K-theory of admissible Zariski-Riemann spaces
aa r X i v : . [ m a t h . K T ] J a n K-THEORY OF ADMISSIBLE ZARISKI-RIEMANN SPACES
CHRISTIAN DAHLHAUSENA
BSTRACT . We study relative algebraic K-theory of admissible Zariski-Riemann spaces andprove that it is equivalent to G-theory and satisfies homotopy invariance. Moreover, we providean example of a non-noetherian abelian category whose negative K-theory vanishes. C ONTENTS
1. Introduction 12. Admissible Zariski-Riemann spaces 23. Modules on admissible Zariski-Riemann spaces 34. K-theory of admissible Zariski-Riemann spaces 8Appendix A. Limits of locally ringed spaces 12Appendix B. Localisations of exact categories 14References 151. I
NTRODUCTION
Under the assumption of resolution of singularities, one can obtain for a non-regular scheme X a regular scheme X ′ which admits a proper, birational morphism X ′ → X . For many pur-poses X ′ behaves similarly to X . Unfortunately, resolution of singularities is not available atthe moment in positive characteristic. From the perspective of K-theory, a good workaroundfor this inconvenience is to work with a Zariski-Riemann type space which is not a schemeanymore, but behaves almost as good as a regular model does. For instance, for a regularnoetherian scheme X one has equivalences K ( Vec ( X )) ≃ K ( Coh ( X )) and K ( X ) ≃ K ( X × A ) for Quillen’s K-theory. For the Zariski-Riemann type space ⟨ X ⟩ defined to be the limit of allschemes which are projective and birational over a scheme X within the category of locallyringed spaces, Kerz-Strunk-Tamme [KST18] established that K ( Vec (⟨ X ⟩)) ≃ K ( Coh (⟨ X ⟩)) andK ( Vec (⟨ X ⟩)) ≃ K ( Vec (⟨ X ⟩ × A k )) . The purpose of this note is to prove the analogous statementfor admissible Zariski-Riemann spaces ⟨ X ⟩ U which are limits of projective morphisms to ascheme X which are isomorphisms over an open subscheme U (Definition 2.1). As one onlymodifies something outside U , one has to pass to the relative K-theory. The main results arethe following ones, see Theorem 4.5 and Theorem 4.15 as well as their corollaries. Theorem.
Let X be a reduced divisorial scheme and let U be a noetherian dense open subset ofX . Denote by ˜ Z the complement of U in ⟨ X ⟩ U and let k ∈ N . Then(i) K (⟨ X ⟩ U on ˜ Z ) ≃ K ( Coh ( ˜ Z )) and(ii) K (⟨ X ⟩ U on ˜ Z ) ≃ K (⟨ X ⟩ U × A k on ˜ Z × A k ) .Moreover, if U is regular, then(iii) K (⟨ X ⟩ U ) ≃ G (⟨ X ⟩ U ) and(iv) K (⟨ X ⟩ U ) ≃ K (⟨ X ⟩ U × A k ) .where G (⟨ X ⟩ U ) ∶ = K ( Mod fp (⟨ X ⟩ U )) (Definition 4.3). This work was supported by the Swiss National Science Foundation (grant number 184613).
The notion of Zariski-Riemann spaces goes back to Zariski [Zar44] who called them “Rie-mann manifolds” and was further studied by Temkin [Tem11]. Recently, Kerz-Strunk-Tamme[KST18] used them to prove that homotopy algebraic K-theory [Wei89] is the cdh-sheafificationof algebraic K-theory, and Elmanto-Hoyois-Iwasa-Kelly applied them on reults on Milnor exci-sion for motivic spectra [EHIK20a, EHIK20b].Combining part (i) of the theorem with a result of Kerz about the vanishing of negativerelative K-theory, we provide an example of a non-noetherian abelian category whose negativeK-theory vanishes (Example 4.19). This gives evidence to a conjecture by Schlichting (whichwas shown to be false at the generality it was stated), see Remark 4.18.
Notation.
A scheme is said to be divisorial iff it admits an ample family of line bundles[TT90, 2.1.1]; such schemes are quasi-compact and quasi-separated.
Acknowledgements.
The author thanks Andrew Kresch for providing a fruitful researchenvironment at the University of Zurich where (most of) this paper was written as well asGeorg Tamme, Matthew Morrow, and Moritz Kerz for helpful conversations.2. A
DMISSIBLE Z ARISKI -R IEMANN SPACES
Notation.
In this section let X be a reduced quasi-compact and quasi-separated scheme andlet U be a quasi-compact open subscheme of X . Definition 2.1. A U-modification of X is a projective morphism X ′ → X of schemes whichis an isomorphism over U . Denote by Mdf ( X , U ) the category of U -modifications of X withmorphisms over X . We define the U-admissible Zariski-Riemann space of X to be the limit ⟨ X ⟩ U = lim X ′ ∈ Mdf ( X , U ) X ′ in the category of locally ringed spaces; it exists due to Proposition A.7. Example 2.2.
Let V be a valuation ring with fraction field K . Then the canonical projection ⟨ Spec ( V )⟩ Spec ( K ) → Spec ( V ) is an isomorphism as every Spec ( K ) -modification of Spec ( V ) issplit according to the valuative criterion for properness. Lemma 2.3.
The full subcategory
Mdf red ( X , U ) spanned by reduced schemes is cofinal in Mdf ( X , U ) .Proof. As U is reduced by assumption, the map X ′ red ↪ X ′ is a U -modification for every X ′ ∈ Mdf ( X , U ) . (cid:3) Lemma 2.4.
The underlying topological space of ⟨ X ⟩ U is coherent and sober (Definition A.4)and for any X ′ ∈ Mdf ( X , U ) the projection map ⟨ X ⟩ U → X ′ is quasi-compact.Proof. This is a special case of Proposition A.7 or [FK18, ch. 0, 2.2.10]. (cid:3)
The notion of a U -modification is convenient as it is stable under base change. In practice,one can restrict to the more concrete notion of a U -admissible blow-up. Definition 2.5. A U-admissible blow-up is a blow-up Bl Z ( X ) → X whith centre Z ⊆ X ∖ U .Denote by Bl ( X , U ) the category of U -admissible blow-ups with morphisms over X . Proposition 2.6.
The inclusion Bl ( X , U ) ↪ Mdf ( X , U ) is cofinal. In particular, the canonicalmorphism ⟨ X ⟩ U = lim X ′ ∈ Mdf ( X , U ) X ′ Ð→ lim X ′ ∈ Bl ( X , U ) X ′ . is an isomorphism of locally ringed spaces.Proof. Since a blow-up is projective and an isomorphism outside its centre, Bl ( X , U ) lies inMdf ( X , U ) . On the other hand, every U -modification is dominated by a U -admissible blow-up[Tem08, Lem. 2.1.5]. Hence the inclusion is cofinal and the limits agree. (cid:3) -THEORY OF ADMISSIBLE ZARISKI-RIEMANN SPACES 3 Remark 2.7.
For the choice of a scheme-structure Z on the closed complement X ∖ U , thecomplement ⟨ X ⟩ U ∖ U comes equipped with the structure of a locally ringed space, namely ⟨ X ⟩ U ∖ U = lim X ′ ∈ Mdf ( X , U ) X ′ Z where X ′ Z ∶ = X ′ × X Z . Remark 2.8.
Let Z be a scheme-structure on X ∖ U . One can define the Zariski-Riemann space ⟨ Z ⟩ to be the limit over all projective and birational morphism Z ′ → Z in the category of locallyringed spaces. In general, there does not exist a canonical morphism ⟨ Z ⟩ /Ð→ ⟨ X ⟩ U ∖ U of locally ringed spaces. Although for X ′ ∈ Mdf ( X , U ) and a U -admissible blow up Bl Y ( X ′ ) → X ′ there exists a canonical morphism Bl Y ( X ′ Z ) Ð→ Bl Y ( X ′ ) ,the problem is that the blow-up Bl Y ( X ′ Z ) → X ′ Z needs not to be birational in case that X ′ Z ∖ Y is not dense in X ′ Z . Comparison to Temkin’s relative Riemann-Zariski spaces.Remark 2.9.
Temkin [Tem11] introduced the notion of a relative Riemann-Zariski space as-sociated with any separated morphism f ∶ Y → X between quasi-compact and quasi-separatedschemes. In loc. cit. a Y -modification is a factorisation Y f i Ð→ X i g i Ð→ X of f where f i is schematically dominant and g i is proper. The family of these Y -modificationsis cofiltered and the relative Riemann-Zariski space of the morphism f ∶ Y → X is definedas the cofiltered limit RZ Y ( X ) ∶ = lim i X i which is indexed by all Y -modifications. Lemma 2.10.
If X is reduced and U is a dense in X , then there exists a canonical morphism RZ U ( X ) Ð→ ⟨ X ⟩ U which is an isomorphism.Proof. If X is reduced, then U is dense in X if and only if it is schematicaly dense in X [Sta21,Tag 056D]. Hence every U -modification in our sense is also a U -modification in Temkin’s sensewith respect to the inclusion U ↪ X . Given a U -modification U g → X ′ p → X in Temkin’s sense,we take a Nagata compactification U j ↪ ¯ X ′ q → X ′ , i.e. j is an open immersion, q is proper, and g = q ○ j [Sta21, Tag 0F41]. Then the base change ( p ○ q ) × X U ∶ ¯ X ′ × X U → U is bijective andsplit by an open immersion, hence an isomorphism. Thus the U -modifications in our sense arecofinal within the U -modifications in Temkin’s sense. (cid:3)
3. M
ODULES ON ADMISSIBLE Z ARISKI -R IEMANN SPACES
The following results depend on Raynaud-Gruson’s platification par éclatement [RG71, 5.2.2].These results and their proofs are modified versions of results of Kerz-Strunk-Tamme whoconsidered birational and projective schemes over X instead of U -modifications, cf. Lemma 6.5and Proof of Proposition 6.4 in [KST18]. Notation.
In this section let X be a reduced quasi-compact and quasi-separated scheme andlet U be a quasi-compact open subscheme of X . The author apologises for the switched order of the names “Zariski” and “Riemann” which tries to be coherentwith the different sources.
CHRISTIAN DAHLHAUSEN
Lemma 3.1.
The canonical functor colim X ′ ∈ Mdf ( X , U ) Mod fp ( X ′ ) Ð→ Mod fp (⟨ X ⟩ U ) . is an equivalence (within the 2-category of categories).Proof. This is a general fact about limits of coherent and sober locally ringes spaces with quasi-compact transition maps, see [FK18, ch. 0, 4.2.1–4.2.3] and Proposition A.7. (cid:3)
This equivalence restricts to the full subcategories of vector bundles, i.e. locally free modules.
Lemma 3.2.
The canonical functor colim X ′ ∈ Mdf ( X , U ) Vec ( X ′ ) Ð→ Vec (⟨ X ⟩ U ) . is an equivalence (within the 2-category of categories).Proof. Clearly, the pullback of a locally free module is locally free again. Hence fully faithful-ness follows from the corresponding statement for finitely presented modules. It remains toshow that the functor is essentially surjective, so let F be a locally free O ⟨ X ⟩ U -module. Sincethe topological space ⟨ X ⟩ U is coherent (Lemma 2.4, we may assume that there exists a finitecover ⟨ X ⟩ U = V ∪ . . . ∪ V k of quasi-compact open subsets such that F ∣ V i ≅ O n i V i for all i and suit-able natural numbers n i . Hence we can argue by induction on k and reduce to the case when k =
2. There exists a U -modification X and quasi-compact open subsets V ′ and V ′ of X suchthat V i = p − ( V ′ i ) for i =
1, 2 and V ∩ V = p − ( V ′ ∩ V ′ ) and there exists O V ′ i -modules G i suchthat p ∗ G i ≅ F ∣ V i where p ∶ ⟨ X ⟩ U → X denotes the canonical projection. By general propertiesof sheaves on limits of locally ringed spaces we have a bijectioncolim X α ∈ Mdf ( X , U ) Hom X α ( p ∗ α ,0 G , p ∗ α ,0 G ) ≅ Ð→ Hom ⟨ X ⟩ U ( F ∣ V , F ∣ V ) .where p α ,0 ∶ X α → X is the transition map [FK18, ch. 0, Thm. 4.2.2]. Hence there exists an α such that p ∗ α ,0 G and p ∗ α ,0 G glue to a locally free sheaf G on X α which further pulls back to F which shows that the functor in question is essentially surjective. (cid:3) Definition 3.3.
Let ( Y , O Y ) be a locally ringed space and let n ≥
0. An O Y -module F is said tohave Tor-dimension ≤ n iff there exists an exact sequence0 → E n → . . . → E → E → F → E n , . . ., E , E are locally free O Y -modules. Denote by Mod ≤ n ( Y ) and Coh ≤ n ( Y ) the fullsubcategories of Mod ( Y ) resp. Coh ( Y ) spanned by O Y -modules of Tor-dimension ≤ n . Lemma 3.4 (cf. [KST18, 6.5 (i)]) . Assume that U is dense in X . Then:(i) For every U -modification p ∶ X ′ → X the pullback functorp ∗ ∶ Mod fp, ≤ ( X ) Ð→ Mod fp, ≤ ( X ′ ) is exact.(ii) For every morphism ϕ ∶ F → G in
Mod fp ( Y ) such that F , G, and coker ( ϕ ) lie in Mod fp, ≤ ( Y ) and for every U -modification p ∶ X ′ → X with X ′ reduced the canonical mapsq ∗ ( ker ( ϕ )) → ker ( q ∗ ϕ ) and q ∗ ( im ( ϕ )) → im ( q ∗ ϕ ) are isomorphisms.Proof. (i) If F is an O X -module of Tor-dimension ≤
1, there exists an exact sequence 0 → E ϕ → E → F → E and E are locally free O X -modules. Then the pulled back sequence0 Ð→ p ∗ E p ∗ ϕ Ð→ p ∗ E Ð→ p ∗ F Ð→ p ∗ E and p ∗ F . We claim that the map p ∗ ϕ is injective. The O X ′ -modules p ∗ E and p ∗ E are locally free, say of rank n and m , respectively. Let η be a generic point of an -THEORY OF ADMISSIBLE ZARISKI-RIEMANN SPACES 5 irreducible component of X ′ . Since U is dense, the map ( p ∗ ϕ ) η ∶ O nX ′ , η → O mX ′ , η is injective sinceit identifies with the injective map ϕ p ( η ) ∶ ( E ) p ( η ) ↪ ( E ) p ( η ) . For every specialisation x of η ,the stalk O X ′ , x embeds into O X ′ , η [GW10, 3.29], hence the induced map ( p ∗ ϕ ) x ∶ O nX ′ , x → O mX ′ , x is injective. Thus p ∗ ϕ is injective at every point of X ′ , hence injective. Now the exactness of p ∗ follows from the nine lemma.(ii) This follows directly from (i). (cid:3) Definition 3.5.
Let ( Y , O Y ) be a locally ringed space, Z a closed subset of Y , and j ∶ ( V , O V ) ↪( Y , O Y ) the inclusion of the open complement. An O Y -module F has support in Z iff j ∗ F vanishes. Denote with a “ Z ” in the lower index the full subcategory of those O Y -moduleswhich have support in Z , e.g. Mod fp Z ( Y ) ⊂ Mod fp ( Y ) . Lemma 3.6.
Assume that X is divisorial and that U is dense in X . Let f ∶ Y → X be a smoothand quasi-projective morphism and let Z = Y ∖ ( Y × X U ) . Given a U -modification p ∶ X ′ → X , wedenote by q ∶ Y ′ = Y × X X ′ → Y the pullback of p along f .(i) For every F ∈ Mod fp Z ( Y ) there exists a U -modification p ∶ X ′ → X such that q ∗ F lies in
Mod fp, ≤ ( Y ′ ) .(ii) For every morphism ϕ ∶ F → G in
Mod fp Z ( Y ) there exists a U -modification p ∶ X ′ → Xsuch that q ∗ F , q ∗ G , ker ( q ∗ ϕ ) , im ( q ∗ ϕ ) , and coker ( q ∗ ϕ ) all lie in Mod fp, ≤ ( Y ′ ) .(iii) For every ( F , ν ) ∈ Nil ( Coh Z ( Y )) there exists a U -modification such that there exists afinite resolution → ( E k , ν k ) → . . . → ( E , ν ) → ( F , ν ) → where all ( E i ) are locally free.Proof. (i) Since X has an ample family of line bundles and the map Y → X is quasi-projective,also Y has an ample family of line bundles [TT90, 2.1.2.(h)]. Hence there exists an exactsequence E ϕ → E → F → E , E are locally free O Y -modules. By our assumptions,im ( ϕ )∣ U = ker ( E → F )∣ U = ker ( E ∣ U → ) = E ∣ U is flat. By platification par éclatement [RG71, 5.2.2] there exists a U -admissible blow-up p ∶ X ′ → X such that the strict transform q st im ( ϕ ) is flat, i.e. locally free. Furthermore, q st im ( ϕ ) = im ( q ∗ ϕ ) , cf. Remark 3.8 below. Hence we obtain an exact sequence0 Ð→ im ( q ∗ ϕ ) Ð→ q ∗ E Ð→ q ∗ F Ð→ q ∗ F ∈ Mod fp, ≤ ( Y ′ ) by Lemma 3.4 (ii).(ii) By (i), there is a U -modification p ∶ X ′ → X such that both q ∗ F and q ∗ G have Tor-dimension ≤
1. Since q ∗ is a left-adjoint, q ∗ ( coker ( ϕ )) = coker ( q ∗ ϕ ) so that we may assume that alsocoker ( q ∗ ϕ ) has Tor-dimension ≤
1. Hence we can apply Lemma 3.4 (ii) and are done.(iii) We argue by induction on k > ν k =
0. The case k = k ≥
2. By (ii) we find a U -modification p ∶ X ′ → X such that q ∗ F , ker ( q ∗ ν k − ) , im ( q ∗ ν k − ) ,and coker ( q ∗ ν k − ) all lie in Mod fp, ≤ ( Y ′ ) . By induction assumption, there exists after further U -modification a finite resolution0 → ( E ′ k , ν ′ l ) → . . . → ( E ′ , ν ′ ) → ( ker ( q ∗ ν k − ) , ν ′ ) → E ′ i are vector bundles and where ν ′ is the restriction of q ∗ ν to ker ( q ∗ ν k − ) . Simi-larly, there exists a finite resolution of ( im ( q ∗ ν k − ) , ν ′′ ) where ν ′′ is the restriction of q ∗ ν toim ( q ∗ ν k − ) . These two finite resolutions now can be patched together to a finite resolution of q ∗ F . (cid:3) Lemma 3.7.
Assume that X is divisorial and that U is dense in X . Then the inclusion
Mod fp, ≤ Z (⟨ X ⟩ U ) Ð→ Mod fp˜ Z (⟨ X ⟩ U ) is an equivalence of categories where ˜ Z ∶ = ⟨ X ⟩ U ∖ U .
CHRISTIAN DAHLHAUSEN
Proof.
It suffices to show that the functor is essentially surjective which follows from Lemma 3.6and Lemma 3.4. (cid:3)
Remark 3.8.
In the situation of the proofs of Lemma 3.6 and Lemma 3.7 we used that im ( q ∗ ϕ ) is the strict transform of im ( ϕ ) along the U -admissible blow-up q ∶ X ′′ → X ′ . By definition, q st im ( ϕ ) is the quotient of q ∗ im ( ϕ ) by its submodule of sections whose support is containedin the centre of the blow-up q [Sta21, Tag 080D]. Since the surjective map q ∗ im ( ϕ ) → q st im ( ϕ ) factors over the map im ( q ∗ ϕ ) ⊂ p ∗ E ′ , the following commutative diagram has exact rows andexact columns. 0 (cid:15) (cid:15) (cid:15) (cid:15) / / ker ( σ ) / / (cid:15) (cid:15) im ( q ∗ ϕ ) σ / / (cid:15) (cid:15) q st im ( ϕ ) (cid:15) (cid:15) / / / / ker ( τ ) / / q ∗ E ′ τ / / q st E ′ / / E ′ is a vector bundle, q ∗ E ′ = q st E [Sta21, Tag 080F] which implies the claim.If a locally ringed space is cohesive, i.e. its structure sheaf is coherent (Definition A.1), thena module is coherent if and only if it is finitely presented (Lemma A.3). Unfortunately, we donot know whether or not this is also true for the Zariski-Riemann space ⟨ X ⟩ U . But passing tothe complement ⟨ X ⟩ U ∖ U we have the following. Proposition 3.9.
Assume that X is divisorial and that U is dense in X . Let ˜ Z be the comple-ment of U in ⟨ X ⟩ U . An O ⟨ X ⟩ U -module with support on ˜ Z is coherent if and only if it is finitelypresented.Proof.
We have to show that every finitely presented O ⟨ X ⟩ U -module with support in ˜ Z is co-herent. Let F be a finitely presented O ⟨ X ⟩ U -module with support on ˜ Z . By definition, F is offinite type. Let V be an open subset of ⟨ X ⟩ U and let ϕ ∶ O nV → F ∣ V be a morphism. We need toshow that ker ( ϕ ) is of finite type. Since this is a local property and ⟨ X ⟩ U is coherent, we mayassume that V is quasi-compact. By passing iteratively to another U -modification, there existsan X ′ ∈ Mdf ( X , U ) with canonical projection p X ′ ∶ ⟨ X ⟩ U → X ′ such that ● F = ( p X ′ ) ∗ F X for some F X ∈ Mod fp Z ′ ( X ′ ) (Proposition A.7 (iv)), ● F X has Tor-dimension ≤ ● V = ( p X ′ ) − ( V ′ ) for some open subset V ′ of X ′ (Proposition A.6 (i)), ● ϕ is induced by a morphism ϕ ′ ∶ O nV ′ → F X ∣ V ′ (Proposition A.7 (v)), and ● coker ( ϕ ) has Tor-dimension ≤ ( ϕ ′ ) is of finite type we may assume that there exists a surjection O V ′ ↠ ker ( ϕ ′ ) (otherwise we have to shrink V ′ ). By Lemma 3.4 ( p X ) ∗ ( ker ( ϕ ′ )) = ker ( ϕ ) , hence it is of finitetype. (cid:3) Theorem 3.10.
Assume that U is dense in X and denote by ˜ Z the complement of U in ⟨ X ⟩ U .Then the canonical functors Coh ≤ Z (⟨ X ⟩ U ) / / (cid:15) (cid:15) Coh ˜ Z (⟨ X ⟩ U ) (cid:15) (cid:15) Mod fp, ≤ Z (⟨ X ⟩ U ) / / Mod fp˜ Z (⟨ X ⟩ U ) are equivalences. In particular, the canonical functor colim X ′ ∈ Mdf ( X , U ) Coh Z ′ ( X ′ ) Ð→ Coh ˜ Z (⟨ X ⟩ U ) -THEORY OF ADMISSIBLE ZARISKI-RIEMANN SPACES 7 is an equivalence of categories where Z ′ = X ′ ∖ U and the colimit is taken in the 2-category ofcategories.Proof.
The lower horizontal functor is an equivalence by Lemma 3.7 and the right verticalfunctor is an equivalence by Lemma 3.9. The other two functors are equivalence since thesquare is a pullback square. The equivalence from Proposition A.7 restricts to an equivalencecolim X ′ ∈ Mdf ( X , U ) Mod fp Z ′ ( X ′ ) Ð→ Mod fp˜ Z (⟨ X ⟩ U ) which yields the desired statement. (cid:3) Proposition 3.11.
Assume that U is noetherian. Denote by ˜ Z the closed complement of U in ⟨ X ⟩ U . Then Mod fp˜ Z (⟨ X ⟩ U ) is a right-s-filtering subcategory of the exact category Mod fp (⟨ X ⟩ U ) and the inclusion j ∶ U ↪ ⟨ X ⟩ U induces an equivalence of categoriesj ∗ ∶ Mod fp (⟨ X ⟩ U )/ Mod fp˜ Z (⟨ X ⟩ U ) ≅ Ð→ Mod fp ( U ) . If we additionally assume that X is divisorial and that U is dense in X , then we have an exactsequence of exact categories
Coh ˜ Z (⟨ X ⟩ U ) Ð→ Mod fp (⟨ X ⟩ U ) Ð→ Coh ( U ) (where the first and the last one are abelian).Proof. Applying Lemma B.6 to the inclusion ˜ Z ⊂ ⟨ X ⟩ U and M ∶ = Mod fp (⟨ X ⟩ U ) we obtain thatMod fp˜ Z (⟨ X ⟩ U ) is a right-s-filtering subcategory. Hence the quotient categoryMod fp (⟨ X ⟩ U )/ Mod fp˜ Z (⟨ X ⟩ U ) is defined [Sch04, Def. 1.14]. By definition, the restriction j ∗ ∶ Mod fp (⟨ X ⟩ U ) → Mod fp ( U ) factorsthrough the canonical functor Mod fp (⟨ X ⟩ U ) → Mod fp (⟨ X ⟩ U )/ Mod fp˜ Z (⟨ X ⟩ U ) . By Corollary B.3 itsuffices to show that the induced functor is essentially surjective and full.For essential surjectivity let F ∈ Mod fp ( U ) = Coh ( U ) . The inclusion j X ∶ U ↪ X induces anequivalence of categories ( j X ) ∗ ∶ Coh ( X )/ Coh Z ( X ) Ð→ Coh ( U ) = Mod fp ( U ) where Z ∶ = X ∖ U [Sch11, 2.3.7]. Thus there exists an F X ∈ Coh ( X ) ⊂ Mod fp ( X ) such that ( j X ) ∗ F X ≅ F . Since j X factors as p X ○ j , it follows that F ≅ j ∗ X F X ≅ ( p X ○ j ) ∗ F X ≅ j ∗ ( p ∗ X F X ) ,i.e. F comes from a finitely presented module p ∗ X F X ∈ Mod fp (⟨ X ⟩ U ) .For fullness let ϕ ∶ j ∗ F → j ∗ G be a morphism of O U -modules with F and G in Mod fp (⟨ X ⟩ U ) .The quotient of Mod fp (⟨ X ⟩ U ) by Mod fp˜ Z (⟨ X ⟩ U ) is the localisation of Mod fp (⟨ X ⟩ U ) along thosemorphisms which are sent to isomorphisms by j ∗ . Consider the pullback diagram( ∆ ) H / / α (cid:15) (cid:15) G β (cid:15) (cid:15) F / / j ∗ j ∗ F j ∗ ( ϕ ) / / j ∗ j ∗ G in Mod fp (⟨ X ⟩ U ) . Since j ∗ is exact, the square j ∗ ( ∆ ) is also a pullback; hence j ∗ ( α ) is anisomorphism (as j ∗ ( β ) is one). Thus the span F ← H → G represents a morphism ψ in thequotient category such that ϕ = j ∗ ( ϕ ) , hence j ∗ is full.The last assertion follows from the equality Coh ( U ) = Mod fp ( U ) and Lemma 3.9. (cid:3) In fact, in loc. cit X is assumed to be noetherian, but this is not necessary.
CHRISTIAN DAHLHAUSEN
4. K-
THEORY OF ADMISSIBLE Z ARISKI -R IEMANN SPACES
In this section we prove for the K-theory of the Zariski-Riemann space a comparison with G-theory (Theorem 4.5, Corollary 4.6), homotopy invariance (Theorem 4.15, Corolary 4.16), anda vanishing statement for negative K-groups (Theorem 4.17). Finally, we provide an exampleof a non-noetherian abelian category whose negative K-theory vanishes (Example 4.19).
Notation.
In this section let X be a reduced quasi-compact and quasi-separated scheme andlet U be a quasi-compact open subscheme of X . Denote by ˜ Z the complement of U within the U -admissible Zariski-Riemann space ⟨ X ⟩ U (Definition 2.1). We equip the closed complement X ∖ U with the reduced scheme structure so that ˜ Z has the structure of a locally ringed space(Remark 2.7). Definition 4.1.
We define the
K-theory of the Zariski-Riemann space asK (⟨ X ⟩ U ) ∶ = K ( Vec (⟨ X ⟩ U )) and the K-theory with support asK (⟨ X ⟩ U on ˜ Z ) ∶ = fib ( K (⟨ X ⟩ U ) Ð→ K ( U )) . Lemma 4.2.
The canonical maps colim X ′ ∈ Mdf ( X , U ) K ( X ′ ) Ð→ K (⟨ X ⟩ U ) colim X ′ ∈ Mdf ( X , U ) K ( X ′ on Z ′ ) Ð→ K (⟨ X ⟩ U on ˜ Z ) are equivalences where Z ′ ∶ = Z × X X ′ .Proof. The first equivalence follows from Lemma 3.2 and the fact that K-theory commutes withfiltered colimits of exact functors. The second one follows since filtered colimits commute withfinite limits. (cid:3)
Comparison with G-theory.
For divisorial noetherian schemes, G-theory is the K-theoryof the abelian category of coherent modules (which is the same as the category of finitely pre-sented modules). Over an arbitrary scheme (or locally ringed space) a finitely generated mod-ule may not be coherent. Since we want the category of vector bundles to be included, we workwith finitely presented modules for defining G-theory for admissible Zariski-Riemann spaces.
Definition 4.3.
We define the
G-theory of the Zariski-Riemann space asG (⟨ X ⟩ U ) ∶ = K ( Mod fp (⟨ X ⟩ U )) .and the G-theory with support asG (⟨ X ⟩ U on ˜ Z ) ∶ = fib ( G (⟨ X ⟩ U ) Ð→ G ( U )) . Proposition 4.4.
Assume that X is divisorial and that U is noetherian and dense in X . Thenthere is a fibre sequence K ( Coh ˜ Z (⟨ X ⟩ U )) Ð→ G (⟨ X ⟩ U ) Ð→ G ( U ) . Moreover, the canonical maps K ( Coh ( ˜ Z )) Ð→ K ( Coh ˜ Z (⟨ X ⟩ U )) Ð→ G (⟨ X ⟩ U on ˜ Z ) are equivalences.Proof. By Propostion 3.11 there is an exact sequenceCoh ˜ Z (⟨ X ⟩ U ) Ð→ Mod fp (⟨ X ⟩ U ) Ð→ Coh ( U ) of additive categories and the first one is a right-s-filtering subcategory of the second one andthe third one is equivalent to the quotient category. Furthermore, the first category is abelian, -THEORY OF ADMISSIBLE ZARISKI-RIEMANN SPACES 9 hence idempotent complete. Thus we can apply Schlichting’s localisation theorem for additivecategories [Sch04, 2.10] which yields the desired fibre sequence and the second equivalence.For the first equivalence note that pushforward yields a fully faithful functor Coh ( ˜ Z ) ↪ Coh ˜ Z (⟨ X ⟩ U ) . We claim that it satisfies the conditions of the Dévissage Theorem [Wei13, V.4.1].Let F ∈ Coh ˜ Z (⟨ X ⟩ U ) . Using iteratively Lemma 3.4, there exist X ′ ∈ Mdf ( X , U ) and a filtration F ′ ⊃ F ′ ⊃ . . . ⊃ F ′ n = F ′ i ∈ Coh ≤ Z ′ ( X ′ ) such that F ≅ p ∗ F ′ and such that the quotients F ′ i / F ′ i − lie in the essentialimage of the functor Coh ( ˜ Z ′ ) → Coh Z ′ ( X ′ ) where Z ′ = Z × X X ′ . Then pulling back to ⟨ X ⟩ U yields a filtration of F such that the successive quotients lie in Coh ( ˜ Z ) as desired. (cid:3) Theorem 4.5.
Assume that X is divisorial and that U is noetherian and dense in X . Then wehave an equivalence of spectra K (⟨ X ⟩ U on ˜ Z ) ≃ K ( Coh ( ˜ Z )) . Proof.
Every U -modification of X has an ample family of line bundles since it is quasi-projectiveover a scheme admitting such a family [TT90, 2.1.2 (h)]. Moreover, those X ′ ∈ Mdf ( X , U ) suchthat X ′ ∖ U is a regular closed immersion form a cofinal subsystem (since U -admissible blow-ups are cofinal). Thus K ( X ′ on Z ′ ) ≃ K ( Coh ≤ Z ′ ( X ′ )) for every X ′ ∈ Mdf ( X , U ) by [TT90, 5.7 (e)].Hence we have thatK (⟨ X ⟩ U on ˜ Z ) ≃ colim X ′ ∈ Mdf ( X , U ) K ( X ′ on Z ′ ) ≃ colim X ′ ∈ Mdf ( X , U ) K ( Coh ≤ Z ′ ( X ′ )) ≃ K (( Coh ≤ Z (⟨ X ⟩ U )) .By Theorem 3.10 and Proposition 4.4 we have equivalencesK (( Coh ≤ Z (⟨ X ⟩ U )) Ð→ K (( Coh ˜ Z (⟨ X ⟩ U )) ←Ð K ( Coh ( ˜ Z )) . (cid:3) Corollary 4.6.
Assume that X is divisorial and that U is regular , noetherian, and dense in X .Then the canonical map K (⟨ X ⟩ U ) Ð → G (⟨ X ⟩ U ) is an equivalence.Proof. We have a commuative diagramK (⟨ X ⟩ U on ˜ Z ) / / (cid:15) (cid:15) K (⟨ X ⟩ U ) / / (cid:15) (cid:15) K ( U ) (cid:15) (cid:15) K ( Coh ( ˜ Z )) / / G (⟨ X ⟩ U ) / / G ( U ) where the upper line is a fibre sequence by design and the lower line is a fibre sequence byProposition 4.4. Furthermore, the first vertical map is an equivalence by Theorem 4.5 and thethird vertical map is an equivalence since U is regular. (cid:3) Homotopy invariance.Definition 4.7.
For an integer n ≥ ⟨ X ⟩ U [ t , . . ., t n ] ∶ = lim X ′ ∈ Mdf ( X , U ) X ′ [ t , . . ., t n ] . Lemma 4.8.
There exists a canonical isomorphism ⟨ X ⟩ U [ t , . . ., t n ] ≅ Ð→ ⟨ X ⟩ U × X X [ t , . . ., t n ] . of locally ringed spaces. Proof.
This follows since different limits commute among each other, namely: ⟨ X ⟩ U [ t , . . ., t n ] = lim X ′ ∈ Mdf ( X , U ) X ′ [ t , . . ., t n ] = lim X ′ ∈ Mdf ( X , U ) lim ( X ′ → X ← X [ t , . . ., X n ]) = lim (( lim X ′ ∈ Mdf ( X , U ) X ′ ) → X ← X [ t , . . ., X n ]) = ⟨ X ⟩ U × X X [ t , . . ., t n ] . (cid:3) Lemma 4.9.
There exists a canonical isomorphism ⟨ X [ t ]⟩ U [ t ] ≅ Ð → ⟨ X ⟩ U [ t ] of locally ringed spaces.Proof. If p ∶ X ′ → X is a U -modification, then the base change p [ t ] ∶ X ′ [ t ] → X [ t ] is a U [ t ] -modification. Hence the projections ⟨ X [ t ]⟩ U [ t ] → X [ t ] induce the desired morphism.For every U [ t ] -modification Y → X [ t ] there exists a U [ t ] -admissible blow-up Bl Z ( X [ t ]) → X [ t ] dominating Y [Tem08, 2.1.5]. Set Z ∶ = Z ∩ X where we consider X as a closed subschemeof X [ t ] via the zero section. By functoriality for blow-ups we get a morphism Bl Z ( X ) Ð→ Bl Z ( X [ t ]) which induces by base change a morphism Bl Z ( X )[ t ] Ð→ Bl Z ( X [ t ]) . As the projec-tion X [ t ] → X is flat, we have that Bl Z [ t ] ( X [ t ]) = Bl Z ( X )[ t ] [GW10, 13.91 (2)]. Thus every U [ t ] -modification is dominated by a base change of a U -modification so that the morphism inquestion is an isomorphism. (cid:3) Definition 4.10.
Let C be a pointed category. We denote by Nil ( C ) the category whose ob-jects are pairs ( F , ν ) where F is an object of C and ν ∶ F → F is a nilpotent endomorphism, i.e.there exists a k such that ν k = C ( F , F ) . The morphisms are given bymorphisms in C which are compatible with the respective endomorphisms.Our cases of interest are the following ones. Definition 4.11.
For a locally ringed space Y we write Nil ( Y ) ∶ = Nil ( Vec ( Y )) and defineNil ( Y ) ∶ = fib ( K ( Nil ( Y )) → K ( Y )) where the map is induced by the forgetful functor Nil ( Y ) → Vec ( Y ) . Lemma 4.12.
Let Y be a quasi-compact and quasi-separated scheme. Then the category
Nil ( Coh ( Y )) is an abelian category and Nil ( Y ) is an exact subcategory. Furthermore the fibre sequence Nil ( Y ) Ð→ K ( Nil ( Y )) Ð→ K ( Y ) splits so that we get a decomposition K ( Nil ( Y )) ≃ K ( Y ) × Nil ( Y ) . Proof.
The first part follows straighforwardly from the fact that Vec ( Y ) is an exact subcategoryof the abelian category Coh ( Y ) . The forgetful funtor Nil (( Y ) → Vec ( Y ) is split by the functorsending a vector bundle E to the pair ( E , 0 ) so that the claim follows. (cid:3) Remark 4.13.
There exists a split fibre sequenceNil (⟨ X ⟩ U ) Ð→ K ( Nil (⟨ X ⟩ U )) Ð→ K (⟨ X ⟩ U ) defined as the colimit of the rescpective fibre sequeces for every X ′ ∈ Mdf ( X , U ) coming fromLemma 4.12. Hence there exists a split fibre sequenceNil (⟨ X ⟩ U on ˜ Z ) Ð→ K ( Nil (⟨ X ⟩ U on ˜ Z )) Ð→ K (⟨ X ⟩ U on ˜ Z ) The author apologises for the possibly confusing notation which tries to be coherent with existing literature. -THEORY OF ADMISSIBLE ZARISKI-RIEMANN SPACES 11 which is defined to be the fibre of the mapNil (⟨ X ⟩ U ) / / (cid:15) (cid:15) K ( Nil (⟨ X ⟩ U )) / / (cid:15) (cid:15) K ( U ) (cid:15) (cid:15) Nil ( U ) / / K ( Nil ( U )) / / K ( U ) Definition 4.14.
We defineNK (⟨ X ⟩ U on ˜ Z ) ∶ = cofib ( K (⟨ X ⟩ U ) → K (⟨ X ⟩ U [ t ] on ˜ Z [ t ]) Theorem 4.15.
Assume that X is divisorial and that U is noetherian and dense in X . Denote ˜ Z ∶ = ⟨ X ⟩ U ∖ U . Then the canonical map K (⟨ X ⟩ U on ˜ Z ) Ð→ K (⟨ X ⟩ U [ t ] on ˜ Z [ t ]) is an equivalence.Proof. As the fibre sequenceNil (⟨ X ⟩ U on ˜ Z ) Ð→ K ( Nil (⟨ X ⟩ U on ˜ Z )) Ð→ K (⟨ X ⟩ U on ˜ Z ) splits, we get a decompositionK n ( Nil (⟨ X ⟩ U on ˜ Z )) ≅ K n (⟨ X ⟩ U on ˜ Z ) × Nil n (⟨ X ⟩ U on ˜ Z ) .Denote by NK (⟨ X ⟩ U on ˜ Z ) the cofibre of the desired equivalence. Then for every n ∈ Z we havethat Nil n (⟨ X ⟩ U on ˜ Z ) ≅ NK n + (⟨ X ⟩ U on ˜ Z ) by the corresponding statement for schemes [Wei13, V.8.1] since K-theory commutes with fil-tered colimits of exact categories. Hence it suffices to show that the mapK n (⟨ X ⟩ U on ˜ Z ) Ð→ K n ( Nil (⟨ X ⟩ U on ˜ Z ) is surjective. This map identifies with the mapK n ( Coh ( ˜ Z )) Ð→ K n ( Nil ( Coh ( ˜ Z ))) which is an isomorphism due to the Dévissage Theorem [Wei13, V.4.1] since every object ( E , ν ) in Nil ( Coh ( ˜ Z )) has a filtration ( E , ν ) ⊃ ( ker ( ν ) , ν ) ⊃ ( ker ( ν ) , ν ) ⊃ . . . ⊃ (cid:3) Corollary 4.16.
Assume that X is divisorial and that U is regular , noetherian, and dense inX . Then the canonical projection ⟨ X ⟩ U [ t , . . ., t n ] → ⟨ X ⟩ U induces an equivalence K (⟨ X ⟩ U ) ≃ Ð→ K (⟨ X ⟩ U [ t , . . ., t n ]) and we have K − n (⟨ X ⟩ U ) ≅ for every n ≥ .Proof. We have a commuative diagram of fibre sequencesK (⟨ X ⟩ U on ˜ Z ) / / (cid:15) (cid:15) K (⟨ X ⟩ U ) / / (cid:15) (cid:15) K ( U ) (cid:15) (cid:15) K (⟨ X ⟩ U [ t ] on ˜ Z [ t ]) / / K (⟨ X ⟩ U [ t ]) / / K ( U [ t ]) where the left map is an equivalence by Theorem 4.15 and the right map is an equivalence byhomotopy invariance of algebraic K-theory for regular noetherian schemes [TT90, 6.8]. Thevanishing of negative homotopy groups follows the same way. (cid:3) Vanishing of negative K-theory.Theorem 4.17 (Kerz) . If X is reduced and divisorial, then K i (⟨ X ⟩ U ) on ˜ Z ) = for i < Proof.
Since K i (⟨ X ⟩ U ) on ˜ Z ) ≃ colim X ′ ∈ Mdf ( X , U ) K ( X ′ on Z ′ ) by design, this follows as the right-hand side vanishes due to a result of Kerz [Ker18, Prop. 7]. (cid:3) Remark 4.18.
There is a conjecture of Schlichting whereby the negative K-theory of an abeliancategory vanishes [Sch06, 9.7]. Schlichting himself proved the vanishing in degree -1 andsubsequently the conjecture in the noetherian case [Sch06, 9.1, 9.3]. A generalised conjec-ture about stable ∞ -categories with noetherian heart was studied by Antieau-Gepner-Heller[AGH19]. Recently, Neeman [Nee20] gave a counterexample to both conjectures, Schlichting’sand Antieau-Gepner-Heller’s. Nevertheless, the following example provides a non-noetherianabelian category whose negative K-theory vanishes. Example 4.19.
Let k be a discretely valued field with valuation ring k ○ , uniformiser π , andresidue field ˜ k = k ○ /( π ) . Consider the scheme X ∶ = Spec ( k ○ ⟨ t ⟩) , its special fibre X / π = Spec ( ˜ k ⟨ t ⟩) and the open complement U ∶ = Spec ( k ⟨ t ⟩) . Let x ∶ Spec ( ˜ k ) → X / π be the zero section. Then wedefine the blow-up X ∶ = Bl { x } ( X ) . Its special fibre X / π is an affine line over ˜ k with an P k attached to the origin. We get a commutative diagram E / / (cid:15) (cid:15) X / π / / (cid:15) (cid:15) X = Bl { x } ( X ) (cid:15) (cid:15) Bl { x } ( X / π ) ≅ x x qqqqqq f f ▼▼▼▼▼▼▼ ❧❧❧❧❧❧❧❧ { x } / / X / π / / X where the two squares are cartesian, all horizontal maps are closed immersions, the blow-upBl { x } ( X / π ) → X / π is an isomorphism, and the map Bl { x } ( X / π ) → X / π is also closed im-mersion. Hence there is a closed immersion X / π ↪ X / π which splits the canonical projection.Now choose a closed point x ∶ Spec ( ˜ k ) → E ∖ X / π and define X ∶ = Bl { x } ( X ) and iterate thisconstruction. Thus we obtain a stricly increasing chain of closed immersions X / π ↪ X / π ↪ . . . ↪ X n / π ↪ . . . . . .and we can consider X i / π as a closed subscheme of X n / π for i < n . In each step, X n + / π isobtained from X n / π by attaching a P k to at a closed point not contained in X n − / π . Passing tothe Zariski-Riemann space ⟨ X ⟩ U , its special fibre ⟨ X ⟩ U / π admits a strictly decreasing chainof closed subschemes ⟨ X ⟩ U / π = p ∗ ( X / π ) ⊋ p ∗ ( X / π ∖ X / π ) ⊋ . . . ⊋ p ∗ n ( X n / π ∖ X n − / π ) ⊋ . . .. . .where p n ∶ ⟨ X ⟩ U / π → X n / π denotes the canonical projection. Thus we obatin a strictly increas-ing chain of ideals 0 = I ⊊ I ⊊ . . . ⊊ I n ⊊ . . .. . .in O ⟨ X ⟩ U / π where I n denotes the ideal sheaf corresponding to p ∗ n ( X n / π ∖ X n − / π ) . By con-struction, they all have support in ˜ Z = ⟨ X ⟩ U / π . Thus the category Coh (⟨ X ⟩ U / π ) is a non-noetherian abelian category whose negative K-theory vanishes due to Theorem 4.17 and Theo-rem 4.5. A PPENDIX
A. L
IMITS OF LOCALLY RINGED SPACES
In this section we collect for the convenience of the reader some facts about locally ringedspaces and filtered limits of those. This is based on the exposition by Fujiwara-Kato [FK18,ch. 0, §4.2]. -THEORY OF ADMISSIBLE ZARISKI-RIEMANN SPACES 13
Definition A.1.
We say that a locally ringed space ( X , O X ) is cohesive iff its structure sheaf O X is coherent. Example A.2.
For a locally noetherian scheme X , an O X -module is coherent if and only if itis finitely presented [Sta21, Tag 01XZ]. Thus a locally noetherian scheme is a cohesive locallyringed space. Lemma A.3 ([FK18, ch. 0, 4.1.8, 4.1.9]) . (i) If ( X , O X ) is cohesive, then an O X -module Fis coherent if and only if it is finitely presented.(ii) If f ∶ ( X , O X ) → ( Y , O Y ) is a morphism of locally ringed spaces and ( X , O X ) is cohesive,then f ∗ ∶ Mod ( Y ) → Mod ( X ) restricts to a functor f ∗ ∶ Coh ( Y ) → Coh ( X ) . Definition A.4.
A topological space is said to be coherent iff it is quasi-compact, quasi-separated, and admits an open basis of quasi-compact subsets. A topological space is called sober iff it is a T -space and any irreducible closed subset has a (unique) generic point. Example A.5.
The underlying topological space of a quasi-compact and quasi-separated schemeis coherent and sober.
Proposition A.6 ([FK18, ch. 0, 2.2.9, 2.2.10]) . Let ( X i , ( p i j ) j ∈ I ) i ∈ I be a filtered system of topo-logical spaces. Denote by X its limit and by p i ∶ X → X i the projection maps.(i) Assume that the topologies of the X i are generated by quasi-compact open subsets andthat the transition maps p i j are quasi-compact. Then every quasi-compact open subsetU ⊂ X is the preimage of a quasi-compact open subset U i ⊂ X i for some i ∈ I.(ii) Assume that all the X i are coherent and sober and that the transition maps p i j arequasi-compact. Then X is coherent and sober and the p i are quasi-compact. Proposition A.7.
Let ( X i , O X i , ( p i j ) j ∈ I ) i ∈ I be a filtered system of locally ringed spaces. Thenits limit ( X , O X ) in the category of locally ringed spaces exists. Let p i ∶ X → X i be the canonicalprojections.(i) The underlying topological space X is the limit of ( X i ) i ∈ I in Top .(ii) O X = colim i ∈ I p − i O X i in Mod ( X ) .(iii) For every x ∈ X have O X , x = colim i ∈ I p − i O X i , p i ( x ) in Ring .Assume additionally that every X i is coherent and sober and that all transitions maps arequasi-compact.(iv) The canonical functor colim i ∈ I Mod fp ( X i ) Ð→ Mod fp ( X ) is an equivalence in the 2-category of categories. In particular, for any finitely presented O X -module F there exists an i ∈ I and a finitely presented O X i -module F i such thatF ≅ p ∗ i F i .(v) For any morphism ϕ ∶ F → G between finitely presented O X -modules there exists ani ∈ I and a morphism ϕ i ∶ F i → G i between finitely presented O X i -modules such that ϕ ≅ p ∗ i ϕ i . Additionally, if ϕ is an isomorphism or an epimorphism, then one canchoose ϕ i to be an isomorphism or an epimorphism, respectively.(vi) For every i ∈ I let F i be an O X i -module and for every i ≤ j in I let ϕ i j ∶ p ∗ i j F i → F j be amorphism of O X j -modules such that ϕ ik = ϕ jk ○ p ∗ jk ϕ i j whenever i ≤ j ≤ k in I. Denoteby F the O X -module colim i p ∗ i F i . Then the canonical map colim i ∈ I H ∗ ( X i , F i ) Ð→ H ∗ ( X , F ) is an isomorphism of abelian groups.Proof. The existence, (i), (ii), and (iii) are [FK18, ch. 0, 4.1.10] and (iv) and (v) are [FK18, ch. 0,4.2.1–4.2.3]. Finally, (vi) is [FK18, ch. 0, 4.4.1]. (cid:3) A PPENDIX
B. L
OCALISATIONS OF EXACT CATEGORIES
Notation.
In this section let X be a locally ringed space and let j ∶ U → X be the inclusion of anopen subspace. Lemma B.1.
Let M be a full exact subcategory of Mod ( X ) and denote by S U the set of thosemorphisms f in M such that j ∗ f is an isomorphism.(i) If j ∗ j ∗ F ∈ M for every F ∈ M , then S U constitutes a left multiplicative system.(ii) If j ! j ∗ F ∈ M for every F ∈ M , then S U constitutes a right multiplicative system.Proof. For (i), we check the conditions of left multiplicative systems [Sta21, Tag 04VC]. Clearly, S U contains the identity morphisms and is closed under composition. Consider a span F ′ s ← F f → G in M with s ∈ S U . We want to show that there exists a commutative diagram F f / / s (cid:15) (cid:15) G t (cid:15) (cid:15) F ′ f ′ / / G ′ with t ∈ S U . We set t to be the unit η G ∶ G → j ∗ j ∗ G and define f ′ to be the composition of the unit η F ′ ∶ F ′ → j ∗ j ∗ F ′ and the morphism ( j ∗ j ∗ f ) ○ ( j ∗ j ∗ s ) − ∶ j ∗ j ∗ F ′ → j ∗ j ∗ G . For the last conditionlet f , g ∶ F → G be two morphisms in M and let s ∶ F ′ → F be in S U such that f ○ s = g ○ s . We haveto show that there exists a morphism t ∶ G → G ′ in S U such that t ○ f = t ○ g . Again, we set t tobe the unit η G ∶ G → j ∗ j ∗ G . Since f ○ s = g ○ s and s ∈ S U , we have that j ∗ f = j ∗ g , hence η G ○ f = ( j ∗ j ∗ f ) ○ η F = ( j ∗ j ∗ g ) ○ η F = η G ○ g which finishes the first part of the proof. Part (ii) works dually by using the counit j ! j ∗ → idinstead. (cid:3) Definition B.2.
In the situation of Lemma B.1 define M U to be the subcategory of of Mod ( U ) whose obects are the modules of the form j ∗ F for F ∈ M and whose morphims are given byHom M U ( j ∗ F , j ∗ G ) ∶ = im ( Hom M ( F , G ) j ∗ Ð→ Hom
Mod ( U ) ( j ∗ F , j ∗ G )) . Corollary B.3.
In the situation of Lemma B.1 there is an equivalence of categories M [ S − U ] ≃ Ð→ M U . Proof.
By the universal property of the localisation M → M [ S − U ] [Sta21, Tag 04VG] the de-sired functor exists. Moreover, on objects it is given by the functor j ∗ . Since M → M U isessentially surjective by design, this also holds for the functor M [ S − U ] → M U . It remains toshow that it is fully faithful. For two objects F , G ∈ M [ S − U ] we have to show that the map Ψ F , G ∶ Hom M[ S − U ] ( F , G ) Ð→ Hom M U ( j ∗ F , j ∗ G ) is a bijection. By construction of the localisation we have thatHom M[ S − U ] ( F , G ) = colim ( s ∶ G → G ′ )∈ S U Hom M ( F , G ′ ) [Sta21, Tag 05Q0] and the map ψ F , G identifies with the map from the colimit induced by j ∗ . Thus Ψ F , G is surjective. For injectivity, let f ∶ F → G be a morphism in M such that j ∗ f =
0. Then the composition F f → G η G → j ∗ j ∗ G is zero, hence f yields the zero element inHom M[ S − U ] ( F , G ) . (cid:3) Reminder B.4.
Recall that an exact category is an additive category A together with a classof conflations A ↣ B ↠ C -THEORY OF ADMISSIBLE ZARISKI-RIEMANN SPACES 15 satisfying certain axioms, cf. [Sch04, 1.1]. The morphism A ↣ B appearing in conflations arecalled inflations and the morphisms B ↠ C are called deflations . Every abelian category isan exact category whose conflations are the the short exact sequences; hence the inflations arethe monomorphisms and the deflations are the epimorphisms. Definition B.5.
Let A be an exact subcategory of an exact category B . We say that A ⊂ B is ...(i) a Serre subcategory iff for every conflation X ′ ↣ X ↠ X ′′ in B we have that X ∈ A if and only if both X ′ and X ′′ lie in A .(ii) right-filtering iff it is a Serre subcategory and if for every morphism f ∶ B → A with B ∈ B and A ∈ A there exists an object A ′ ∈ A such that f can be factored as a composi-tion B → A ′ ↠ A where the morphism B ↠ A is a deflation.(iii) right-s-filtering iff it is right-filtering and if for every inflation A ↣ B in B thereexists deflation B ↠ A ′ with A ′ ∈ A such that the composition A ↣ B ↠ A ′ is aninflation. Lemma B.6.
In the situation of Lemma B.1, let Z be the closed complement of U in X . Denoteby M Z the full subcategory of M which is spanned by modules with support in Z. Then M Z is a right-s-filtering subcategory of M and the quotient category M/M Z is equivalent to thelocalisation M[ S − U ] .Proof. The inclusion M Z ↪ M is the kernel of the exact restriction functor M → Mod ( U ) ,hence it is closed under subobjects, quotients, and extension, thus a Serre subcategory. Let i ∶ Z ↪ X be the inclusion map. Given a morphism f ∶ B → A in M with A ∈ M Z , we have afactorisation of f as B ε B ↠ i ∗ i ∗ B i ∗ i ∗ f → i ∗ i ∗ A ( ε A ) − → A .and the composition ( ε A ) − ○ ( i ∗ i ∗ f ) is a deflation. Thus M Z is right-filtering. Now let A ↣ B be an inflation (i.e. a monomorphism whose cokernel lies in M ) with A ∈ M Z . Then the unit B ↠ i ∗ i ∗ B is a deflation and the composition A ↣ B η B → i ∗ i ∗ B is an inflation as it equals thecomposition A η A → i ∗ i ∗ A ↣ i ∗ i ∗ B . For the last part note that the quotient M/M Z is defined asthe localisation along those morphisms which are finite compositions of inflations with cokernelin M Z and deflations with kernel in M Z . It is straightforward to check that these are preciselythe morphisms in S U . (cid:3) R EFERENCES [AGH19] Benjamin Antieau, David Gepner, and Jeremiah Heller,
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