − 1, the map δ − : K − → ker δ is even a homeomorphism because K − = ker δ and topology on K − was defined to be the subspace topology.We consider the restriction δ i : K i → ker δ i +1 , where the target is endowed with the subspacetopology. This map is open if and only if, for each k , there is m such thatker δ i +1 ∩ I m K i +1 ⊂ I k δ i ( K i ) = δ i ( I k K i )Now we note that I m K • ≃ I m ˇ C • aug ( P , O P ) ≃ ˇ C • aug ( P , I m O P ) =: ( K • m , δ m ) . Then it suffices to show that, for any k , there is m such thatker δ i +1 ∩ K i +1 m ⊂ δ ik ( K ik )that is equivalent to ker δ i +1 m ⊂ δ ik ( K ik )This means that we need to find m such thatH i +1 ( K m ) → H i +1 ( K k ) Note that A is universally pseudo-adhesive by Theorem 2.15 HEAFINESS OF STRONGY RIGID-NOETHERIAN HUBER PAIRS 9 is zero. Unravelling the definitions, we get that this is equivalent to find m such thatH i +1 ( P, I m O P ) → H i +1 ( P, I k O P )is zero.Now we prove that claim under the assumption that I c H i +1 ( P, I k O P ) = 0 for some c (dependingon k and i ≥ 0) and then we show that this assumption always holds. We firstly observe thatIm (cid:16) H i +1 ( P, I m O P ) → H i +1 ( P, I k O P ) (cid:17) = F m − k H i +1 ( P, I k O P ) , where F • stands for the natural I -filtration (see Definition A.7). Now we note that I k O P is afinitely generated, quasi-coherent O P -module, so it is FP-approximated by Lemma A.3. Therefore,Theorem A.13 ensures that the natural I -topology on H i +1 ( P, I k O P ) is the I -adic topology. Thenthere is some d such that F d H i +1 (cid:16) P, I k O P (cid:17) ⊂ I c H i +1 (cid:16) P, I k O P (cid:17) = 0 . Claim 2 below ensures that H i +1 ( P, I k O P ) is indeed annihilated by some I c for some c dependingon k and i ≥ Claim 1. The morphism g : P → S is an isomorphism away from V( I ). Proof. It suffices to show that g is isomorphism over D( f ) for any f ∈ I . We note that (cid:16) Proj M J m (cid:17) × Spec A Spec( A ) f ≃ Proj M (cid:16) J ( A ) f (cid:17) m as A → ( A ) f is flat. Therefore, it suffices to show that ( A ) f ≃ A f as then J ( A ) f = J A f = ( J A ) A f = A f = ( A ) f , and so g is an isomorphism over ( A ) f . Now (the proof of) [Hub93, Lemma 3.7] implies that thenatural map ( A ) f → A f is an isomorphism. (cid:3) Claim 2. For any i, k ≥ 0, there is c such that I c H i +1 ( P, I k O P ) = 0. Proof. We note that g is quasi-compact and separated, soR i +1 g ∗ (cid:16) I k O P (cid:17) ≃ ^ H i +1 ( P, I k O P ) . Now Claim 1 says that g is an isomorphism over Spec A \ V( I ), so since i ≥ i +1 g ∗ ( I n O P ) | Spec A \ V( I ) ≃ . Since I is finitely generated, this saysH i +1 ( P, I k O P ) = H i +1 ( P, I k O P )[ I ∞ ] . As g is projective, Theorem A.5 implies that H i +1 ( P, I k O P ) is an FP-approximated A -module.Therefore, Lemma A.2 ensures that for some c ≥ i +1 ( P, I k O P ) = H i +1 ( P, I k O P )[ I ∞ ] = H i +1 ( P, I k O P )[ I c ];i.e. H i +1 ( P, I k O P ) is annihilated by I c . (cid:3)(cid:3) AppendixAppendix A. FP-Approximated Sheaves This section is a summary of the results from [FK18, Appendix C to Chapter I]. However,some of them were only announced in that Appendix, but no proof was given. Since these resultsare crucial for our proof of Theorem 3.5, we decided to provide the reader with the proofs in thegenerality we need in this paper. All main ideas are already present in [FK18]For the rest of the appendix, we fix a universally pseudo-adhesive pair ( R, I ) (see Definition 2.13).In particular, it Spec R noetherian outside V( I ) and I is finitely generated.We recall that an R -scheme is universally I -adically pseudo-adhesive (or simply universallypseudo-adhesive ) if it has a covering by open affines Spec A i such that each A i is I -adically univer-sally pseudo-adhesive (see [FK18, § R -scheme is universally pseudo-adhesive by Remark 2.14. In particular, any quasi-coherent O X -module of finite type F has bounded I -power torsion, i.e. F [ I ∞ ] = F [ I n ] for some n .Let us mention that the main reason to bring in the pseudo-adhesive assumption is to rescuenoetherian techniques for non-noetherian situations with suitable finitely generated ideals. Forinstance, we will need to ensure that a submodule of a finite A -module has some precise finitenessproperty (Lemma A.3) and its subspace topology coincides with the I -adic topology (Lemma A.10). Definition A.1. (1) A morphism of O X -modules ϕ : F → G is a weak isomorphism if coker ϕ and ker ϕ are annihilated by I n for some n .(2) An FP-approximation of a quasi-coherent O X -module F is a weak isomorphism ϕ : G → F from a finitely presented O X -module G .(3) An FP-thickening of a quasi-coherent O X -module F is a surjective FP-approximation ϕ : G → F .(4) A quasi-coherent O X -module is FP-approximated if there is an FP-approximation ϕ : G → F .(5) An R -module M is FP-approximated if f M is an FP-approximated sheaf on Spec R . Lemma A.2. Let M be an FP-approximated R -module. Then its I ∞ -torsion is bounded, i.e. M [ I ∞ ] = M [ I n ] for some n ≥ Proof. The definition of FP-approximated modules implies that there is a finite type R -submodule N ⊂ M such that M/N is killed by I m for some m . So we may and do assume that M is an R -finite module. This case follows from the definition of pseudo-adhesive pairs. (cid:3) Lemma A.3. Let X be a finite type R -scheme. Then(1) any quasi-coherent O X -module of finite type admits an FP-thickening,(2) the category of FP-approximated sheaves is a Weak Serre abelian subcategory of the cate-gory of O X -modules,(3) Any quasi-coherent sub or quotient sheaf of an FP-approximated F is FP-approximated. Proof. Part (1) is [FK18, Proposition I.C.2.2]. Part (2) is [FK18, Theorem I.C.2.5].We firstly prove Part (3) for quasi-coherent quotients of F . The definition of FP-approximatedsheaves easily implies that there is finite type quasi-coherent O X -submodule G ⊂ F such that F / G is annihilated by some I n . Then if π : F → F ′ is a surjective map of quasi-coherent O X -modules, wedefine G ′ := π ( G ). Clearly, G ′ is a quasi-coherent O X -module of finite type, and F ′ / G ′ is annihilatedby I n . Therefore, Part (1) implies that F ′ is FP-approximated. HEAFINESS OF STRONGY RIGID-NOETHERIAN HUBER PAIRS 11 Now if F ′ is a quasi-coherent subsheaf of F , it is clear that F ′′ := F / F ′ is quasi-coherent. So F ′′ is FP-approximated by the discussion above. Thus, Part (2) implies that F ′ is FP-approximatedbecause FP-approximated sheaves are closed under kernels. (cid:3) Corollary A.4. Let i : X → Y be a closed immersion of finite type R -schemes, and let F be anFP-approximated O X -module. Then i ∗ F is an FP-approximated O Y -module. Proof. As i ∗ is exact, it suffices to show that the corollary holds for finitely presented O X -modules.So we may and do assume that F is finitely presented. Then i ∗ F is clearly a quasi-coherent O X -module of finite type. Therefore, it is FP-approximated by Lemma A.3(1). (cid:3) Now we want to study cohomology groups of FP-approximated sheaves on projective R -schemes.We show that these cohomology are always FP-approximated R -modules, and a certain naturaltopology on these modules coincides with the I -adic topology. These results were announced inthe proper case in [FK18, Appendix C to Chapter I]. We do not discuss this generalization as theprojective case is sufficient for our purposes. Theorem A.5. Let X be a projective R -scheme, and let F be an FP-approximated O X -module.Then H i ( X, F ) is an FP-approximated R -module for any i ≥ Remark A.6. We do not impose the finite presentation assumption on X . The finite presentationversion of Theorem A.5 will be inadequate for the purpose of proving Theorem 3.5. Proof. We firstly reduce to the case X = P nR . Namely, there is a closed immersion i : X → P nR as X is projective. Since i ∗ is exact, it suffices to show the claim for the sheaf i ∗ F that is FP-approximatedby Corollary A.4.Now we argue that H i ( P nR , F ) is an FP-approximated R -module by descending induction on i .We claim that H i ( P nR , F ) = 0 if i > n . Indeed, P nR admits the standard affine covering U = { U i } by n + 1 opens. So the cohomology groups of any quasi-coherent sheaf can be computated by the alternating Cech complex with respect to that covering. Thus, H i ( P nR , F ) = 0 for any i > n .Now we do the induction step. Suppose we know the claim for all FP-approximated sheaves F and all i > k , we conclude the statement for i = k . By definition, we can find a weak isomorphism G → F with a finitely presented O X -module G . It is clear that morphisms H i ( P nR , G ) → H i ( P nR , F )are weak isomorphisms for any i . Thus it suffices to prove the claim for a finitely presented O X -module F .We invoke the ample line bundle O P nR (1) to say that there is always a short exact sequence0 → F ′ → O P nR ( r ) m → F → negative r . Lemma A.3(2) implies that F ′ is FP-approximated, so H i ( P nR , F ′ ) are FP-approximated for any i > k by the induction assumption.Firstly we consider the case k = n . Then we know that H n +1 ( P nR , F ′ ) = 0 by the discussion above.So the natural morphism H k ( P nR , O P nR ( r )) m → H k ( P nR , F ) is surjective. This implies that H k ( P nR , F )is a finite R -module by Serre’s computation. Therefore, it is FP-approximated by Lemma A.3(1).Now suppose that k < n . Then we know that H k ( P nR , O P nR ( r )) m = 0 by Serre’s computa-tion . Therefore, we conclude that the natural map H k ( P nR , F ) → H k +1 ( P nR , F ′ ) is injective. Thus,H k ( P nR , F ) is FP-approximated by Lemma A.3(3) and the induction assumption. (cid:3) Now we try to understand a topology on H i ( X, F ). We use here that r < Definition A.7. The natural I -filtration F • H i ( X, F ) isF n H i ( X, F ) := Im (cid:0) H i ( X, I n F ) → H i ( X, F ) (cid:1) The natural I -topology on H i ( X, F ) is the topology induced by the filtration F • H i ( X, F ). Remark A.8. Suppose X is a separated quasi-compact R -scheme, F a quasi-coherent O X -module,and U = { U , . . . , U n } an open affine covering of X . Then the natural I -topology on H i ( X, F )coincides with the subquotient topology on H i ( X, F ) ≃ ˇH i ( U , F ) induced from the I -adic topologyon the (alternating) ˇCech complex ˇ C i ( U , F ).Clearly I n H i ( X, F ) ⊂ F n H i ( X, F ) for any n . These two filtrations on H i ( X, F ) are usuallydifferent, but we claim that the induced topologies are the same for any FP-approximated sheaf F on a projective R -scheme X .Before proving this claim, we need the following lemma: Lemma A.9. Let M be an FP-approximated R -module, and N ⊂ M be any submodule. The the I -adic topology on M restricts to the I -adic topology on N . Proof. If M is a finitely generated this is proven in [FK18, Proposition 0.8.5.6].Now we deal with the case of any FP-approximated R -module M . Clearly, I n N ⊂ I n M ∩ N forany n . So it suffices to show that, for any n , there is m such that I m M ∩ N ⊂ I n N .We can find a finite R -submodule M ′ ⊂ M such that M/M ′ is annihilated by I c . Then we knowthat the I -adic topology on M ′ restricts to the I -adic topology on N ′ := M ′ ∩ N by the case offinite R -modules. This means that there is an integer p such that I p M ′ ∩ N ⊂ I n N ′ . Then I c + p M ∩ N ⊂ I p M ′ ∩ N ⊂ I n N ′ ⊂ I n N. So m = c + p does the job. (cid:3) Corollary A.10. Let X be a finite type R -scheme, F an FP-approximated sheaf, G ⊂ F be aquasi-coherent O X -submodule of F . Then, for any n , there is m such that I m F ∩ G ⊂ I n G . Proof. It suffices to assume that X is affine, in which case it follows from Lemma A.9. (cid:3) Corollary A.11. Let X be a finite type R -scheme, G an FP-approximated sheaf, and ϕ : G → F a weak isomorphism of quasi-coherent O X -modules. Then, for every i ≥ 0, the natural I -topologyon H i ( X, F ) coincides with the topology induced by the filtrationFil n G H i ( X, F ) = Im(H i ( X, I n G ) → H i ( X, F )) . Proof. Consider the short exact sequences0 → K → G → H → , → H → F → Q → , where K and Q are annihilated by I n for some n . The first short exact sequence induced the shortexact sequence 0 → K ∩ I m G → I m G → I m H → m ≥ 0. Corollary A.10 implies that K ∩ I m G ⊂ I n K = 0 for large enough m . Therefore,the natural map I m G → I m H is an isomorphism for large enough m . So we can replace G with H to assume that ϕ is injective (since H is FP-approximated by Lemma A.3(3)).Now clearly Fil k G H i ( X, F ) ⊂ F k H i ( X, F ) for every k . So it suffices to show that, for any k , there m such that F m H i ( X, F ) ⊂ Fil k G H i ( X, F ). We consider the short exact sequence0 → G ∩ I m F → I m F → I m Q → . HEAFINESS OF STRONGY RIGID-NOETHERIAN HUBER PAIRS 13 If m ≥ n we get that G ∩ I m F = I m F because I m Q ≃ 0. Now we use Corollary A.10 to concludethere is m ≥ n such that I m F = G ∩ I m F ⊂ I k G Therefore, F m H i ( X, F ) ⊂ Fil k G H i ( X, F ). (cid:3) Lemma A.12. Let X be a finite type R -scheme, G an FP-approximated sheaf, and G → F a weakisomorphism of quasi-coherent O X -modules. Suppose that the natural I -topology on H i ( X, G ) isthe I -adic topology. Then the same holds for H i ( X, F ). Proof. Clearly, I n H i ( X, F ) ⊂ F n ( X, F ). So it suffices to show that, for every n , there is an m suchthat F m H i ( X, F ) ⊂ I n H i ( X, F ).The assumption that the natural I -topology on G coincides with the I -adic topology guaranteesthat H i ( X, I k G ) ⊂ I n H i ( X, G ) for large enough k . Pick such a k . Corollary A.11 implies thatF m H i ( X, F ) ⊂ Im(H i ( X, I k G ) → H i ( X, F ))for large enough m . So we get, for such m , thatF m H i ( X, F ) ⊂ Im (cid:16) H i ( X, I k G ) → H i ( X, F ) (cid:17) ⊂ Im (cid:0) I n H i ( X, G ) → H i ( X, F ) (cid:1) ⊂ I n H i ( X, F )for a large enough m . (cid:3) Theorem A.13. Let X be a projective R -scheme, and F be an FP-approximated O X -module.Then the natural I -topology on H i ( X, F ) coincides with the I -adic topology for any i .The proof follows the idea of the proof of the Formal Function Theorem in rigid geometry.Namely, we give a relatively simple argument in the case I is generated by one element, and thenargue by induction on the number of generators. See [Kie67, Theorem 3.4] or [Bos14, Proposition6.4/8] for an example of a classical argument of this form. However, it would be nice to give aproof of Theorem A.13 as a formal consequence of Theorem A.5 similar to what happens in [FK18,Proposition I.8.5.2]. Proof. Step 1. Case of a principal ideal I : Suppose that I is a generated by one element a . Choosea finite open affine covering X = ∪ ni =1 U i that we denote by U . Then we define C • := ˇ C • ( U , F )to be the (alternating) Cech complex of F with respect to the covering U . We note that I n C • =ˇ C • ( U , I n F ). So we conclude thatF n H i ( X, F ) = Im(H i ( I n C • ) → H i ( C • )) . Since the natural I -topology on H i ( X, F ) is induced from the subspace topology on ker d i , it sufficesto show subspace topology on ker d i ⊂ C i coincides with the I -adic topology. [FK18, Lemma0.8.2.14] ensures that it suffices to verify that C i / ker d i has bounded a ∞ -torsion. Since C i / ker d i is naturally a submodule of C i +1 , it suffices to justify the claim for C i +1 . Now we recall that C i +1 = ˇ C i +1 ( U , F ), so it suffices to show that F ( U j ∩ . . . U j i +1 ) has bounded a ∞ -torsion for allpossible j , . . . , j i +1 ∈ [1 , n ]. This follows from affinness of each intersection U j ∩ · · · ∩ U j i +1 andLemma A.2 since F is FP-approximated. Step 2. The General Case : We argue by induction on the number of generators I = ( a , . . . , a r )over all such F . The claim for r = 1 was proven in Step 1. So we assume that the claim is knownfor any i < r and all such F , we show that this implies the claim for r . Clearly, I n H i ( X, F ) ⊂ F n H i ( X, F ), so it suffices to show that, for any n , there is an m such thatF m H i ( X, F ) ⊂ I n H i ( X, F ) . Lemma A.12 ensures that it suffices to prove the claim under the assumption that F is a quasi-coherent O X -module of finite type. In particular, F is FP-approximated with respect to I =( a , . . . , a r − ) and a r by Lemma A.3(1). We also note that both pairs ( R, I ) and ( R, a r ) areuniversally pseudo-adhesive.Indeed, [FK18, Proposition 0.8.2.16] implies that they satisfy the ( BT ) property, i.e. any finite R -module M has bounded a r -power and I -power torsion. Clearly, Spec R is noetherian outsideV( a r ) and V( I ) as ( a r ) , I ⊂ I . Applying the same argument to R [ T , . . . , T d ] for every d , weget that R is universally I -adically and a r -adically pseudo-adhesive. Therefore, the inductionhypothesis can be applied to both ( R, I ) and ( R, a r ).Now we consider the short exact sequence0 → a kr F → F → F /a kr F → i := Im (cid:16) H i ( X, F ) → H i ( X, F /a kr F ) (cid:17) with the topology induced from the natural I -topology on H i ( X, F ). More precisely, it is topologydefined by the filtrationF n H i := Im (cid:0) F n H i ( X, F ) → H i (cid:1) = Im (cid:16) H i ( X, I n F ) → H i ( X, F /a kr F ) (cid:17) The following two claims finish the proof. Claim 1. It suffices to show that the topology on H i coincides with the I -adic topology for any k ≥ Proof. Step 1 justifies that there is d such that Im (cid:0) H i (cid:0) X, a dr F (cid:1) → H i ( X, F ) (cid:1) ⊂ a nr H i ( X, F ). Thenwe use the assumption for k = d to see that there is m such thatF m H i ⊂ I n H i . This implies thatF m H i ( X, F ) ⊂ I n H i ( X, F )+Im (cid:16) H i ( X, a dr F ) → H i ( X, F ) (cid:17) ⊂ I n H i ( X, F )+ a nr H i ( X, F ) ⊂ I n H i ( X, F ) . So this constructs the desired m . (cid:3) Claim 2. The topology on H i coincides with the I -adic topology for any k ≥ Proof. Clearly, F n H i ⊂ I n H i . Thus, we only need to show that, for any n , there is m such thatF m H i ⊂ I n H i . Now we note that the I -adic topology on H i coincides with the I -adic topology onH i . Therefore, it suffices to show that, for any n , there is m such thatF m H i ⊂ I n H i . Now Theorem A.5 H i ( X, F /a kr F ) is an FP-approximated module for the pair ( R, I ). Therefore,H i is also FP-approximated as a submodule of an FP-approximated module H i ( X, F /a kr F ). NowLemma A.9 says that the subspace topology on H i coincides with the I -adic topology. Thus, itsuffices to show that, for any n , there is m such thatF m H i ⊂ I n H i (cid:16) X, F /a kr F (cid:17) . EFERENCES 15 However, there is an evident inclusionF m H i ⊂ F mI H i (cid:16) X, F /a kr (cid:17) . Now we invoke the induction hypotheses to say that the natural I -adic topology on H i ( X, F /a kr F )coincides with the I -adic topology in H i ( X, F /a kr F ). This, in turn, implies that there is m suchthat F m H i ⊂ F mI H i (cid:16) X, F /a kr F (cid:17) ⊂ I n H i (cid:16) X, F /a kr F (cid:17) . (cid:3)(cid:3) References [Bos14] Siegfried Bosch. Lectures on formal and rigid geometry . Vol. 2105. Lecture Notes inMathematics. Springer, Cham, 2014, pp. viii+254.[Bou98] Nicolas Bourbaki. Commutative algebra. Chapters 1–7 . 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Sheafiness Criteria For Huber Rings . https://kskedlaya.org/papers/criteria.pdf .2017.[Kie67] Reinhardt Kiehl. “Der Endlichkeitssatz f¨ur eigentliche Abbildungen in der nichtarchimedis-chen Funktionentheorie”. In: Invent. Math. Publ. Math. Inst. Hautes ´Etudes Sci. 116 (2012),pp. 245–313. issnissn