Algebraic K-theory of quasi-smooth blow-ups and cdh descent
aa r X i v : . [ m a t h . K T ] F e b ALGEBRAIC K-THEORY OF QUASI-SMOOTH BLOW-UPSAND CDH DESCENT
ADEEL A. KHAN
Abstract.
We construct a semi-orthogonal decomposition on the cat-egory of perfect complexes on the blow-up of a derived Artin stack in aquasi-smooth centre. This gives a generalization of Thomason’s blow-upformula in algebraic K-theory to derived stacks. We also provide a newcriterion for descent in Voevodsky’s cdh topology, which we use to givea direct proof of Cisinski’s theorem that Weibel’s homotopy invariantK-theory satisfies cdh descent.
1. Introduction 12. Preliminaries 52.1. Derived algebraic geometry 52.2. Semi-orthogonal decompositions 72.3. Additive and localizing invariants 93. The projective bundle formula 93.1. Projective bundles 93.2. Semi-orthogonal decomposition on Qcoh ( P (E)) ( Bl Z / X ) Introduction i ∶ Z → X a regular closed immersion. Thismeans that Z is, Zariski-locally on X, the zero-locus of some regular sequence
Date : 2020-02-01.Author partially supported by SFB 1085 Higher Invariants, Universit¨at Regensburg.
ADEEL A. KHAN of functions f , . . . , f n ∈ Γ ( X , O X ) . Then the blow-up Bl Z / X fits into a square(1.1) P (N Z / X ) Bl Z / X Z X , i D q pi where the exceptional divisor is the projective bundle associated to the conor-mal sheaf N Z / X , which under the assumptions is locally free of rank n . Aresult of Thomason [Tho93b] asserts that after taking algebraic K-theory,the induced square of spectraK ( X ) K ( Z ) K ( Bl Z / X ) K ( P (N Z / X )) i ∗ p ∗ is homotopy cartesian. Here K ( X ) denotes the Bass–Thomason–Trobaughalgebraic K-theory spectrum of perfect complexes on a scheme X. We maysummarize this property by saying that algebraic K-theory satisfies descent with respect to blow-ups in regularly immersed centres.Now suppose that i is more generally a quasi-smooth closed immersion ofderived schemes. This means that Z is, Zariski-locally on X, the derived zero-locus of some arbitrary sequence of functions f , . . . , f n ∈ Γ ( X , O X ) . (WhenX is a classical scheme and the sequence is regular, this is the same as theclassical zero-locus, and we are in the situation discussed above.) In thederived setting there is still a conormal sheaf N Z / X on Z, locally free of rank n , and one may still form the blow-up square (1.1), see [KR18a]. Our goalin this paper is to generalize Thomason’s result above to this situation. Atthe same time we also allow X to be a derived Artin stack , and considerany additive invariant of stable ∞ -categories (see Definition 2.6). Examplesof additive invariants include algebraic K-theory K, connective algebraic K-theory K cn , topological Hochschild homology THH, and topological cyclichomology TC. Theorem A.
Let E be an additive invariant of stable ∞ -categories. Then E satisfies descent by quasi-smooth blow-ups. That is, given a derived Artinstack X and a quasi-smooth closed immersion i ∶ Z → X of virtual codimen-sion n ⩾ , form the blow-up square (1.1). Then the induced commutativesquare E ( X ) E ( Z ) E ( Bl Z / X ) E ( P (N Z / X )) i ∗ p ∗ is homotopy cartesian. We deduce Theorem A from an analysis of the categories of perfect com-plexes on Bl Z / X and on the exceptional divisor P (N Z / X ) . The relevant notionis that of a semi-orthogonal decomposition , see Definition 2.2. LGEBRAIC K-THEORY OF QUASI-SMOOTH BLOW-UPS AND CDH DESCENT 3
Theorem B.
Let X be a derived Artin stack. For any locally free O X -module E of rank n + , n ⩾ , consider the projective bundle q ∶ P (E) → X . Thenwe have: (i) For each ⩽ k ⩽ n , the assignment F ↦ q ∗ (F ) ⊗ O(− k ) defines a fullyfaithful functor Perf ( X ) → Perf ( P (E)) , whose essential image we denote A (− k ) . (ii) The sequence of full subcategories ( A ( ) , . . . , A (− n )) forms a semi-orthogonaldecomposition of Perf ( P (E)) . Theorem C.
Let X be a derived Artin stack. For any quasi-smooth closedimmersion i ∶ Z → X of virtual codimension n ⩾ , form the blow-up square(1.1). Then we have: (i) The assignment
F ↦ p ∗ (F ) defines a fully faithful functor Perf ( X ) → Perf ( Bl Z / X ) ,whose essential image we denote B ( ) . (ii) For each ⩽ k ⩽ n − , the assignment F ↦ ( i D ) ∗ ( q ∗ (F ) ⊗ O(− k )) definesa fully faithful functor Perf ( Z ) → Perf ( Bl Z / X ) , whose essential image wedenote B (− k ) . (iii) The sequence of full subcategories ( B ( ) , . . . , B (− n + )) forms a semi-orthogonaldecomposition of Perf ( Bl Z / X ) . We immediately deduce the projective bundle and blow-up formulasE ( P (E)) ≃ n ⊕ m = E ( X ) , E ( Bl Z / X ) ≃ E ( X ) ⊕ n − ⊕ k = E ( Z ) , for any additive invariant E, see Corollaries 3.6 and 4.4, from which Theo-rem A immediately follows (see Subsect. 4.5).1.2. The results mentioned above admit the following interesting specialcases:(a) Suppose that X is a smooth projective variety over the field of complexnumbers. This case of Theorem B was proven by Orlov in [Orl92]. He alsoproved Theorem C for any smooth subvariety Z ↪ X.(b) More generally suppose that X is a quasi-compact quasi-separated classi-cal scheme. Then the projective bundle formula (Corollary 3.6) for alge-braic K-theory was proven by Thomason [TT90, Tho93a]. Similarly sup-pose that i ∶ Z → X is a quasi-smooth closed immersion of quasi-compactquasi-separated classical schemes. Then it is automatically a regular closedimmersion, and in this case Thomason also proved Corollary 4.4 for algebraicK-theory [Tho93b]. In fact, the papers [Tho93a] and [Tho93b] essentiallycontain under these assumptions proofs of Theorems B and C, respectively,even if the term “semi-orthogonal decomposition” is not used explicitly. ForTHH and TC, these cases of Corollaries 3.6 and 4.4 were proven by Blumbergand Mandell [BM12].(c) More generally still, let X and Z be classical Artin stacks. These cases ofTheorems B and C are proven by by Bergh and Schn¨urer in [BS17]. However
ADEEL A. KHAN we note that Corollaries 3.6 and 4.4 were obtained earlier by Krishna andRavi in [KR18b], and their arguments in fact prove Theorems B and C forclassical Artin stacks.(d) Let X be a noetherian affine classical scheme, and let Z be the derived zero-locus of some functions f , . . . , f n ∈ Γ ( X , O X ) . Then the canonical morphism i ∶ Z → X is a quasi-smooth closed immersion. In this case, Theorem Afor algebraic K-theory was proven by Kerz–Strunk–Tamme [KST18] (wherethe blow-up Bl Z / X was explicitly modelled as the derived fibred productX × A n Bl { }/ A n ), as part of their proof of Weibel’s conjecture on negativeK-theory.1.3. Let KH denote homotopy invariant K-theory. Recall that this is the A -localization of the presheaf X ↦ K ( X ) . That is, it is obtained by forcingthe property of A -homotopy invariance: for every quasi-compact quasi-separated algebraic space X, the mapKH ( X ) → KH ( X × A ) is invertible (see [Wei89, Cis13]). As an application of Theorem A, we givea new proof of the following theorem of Cisinski [Cis13]: Theorem D.
The presheaf of spectra S ↦ KH ( S ) satisfies cdh descent onthe site of quasi-compact quasi-separated algebraic spaces. This was first proven by Haesemeyer [Hae04] for schemes over a field ofcharacteristic zero, using resolution of singularities. Cisinski’s proof overgeneral bases (noetherian schemes of finite dimension) relies on Ayoub’sproper base change theorem in motivic homotopy theory. A different proofof Theorem D (also in the noetherian setting) was recently given by Kerz–Strunk–Tamme [KST18, Thm. C], as an application of pro-cdh descent andtheir resolution of Weibel’s conjecture on negative K-theory. The proofwe give here is more direct and uses a new criterion for cdh descent (seeTheorem 5.6 for a more precise statement):
Theorem E.
Let F be a Nisnevich sheaf of spectra on the category of quasi-compact quasi-separated algebraic spaces. Then F satisfies cdh descent if andonly if it sends closed squares and quasi-smooth blow-up squares to cartesiansquares. Theorem E can be compared to a similar criterion due to Haesemeyer,implicit in [Hae04], which applies to Nisnevich sheaves of spectra on thecategory of schemes over a field k of characteristic zero. It asserts that forsuch a sheaf, cdh descent is equivalent to descent for finite cdh squares andregularly immersed blow-up squares. Note that the first condition is strongerthan descent for closed squares, while the second is weaker than descentfor quasi-smooth blow-up squares: regularly immersed blow-up squares areprecisely those quasi-smooth blow-up squares where all schemes appearingare underived. For invariants of stable ∞ -categories, a similar cdh descentcriterion was noticed independently by Land and Tamme [LT18, Thm. A.2].Theorem D was extended to certain nice Artin stacks recently by Hoyoisand Hoyois–Krishna [Hoy16, HK17]. Our cdh descent criterion also applies LGEBRAIC K-THEORY OF QUASI-SMOOTH BLOW-UPS AND CDH DESCENT 5 in that setting (Remark 5.11(iii)) and gives another potential approach tosuch results.1.4. The organization of this paper is as follows. We begin in Sect. 2 withsome background on derived algebraic geometry and on semi-orthogonaldecompositions of stable ∞ -categories.Sect. 3 is dedicated to the proof of Theorem B. We first show thatthe semi-orthogonal decomposition exists on the larger stable ∞ -categoryQcoh ( P (E)) (Theorem 3.3). Then we show that it restricts to Perf ( P (E)) (Subsect. 3.3), and deduce the projective bundle formula (Corollary 3.6) forany additive invariant.We follow a similar pattern in Sect. 4 to prove Theorem C. There is asemi-orthogonal decomposition on Qcoh ( Bl Z / X ) (Theorem 4.3) which thenrestricts to Perf ( Bl Z / X ) (Subsect. 4.4). This gives both the blow-up formula(Corollary 4.4) as well as Theorem A (4.5.2) for additive invariants. Asinput we prove a Grothendieck duality statement for virtual Cartier divisors(Proposition 4.2) that should be of independent interest.Sect. 5 contains our results on cdh descent and KH. We first give thegeneral cdh descent criterion (Theorem 5.6). We apply this criterion to KHto give our proof of Theorem D (5.4.3).1.5. I would like to thank Marc Hoyois, Charanya Ravi, and David Rydhfor helpful discussions and comments on previous revisions. I am especiallygrateful to David Rydh for pointing out the relevance of the resolution prop-erty in Sect. 5. 2. Preliminaries
Throughout the paper we work with the language of ∞ -categories as in[HTT, HA].2.1. Derived algebraic geometry.
This paper is set in the world of de-rived algebraic geometry, as in [TV08, SAG, GR17].2.1.1. Let SCRing denote the ∞ -category of simplicial commutative rings.A derived stack is an ´etale sheaf of spaces X ∶ SCRing → Spc. If X iscorepresentable by a simplicial commutative ring A, we write X = Spec ( A ) and call X an affine derived scheme . A derived scheme is a derived stack Xthat admits a Zariski atlas by affine derived schemes, i.e., a jointly surjectivefamily ( U i → X ) i of Zariski open immersions with each U i an affine derivedscheme. Allowing Nisnevich, ´etale or smooth atlases, respectively, gives riseto the notions of derived algebraic space , derived Deligne–Mumford stack ,and derived Artin stack . The precise definition is slightly more involved,see e.g. [GR17, Vol. I, Sect. 4.1]. That this agrees with the classical notion of algebraic space (at least under quasi-compactness and quasi-separatedness hypotheses) follows from [RG71, Prop. 5.7.6]. Thatit agrees with Lurie’s definition follows from [SAG, Ex. 3.7.1.5].
ADEEL A. KHAN
Any derived stack X admits an underlying classical stack which we denoteX cl . If X is a derived scheme, algebraic space, Deligne–Mumford or Artinstack, then X cl is a classical such. For example, Spec ( A ) cl = Spec ( π ( A )) for a simplicial commutative ring A.2.1.2. Let X be a derived scheme and let f , . . . , f n ∈ Γ ( X , O X ) be functionsclassifying a morphism f ∶ X → A n to affine space. The derived zero-locus of these functions is given by the derived fibred product(2.1) Z X { } A n . f If X is classical, then Z is classical if and only if the sequence ( f , . . . , f n ) is regular in the sense of [SGA 6], in which case Z is regularly immersed.A closed immersion of derived schemes i ∶ Z → X is called quasi-smooth (ofvirtual codimension n ) if it is cut out Zariski-locally as the derived zero-locusof n functions on X. Equivalently, this means that i is of finite presentationand its shifted cotangent complex N Z / X ∶ = L Z / X [ − ] is locally free (of rank n ). A closed immersion of derived Artin stacks is quasi-smooth if it satisfiesthis condition smooth-locally.A morphism of derived schemes f ∶ Y → X is quasi-smooth if it can befactored, Zariski-locally on Y, through a quasi-smooth closed immersion i ∶ Y → X ′ and a smooth morphism X ′ → X. A morphism of derived Artinstacks is quasi-smooth if it satisfies this condition smooth-locally on Y. Werefer to [KR18a] for more details on quasi-smoothness.2.1.3. Important for us is the following construction from [KR18a]. Givenany quasi-smooth closed immersion i ∶ Z → X of derived Artin stacks, thereis an associated quasi-smooth blow-up square :(2.2) D Bl Z / X Z X . i D q pi Here Bl Z / X is the blow-up of X in Z, which is a quasi-smooth proper derivedArtin stack over X, and D = P ( N Z / X ) is the projectivized normal bundle,which is a smooth proper derived Artin stack over X. This square is universalwith the following properties: (a) the morphism i D is a quasi-smooth closedimmersion of virtual codimension 1, i.e., a virtual effective Cartier divisor;(b) the underlying square of classical Artin stacks is cartesian; and (c) thecanonical map q ∗ N Z / X → N D / Bl Z / X is surjective on π . When X is a derivedscheme (resp. derived algebraic space, derived Deligne–Mumford stack),then so is Bl Z / X . LGEBRAIC K-THEORY OF QUASI-SMOOTH BLOW-UPS AND CDH DESCENT 7 ∞ -category of quasi-coherentsheaves Qcoh ( X ) is the limitQcoh ( X ) = lim ←Ð Spec ( A ) → X Qcoh ( Spec ( A )) taken over all morphisms Spec ( A ) → X with A ∈ SCRing. Here Qcoh ( Spec ( A )) is the stable ∞ -category Mod A of A-modules in the sense of Lurie. In-formally speaking, a quasi-coherent sheaf F on X is thus a collection ofquasi-coherent sheaves x ∗ ( F ) ∈ Qcoh ( Spec ( A )) , for every simplicial commu-tative ring A and every A-point x ∶ Spec ( A ) → X, together with a homotopycoherent system of compatibilities.The full subcategory Perf ( X ) ⊂ Qcoh ( X ) is similarly the limitPerf ( X ) = lim ←Ð Spec ( A ) → X Perf ( Spec ( A )) , where Perf ( Spec ( A )) is the stable ∞ -category Mod perfA of perfect A-modules.In other words, F ∈ Qcoh ( X ) belongs to Perf ( X ) if and only if x ∗ ( F ) isperfect for every simplicial commutative ring A and every morphism x ∶ Spec ( A ) → X.2.1.5. There is an inverse image functor f ∗ ∶ Qcoh ( X ) → Qcoh ( Y ) for anymorphism of derived stacks f ∶ Y → X. It preserves perfect complexes andinduces a functor f ∗ ∶ Perf ( X ) → Perf ( Y ) . Regarded as presheaves of ∞ -categories, the assignments X ↦ Qcoh ( X ) and X ↦ Perf ( X ) satisfy descent for the fpqc topology ([SAG, Cor. D.6.3.3], [GR17, Thm. 1.3.4]). This meansin particular that given any fpqc covering family ( f α ∶ X α → X ) α , the familyof inverse image functors f ∗ α ∶ Qcoh ( X ) → Qcoh ( X α ) is jointly conservative.If f ∶ Y → X is quasi-compact and schematic , in the sense that its fibre overany affine derived scheme is a derived scheme, then there is a direct imagefunctor f ∗ , right adjoint to f ∗ , which commutes with colimits and satisfiesa base change formula against inverse images ([SAG, Prop. 2.5.4.5], [GR17,Vol. 1, Chap. 3, Prop. 2.2.2]). If f is proper, locally of finite presentation,and of finite tor-amplitude, then f ∗ also preserves perfect complexes [SAG,Thm. 6.1.3.2].2.2. Semi-orthogonal decompositions.
The following definitions wereoriginally formulated by [BK89] in the language of triangulated categoriesand are standard.
Definition 2.1.
Let C be a stable ∞ -category and D a stable full subcat-egory. An object x ∈ C is left orthogonal , resp. right orthogonal , to D ifthe mapping space Maps C ( x, d ) , resp. Maps C ( d, x ) , is contractible for allobjects d ∈ D . We let ⊥ D ⊆ C and D ⊥ ⊆ C denote the full subcategories ofleft orthogonal and right orthogonal objects, respectively. Note that if A is discrete (an ordinary commutative ring), then this is not the abeliancategory of discrete A-modules, but rather the derived ∞ -category of this abelian categoryas in [HA, Chap. 1]. ADEEL A. KHAN
Definition 2.2.
Let C be a stable ∞ -category and let C ( ) , . . . , C ( − n ) befull stable subcategories. Suppose that the following conditions hold:(i) For all integers i > j , there is an inclusion C ( i ) ⊆ ⊥ C ( j ) .(ii) The ∞ -category C is generated by the subcategories C ( ) , . . . , C ( − n ) , underfinite limits and finite colimits.Then we say that the sequence ( C ( ) , . . . , C ( − n )) forms a semi-orthogonaldecomposition of C .Semi-orthogonal decompositions of length 2 come from split short exactsequences of stable ∞ -categories, as in [BGT13]. Definition 2.3. (i) A short exact sequence of small stable ∞ -categories is a diagram C ′ i Ð→ C p Ð→ C ′′ , where i and p are exact, the composite p ○ i is null-homotopic, i is fullyfaithful, and p induces an equivalence ( C / C ′ ) idem ≃ ( C ′′ ) idem (where ( − ) idem denotes idempotent completion).(ii) A short exact sequence of small stable ∞ -categories C ′ i Ð→ C p Ð→ C ′′ is split if there exist functors q ∶ C → C ′ and j ∶ C ′′ → C , right adjoint to i and p , respectively, such that the unit id → q ○ i and co-unit p ○ j → id areinvertible. Remark 2.4.
Let C be a small stable ∞ -category, and let ( C ( ) , C ( − )) be a semi-orthogonal decomposition. Then for any object x ∈ C , there existsan exact triangle x ( ) → x → x ( − ) , where x ( ) ∈ C ( ) and x ( − ) ∈ C ( − ) . To see this, simply observe thatthe full subcategory spanned by objects x for which such a triangle exists,is closed under finite limits and colimits, and contains C ( ) and C ( − ) .Moreover, the assignments x ↦ x ( ) and x ↦ x ( − ) determine well-definedfunctors q ∶ C → C ( ) and p ∶ C → C ( − ) , respectively, which are right andleft adjoint, respectively, to the inclusions (see e.g. [SAG, Rem. 7.2.0.2]).It follows from this that any semi-orthogonal decomposition ( C ( ) , C ( − )) induces a split short exact sequence C ( ) → C p Ð→ C ( − ) . Lemma 2.5.
Let C be a stable ∞ -category, and let ( C ( ) , . . . , C ( − n )) be asequence of full stable subcategories forming a semi-orthogonal decompositionof C . For each ⩽ m ⩽ n , let C ⩽− m ⊆ C denote the full stable subcategorygenerated by objects in the union C ( − m ) ∪ ⋯ ∪ C ( − n ) , and let C ⩽− n − ⊆ C denote the full subcategory spanned by the zero object. Then there are splitshort exact sequences C ⩽− m − ↪ C ⩽− m → C ( − m ) LGEBRAIC K-THEORY OF QUASI-SMOOTH BLOW-UPS AND CDH DESCENT 9 for each ⩽ m ⩽ n .Proof. It follows from the definitions that for each 0 ⩽ m ⩽ n , the sequence ( C ( − m ) , C ⩽− m − ) forms a semi-orthogonal decomposition of C . Thereforethe claim follows from Remark 2.4. (cid:3) Additive and localizing invariants.
The following definition is from[BGT13], except that we do not require commutativity with filtered colimits.
Definition 2.6.
Let A be a stable presentable ∞ -category. Let E be an A -valued functor from the ∞ -category of small stable ∞ -categories and exactfunctors.(i) We say that E is an additive invariant if for any split short exact sequence C ′ i Ð→ C p Ð→ C ′′ , the induced map E ( C ′ ) ⊕ E ( C ′′ ) ( i,j ) ÐÐ→ E ( C ) is invertible, where j is a right adjoint to p .(ii) We say that E is a localizing invariant if for any short exact sequence C ′ i Ð→ C p Ð→ C ′′ , the induced diagram E ( C ′ ) → E ( C ) → E ( C ′′ ) is an exact triangle. Remark 2.7.
Any localizing invariant is also additive.
Lemma 2.8.
Let C be a stable ∞ -category, and let ( C ( ) , . . . , C ( − n )) be asequence of full stable subcategories forming a semi-orthogonal decompositionof C . Then for any additive invariant E there is a canonical isomorphism E ( C ) ≃ n ⊕ m = E ( C ( − m )) . Proof.
Follows immediately from Lemma 2.5. (cid:3) The projective bundle formula
Projective bundles.
Let X be a derived stack and E a locally free O X -module of finite rank. Recall that the projective bundle associated to E is a derived stack P ( E ) over X equipped with an invertible sheaf O ( ) together with a surjection E → O ( ) . More precisely, for any derived schemeS over X, with structural morphism x ∶ S → X, the space of S-points of P ( E ) is the space of pairs ( L , u ) , where L is a locally free O S -module of rank 1,and u ∶ x ∗ ( E ) → L is surjective on π . We recall the standard properties ofthis construction: Proposition 3.1. (i) If f ∶ X ′ → X is a morphism of derived stacks, then there is a canonicalisomorphism P ( f ∗ ( E )) → P ( E ) × X X ′ of derived stacks over X ′ . (ii) The projection P ( E ) → X is proper and schematic. In particular, if X isa derived scheme (resp. derived algebraic space, derived Deligne–Mumfordstack, derived Artin stack), then the same holds for the derived stack P ( E ) . (iii) The relative cotangent complex L P ( E )/ X is canonically isomorphic to F ⊗O ( − ) , where the locally free sheaf F is the fibre of the canonical map E → O ( ) . In particular, the morphism P ( E ) → X is smooth of relativedimension equal to rk ( E ) − . Proposition 3.2 (Serre) . Let X be a derived Artin stack, and E a locallyfree sheaf of rank n + , n ⩾ . If q ∶ P ( E ) → X denotes the associatedprojective bundle, then we have canonical isomorphisms q ∗ ( O ( )) ≃ O X , q ∗ ( O ( − m )) ≃ ( ⩽ m ⩽ n ) in Qcoh ( X ) .Proof. There is a canonical map O X → q ∗ ( O ( )) , the unit of the adjunction ( q ∗ , q ∗ ) , and there is a unique map 0 → q ∗ ( O ( − m )) for each m . To showthat these are invertible, we may use fpqc descent and base change to thecase where X is affine and E is free. Then this is Serre’s computation, asgeneralized to the derived setting by Lurie [SAG, Thm. 5.4.2.6]. (cid:3) Semi-orthogonal decomposition on
Qcoh ( P ( E )) . In this subsec-tion we will show that the stable ∞ -category Qcoh ( P ( E )) admits a canonicalsemi-orthogonal decomposition. Theorem 3.3.
Let X be a derived Artin stack. Let E be a locally free O X -module of rank n + , n ⩾ , and q ∶ P ( E ) → X the associated projectivebundle. Then we have: (i) For every integer k ∈ Z , the assignment F ↦ q ∗ ( F ) ⊗ O ( k ) defines a fullyfaithful functor Qcoh ( X ) → Qcoh ( P ( E )) . (ii) For every integer k ∈ Z , let C ( k ) ⊂ Qcoh ( P ( E )) denote the essential imageof the functor in (i). Then the subcategories C ( k ) , . . . , C ( k − n ) form asemi-orthogonal decomposition of Qcoh ( P ( E )) . We will need the following facts (see Lemmas 7.2.2.2 and 5.6.2.2 in [SAG]):
Lemma 3.4.
Let R be a simplicial commutative ring and X = Spec ( R ) .Denote by P n R = P ( O n + ) the n -dimensional projective space over R . Thenfor every integer m ∈ Z , there is a canonical isomorphism lim Ð→ J ⊊ [ n ] O ( m + ∣ J ∣) ∼ Ð→ O ( m + n + ) in Qcoh ( P n R ) , where the colimit is taken over the proper subsets J of the set [ n ] = { , , . . . , n } , and ⩽ ∣ J ∣ ⩽ n denotes the cardinality of such a subset. LGEBRAIC K-THEORY OF QUASI-SMOOTH BLOW-UPS AND CDH DESCENT 11
Lemma 3.5.
Let R be a simplicial commutative ring and X = Spec ( R ) .Denote by P n R = P ( O n + ) the n -dimensional projective space over R . Thenfor any connective quasi-coherent sheaf F ∈ Qcoh ( P n R ) , there exists a map ⊕ α O ( d α ) → F , with d α ∈ Z , which is surjective on π .Proof of Theorem 3.3. Since the functors − ⊗ O ( k ) are equivalences, it willsuffice to take k = F → q ∗ q ∗ ( F ) is invertible for all F ∈ Qcoh ( X ) . By fpqc descentand base change (2.1.5), we may reduce to the case where X = Spec ( R ) isaffine and E = O n + is free. Now both functors q ∗ and q ∗ are exact andmoreover commute with arbitrary colimits (the latter by 2.1.5 since q isquasi-compact and schematic), and Qcoh ( X ) ≃ Mod R is generated by O X under colimits and finite limits. Therefore we may assume F = O X , in whichcase the claim holds by Proposition 3.2.For claim (ii), let us first check the orthogonality condition in Defini-tion 2.2. Thus take F , G ∈ Qcoh ( X ) and consider the mapping spaceMaps ( q ∗ ( F ) , q ∗ ( G ) ⊗ O ( − m )) ≃ Maps ( F , q ∗ ( O ( − m )) ⊗ G ) for 1 ⩽ m ⩽ n , where the identification results from the projection formula.Since q ∗ ( O ( − m )) ≃ F ∈ Qcoh ( P ( E )) belongs to the fullsubcategory ⟨ C ( ) , . . . , C ( − n )⟩ ⊆ Qcoh ( P ( E )) generated under finite colim-its and limits by the subcategories C ( ) , . . . , C ( − n ) . Set G − = F ⊗ O ( − ) and define G m , for m ⩾
0, so that we have exact triangles(3.1) q ∗ q ∗ ( G m − ⊗ O ( )) counit ÐÐÐ→ G m − ⊗ O ( ) → G m . For each m ⩾ −
1, we claim that G m is right orthogonal to each of the subcate-gories C ( ) , C ( ) , . . . , C ( m ) . For m = − m ⩾ m −
1. Since q ∗ q ∗ ( G m − ⊗ O ( )) iscontained in C ( ) , it follows that G m is right orthogonal to C ( ) . To showthat G m is right orthogonal to C ( i ) , for 1 ⩽ i ⩽ m , it will suffice to show thatthe left-hand and middle terms of the exact triangle (3.1) are both rightorthogonal to C ( i ) . For the left-hand term this follows from the inclusion C ( ) ⊂ C ( i ) ⊥ , demonstrated above. For the middle term G m − ⊗ O ( ) , theclaim follows by the induction hypothesis.Now we claim that G n is zero. Using fpqc descent again, we may assumethat X = Spec ( R ) and E = O ⊕ n + is free (since the sequence ( G − , G , . . . , G n ) is stable under base change). Using Lemma 3.5 we can build a map ϕ ∶ ⊕ α O ( m α )[ k α ] → G n which is surjective on all homotopy groups. From Lemma 3.4 it followsthat G n is right orthogonal to all C ( i ) , i ∈ Z . Thus ϕ must be null-homotopic, so G n ≃ G n − ∈ C ( − ) , ..., G ∈ ⟨ C ( − ) , . . . , C ( − n )⟩ , and then finally that F ∈ ⟨ C ( ) , C ( − ) , . . . , C ( − n )⟩ as claimed. (cid:3) Proof of Theorem B.
We now deduce Theorem B from Theorem 3.3.First note that the fully faithful functor
F ↦ q ∗ ( F ) ⊗O ( k ) of Theorem 3.3(i)restricts to a fully faithful functor Perf ( X ) → Perf ( P ( E )) , since q ∗ preservesperfect complexes. This shows Theorem B(i).For part (ii) we argue again as in the proof of Theorem 3.3. The point isthat if F ∈ Qcoh ( P ( E )) is perfect, then so is each G m ∈ Qcoh ( P ( E )) , since q ∗ and q ∗ preserve perfect complexes (the latter because q is smooth andproper).3.4. Projective bundle formula.
From Theorem B and Lemma 2.8 wededuce:
Corollary 3.6.
Let X be a derived Artin stack, E a locally free O X -moduleof rank n + , n ⩾ , and q ∶ P ( E ) → X the associated projective bundle. Thenfor any additive invariant E , there is a canonical isomorphism E ( P ( E )) ≃ n ⊕ k = E ( X ) induced by the functors q ∗ ( − ) ⊗ O ( − k ) ∶ Perf ( X ) → Perf ( P ( E )) . The blow-up formula
Virtual Cartier divisors.
Recall from [KR18a] that a virtual (effec-tive) Cartier divisor on a derived Artin stack X is a quasi-smooth closedimmersion i ∶ D → X of virtual codimension 1. For any such i ∶ D → X, thereis a canonical exact triangle O X ( − D ) → O X → i ∗ ( O D ) , where O X ( − D ) is a locally free sheaf of rank 1, equipped with a canonicalisomorphism i ∗ ( O X ( − D )) ≃ N D / X (see 3.2.3 and 3.2.9 in [KR18a]). Lemma 4.1.
Let X be a derived Artin stack and i ∶ D → X a virtual Cartierdivisor. Then there is a canonical isomorphism i ∗ i ∗ ( O D ) ≃ O D ⊕ N D / X [ ] . Proof.
Applying i ∗ to the exact triangle above (and rotating), we get theexact triangle O D → i ∗ i ∗ ( O D ) → N D / X [ ] . The map O D → i ∗ i ∗ ( O D ) is induced by the natural transformation i ∗ ( η ) ∶ i ∗ → i ∗ i ∗ i ∗ (where η is the adjunction unit), so by the triangle identities ithas a retraction given by the co-unit map i ∗ i ∗ ( O D ) → O D . In other words,the triangle splits. (cid:3) Grothendieck duality.
Let i ∶ Z → X be a quasi-smooth closed im-mersion of derived Artin stacks. The functor i ∗ admits a right adjoint i ! ,which for formal reasons can be computed by the formula i ! ( − ) ≃ i ∗ ( − ) ⊗ ω D / X , LGEBRAIC K-THEORY OF QUASI-SMOOTH BLOW-UPS AND CDH DESCENT 13 where ω D / X ∶ = i ! ( O X ) is called the relative dualizing sheaf . See [SAG, Cor. 6.4.2.7].When i is a virtual Cartier divisor, ω D / X can be computed as follows: Proposition 4.2 (Grothendieck duality) . Let X be a derived Artin stack.Then for any virtual Cartier divisor i ∶ D → X , there is a canonical isomor-phism N ∨ D / X [ − ] ∼ Ð→ ω D / X of perfect complexes on D . In particular, there is a canonical identification i ! ≃ i ∗ ( − ) ⊗ N ∨ D / X [ − ] .Proof. Write
L ∶ = O X ( − D ) and consider again the exact triangle L → O X → i ∗ ( O D ) . By the projection formula, this can be refined to an exact triangleof natural transformations id ⊗ L → id → i ∗ i ∗ , or, passing to right adjoints,an exact triangle i ∗ i ! → id → id ⊗ L ∨ . In particular we get the exact triangle(4.1) i ∗ i ! ( O X ) → O X → L ∨ . The associated map L ∨ [ − ] → i ∗ i ! ( O X ) gives by adjunction a canonicalmorphism N ∨ D / X [ − ] ≃ i ∗ ( L ∨ )[ − ] → i ! ( O X ) , which we claim is invertible. By fpqc descent and the fact that i ! commuteswith the operation f ∗ , for any morphism f [SAG, Prop. 6.4.2.1], we mayassume that X is affine. In this case the functor i ∗ is conservative, so it willsuffice to show that the canonical map i ∗ ( N ∨ D / X )[ − ] → i ∗ i ! ( O X ) is invertible. Considering again the triangle F ⊗ L → F → i ∗ i ∗ ( F ) aboveand taking F = L ∨ , we get the exact triangle O X → L ∨ → i ∗ i ∗ ( L ∨ ) ≃ i ∗ ( N ∨ D / X ) , since L is invertible. Comparing with (4.1) yields the claim. (cid:3) Semi-orthogonal decomposition on
Qcoh ( Bl Z / X ) . In this subsec-tion we prove:
Theorem 4.3.
Let X be a derived Artin stack and i ∶ Z → X a quasi-smoothclosed immersion of virtual codimension n ⩾ . Let ̃ X = Bl Z / X and considerthe quasi-smooth blow-up square (2.2) D ̃ XZ X i D q pi Then we have: (i)
The functor p ∗ ∶ Qcoh ( X ) → Qcoh (̃ X ) is fully faithful. We denote its essen-tial image by D ( ) ⊂ Qcoh (̃ X ) . (ii) The functor ( i D ) ∗ ( q ∗ ( − ) ⊗ O ( − k )) ∶ Qcoh ( Z ) → Qcoh (̃ X ) is fully faithful,for each ⩽ k ⩽ n − . We denote its essential image by D ( − k ) ⊂ Qcoh (̃ X ) . (iii) For each ⩽ k ⩽ n − , the full stable subcategory D ( − k ) ⊂ Qcoh (̃ X ) is rightorthogonal to each of D ( ) , . . . , D ( − k + ) . (iv) The stable ∞ -category Qcoh (̃ X ) is generated by the full subcategories D ( ) , D ( − ) , . . . , D ( − n + ) under finite colimits and finite limits. In partic-ular, the sequence ( D ( ) , D ( − ) , . . . , D ( − n + )) forms a semi-orthogonaldecomposition of Qcoh (̃ X ) . Proof of (i) . The claim is that for any F ∈ Qcoh ( X ) , the unit map F → p ∗ p ∗ ( F ) is invertible. By fpqc descent we may reduce to the case whereX is affine and i fits in a cartesian square of the form (2.1). Since Qcoh ( X ) is then generated under colimits and finite limits by O X , and p ∗ commuteswith colimits since p is quasi-compact and schematic (2.1.5), we may assumethat F = O X . In other words, it suffices to show that the canonical map O X → p ∗ ( O ̃ X ) is invertible.D ̃ XZ X i D q pi P n − Bl { }/ A n { } A n , p i Since the left-hand square is the (derived) base change of the right-handsquare along the morphism f ∶ X → A n , it follows that the map O X → p ∗ ( O ̃ X ) is the inverse image of the canonical map O A n → ( p ) ∗ ( O Bl { }/ A n ) .Thus we reduce to the case where i is the immersion { } → A n . This iswell-known, see [SGA 6, Exp. VII].4.3.2. Proof of (ii) . It suffices to show the unit map
F → q ∗ ( i D ) ! ( i D ) ∗ q ∗ ( F ) is invertible for all F ∈ Qcoh ( Z ) . As in the previous claim we may assume Xis affine and that F = O Z . Using Proposition 4.2, the canonical identification N D /̃ X ≃ O D ( ) , and Lemma 4.1, the unit map is identified with O Z → q ∗ (( i D ) ∗ ( i D ) ∗ ( O D ) ⊗ O D ( − ))[ − ] ≃ q ∗ ( O D ( − )) ⊕ q ∗ ( O D ) . Since q ∶ D → Z is the projection of the projective bundle P ( N Z / X ) , itfollows from Proposition 3.2 that we have identifications q ∗ ( O D ( − )) ≃ q ∗ ( O D ) ≃ O Z , under which the map in question is the identity.4.3.3. Proof of (iii) . To see that D ( − k ) is right orthogonal to D ( ) , observethat by Theorem 3.3, the mapping spaceMaps ( p ∗ ( F X ) , ( i D ) ∗ ( q ∗ ( F Z ) ⊗ O ( − k ))) ≃ Maps ( q ∗ i ∗ ( F X ) , q ∗ ( F Z ) ⊗ O ( − k )) is contractible for every F X ∈ Qcoh ( X ) and F Z ∈ Qcoh ( Z ) .To see that D ( − k ) is right orthogonal to D ( − k ′ ) , for 1 ⩽ k ′ < k , considerthe mapping spaceMaps (( i D ) ∗ ( q ∗ ( F Z ) ⊗ O ( − k ′ )) , ( i D ) ∗ ( q ∗ ( F ′ Z ) ⊗ O ( − k ))) , for F Z , F ′ Z ∈ Qcoh ( Z ) . Using fpqc descent and base change for ( i D ) ∗ against f ∗ for any morphism f ∶ U → ̃ X, we may reduce to the case where X is affine.
LGEBRAIC K-THEORY OF QUASI-SMOOTH BLOW-UPS AND CDH DESCENT 15
Since Qcoh ( Z ) is then generated under colimits and finite limits by O Z , wemay assume that F Z = F ′ Z = O Z . Then we haveMaps (( i D ) ∗ ( O ( − k ′ )) , ( i D ) ∗ ( O ( − k ))) ≃ Maps (( i D ) ∗ ( i D ) ∗ ( O ( − k ′ )) , O ( − k )) ≃ Maps ( O ( − k ′ ) ⊕ O ( − k ′ + )[ ] , O ( − k )) by Lemma 4.1 and the projection formula, and this space is contractible byTheorem 3.3.4.3.4. Proof of (iv) . Denote by D the full subcategory of Qcoh (̃ X ) generatedby D ( ) , D ( − ) , . . . , D ( − n + ) under finite colimits and finite limits. Theclaim is that the inclusion D ⊆ Qcoh (̃ X ) is an equality. Note that O ̃ X ∈ D ( ) ⊂ D and ( i D ) ∗ ( O D ( − k )) ∈ D ( − k ) ⊂ D for 1 ⩽ k ⩽ n −
1. Consider theexact triangle O ̃ X ( − D ) → O ̃ X → ( i D ) ∗ ( O D ) and recall that O ̃ X ( − D ) ≃ O ̃ X ( ) .Tensoring with O ( − k ) and using the projection formula, we get the exacttriangle O ̃ X ( − k + ) → O ̃ X ( − k ) → ( i D ) ∗ ( O D ( − k )) for each 1 ⩽ k ⩽ n −
1. Taking k = O ̃ X ( − ) ∈ D . Continuingrecursively we find that O ̃ X ( − k ) ∈ D for all 1 ⩽ k ⩽ n − F ∈ Qcoh (̃ X ) . Denote by G ∈ Qcoh (̃ X ) the cofibre of the co-unit map p ∗ p ∗ ( F ) → F . Note that G is right orthogonal to D ( ) . For1 ⩽ m ⩽ n − G m recursively by the exact triangles ( i D ) ∗ ( q ∗ q ∗ (( i D ) ! ( G m − ) ⊗ O ( m )) ⊗ O ( − m )) counit ÐÐÐ→ G m − → G m . Just as in the proof of Theorem 3.3, a simple induction argument shows thateach G m is right orthogonal to all of the subcategories D ( ) , . . . , D ( m − ) .We now claim that G n − is zero; it will follow by recursion that F belongsto D , as desired.Since the objects G k are stable under base change, we may use fpqc descentand base change to assume that X is affine. Moreover we may assume that i ∶ Z → X fits in a cartesian square of the form (2.1). By [KR18a, 3.3.6], p ∶ ̃ X → X factors through a quasi-smooth closed immersion i ′ ∶ ̃ X → P n − . Recallfrom Lemma 3.4 that there is a canonical isomorphism lim Ð→ J ⊊ [ n − ] O (∣ J ∣) ≃ O ( n ) in Qcoh ( P n − ) . Applying ( i ′ ) ∗ , we get lim Ð→ J ⊊ [ n − ] O ̃ X (∣ J ∣) ≃ O ̃ X ( n ) inQcoh (̃ X ) . In particular, every O ̃ X ( k ) belongs to D for all k ∈ Z . Recall alsothat we may find a map ⊕ α O ( d α )[ n α ] → i ′∗ ( G n − ) which is surjective on allhomotopy groups (Lemma 3.5). By adjunction this corresponds to a map ⊕ α O ( d α )[ n α ] → G n − (which is also surjective on homotopy groups). Butthe source belongs to D , and the target is right orthogonal to D , so thismap is null-homotopic. Thus G n − is zero.4.4. Proof of Theorem C.
We now deduce Theorem C from Theorem 4.3.First note that the fully faithful functor
F ↦ p ∗ ( F ) of Theorem 4.3(i) pre-serves perfect complexes and therefore restricts to a fully faithful functorPerf ( X ) → Perf ( Bl Z / X ) . This shows Theorem C(i). Similarly, part (ii) follows from the fact that the functors q ∗ and ( i D ) ∗ preserve perfect complexes. For the latter, this is because i D is quasi-smooth(and hence of finite presentation and of finite tor-amplitude).For part (iii) we argue again as in the proof of Theorem 4.3(iv). The pointis that if F ∈ Qcoh ( Bl Z / X ) is perfect, then so is each G m ∈ Qcoh ( P ( E )) , since q ∗ , q ∗ , ( i D ) ∗ and ( i D ) ! all preserve perfect complexes. For the latter thisfollows from Proposition 4.2.4.5. Blow-up formula.
Corollary 4.4.
Let X be a derived Artin stack and i ∶ Z → X a quasi-smooth closed immersion of virtual codimension n ⩾ . Then for any additiveinvariant E , there is a canonical isomorphism E ( Bl Z / X ) ≃ E ( X ) ⊕ n − ⊕ k = E ( Z ) . Proof of Theorem A.
Combine Corollaries 4.4 and 3.6 (with E = N Z / X ). 5. The cdh topology
The cdh topology.
The following notion was introduced by Voevod-sky [Voe10b] for noetherian schemes:
Definition 5.1.
Suppose given a cartesian square Q of algebraic spaces(5.1) B YA X . pe (i) We say that Q is a Nisnevich square if e is an open immersion, and p is an´etale morphism inducing an isomorphism ( Y ∖ B ) red ≃ ( X ∖ A ) red .(ii) We say that Q is a proper cdh square , or abstract blow-up square , if e is aclosed immersion of finite presentation, and p is a proper morphism inducingan isomorphism ( Y ∖ B ) red ≃ ( X ∖ A ) red .(iii) We say that Q is a cdh square if it is either a Nisnevich square or a propercdh square.5.1.1. Given any class of commutative squares of algebraic spaces, we saythat a presheaf satisfies descent for this class if it sends all such squares tohomotopy cartesian squares, and the empty scheme to a terminal object. Incase of the three classes considered in Definition 5.1, it follows from a theo-rem of Voevodsky [Voe10a, Cor. 5.10] that descent in this sense is equivalentto ˇCech descent with respect to the associated Grothendieck topology. LGEBRAIC K-THEORY OF QUASI-SMOOTH BLOW-UPS AND CDH DESCENT 17
Example 5.2.
Every localizing invariant E satisfies Nisnevich descent whenregarded as a presheaf on quasi-compact quasi-separated algebraic spaceswith E ( X ) = E ( Perf ( X )) . This is essentially due to Thomason [TT90] and inthe asserted generality is a consequence of the study of compact generationproperties of the ∞ -categories Qcoh ( X ) carried out by Bondal–Van denBergh [BVdB03]. Example 5.3.
Any quasi-smooth blow-up square (2.2) induces a propercdh square P ( N Z / X ∣ Z cl ) ( Bl Z / X ) cl Z cl X cl on underlying classical algebraic spaces. Example 5.4.
Consider the class of proper cdh squares (5.1) where theproper morphism p is a closed immersion (with quasi-compact open comple-ment). The associated Grothendieck topology is the same as the one gen-erated by closed squares , i.e. cartesian squares as in (5.1) such that e and p are closed immersions, e is of finite presentation and p has quasi-compactopen complement, and A ⊔ Y → X is surjective on underlying topologicalspaces.
Example 5.5.
Note that for any algebraic space X, the square ∅ X red ∅ Xis a closed square as in Example 5.4.5.2.
A cdh descent criterion.Theorem 5.6.
Let F be a presheaf on the category C of algebraic spaces,with values in a stable ∞ -category. Then F satisfies cdh descent if and onlyif it satisfies the following conditions: (i) It sends the empty scheme to a zero object. (ii)
It sends Nisnevich squares to cartesian squares. (iii)
It sends closed squares to cartesian squares. (iv)
For every X ∈ C and every quasi-smooth closed immersion Z → X , it sendsthe square (Example 5.3) P ( N Z / X ∣ Z cl ) ( Bl Z / X ) cl Z cl X to a cartesian square. Moreover, the same holds if C is replaced by the full subcategory of (a) quasi–compact quasi-separated (qcqs) algebraic spaces, (b) schemes, (c) or qcqsschemes. Remark 5.7.
Any presheaf F on algebraic spaces can be trivially extendedto derived algebraic spaces, by setting Γ ( X , F ) = Γ ( X cl , F ) for every derivedalgebraic space X. The condition (iv) in Theorem 5.6 is equivalent to re-quiring this extension to satisfy descent for quasi-smooth blow-up squares(2.2). Example 5.8.
Let E be a localizing invariant of stable ∞ -categories. Thenit satisfies Nisnevich descent on qcqs algebraic spaces (Example 5.2) andquasi-smooth blow-up descent (Theorem A). Assume that E also satisfies derived nilpotent invariance , i.e., that the canonical map E ( X ) → E ( X cl ) isinvertible for every derived algebraic space X. Then the condition (iv) inTheorem 5.6 holds. Therefore, E satisfies cdh descent if and only if it satisfiesclosed descent. Moreover, by Nisnevich descent it suffices to consider closedsquares of affine schemes. Example 5.9.
In the presence of A -homotopy invariance, the Morel–Voevodsky localization theorem [MV99, Theorem 3.2.21] provides the fol-lowing sufficient condition for closed descent. Let F be an A -invariantNisnevich sheaf on the category of algebraic spaces. Suppose that, for everyalgebraic space S, its restriction F S to the site of smooth algebraic spacesover S is stable under arbitrary base change. That is, for every morphismof algebraic spaces f ∶ T → S, the canonical map f ∗ ( F S ) → F T is invertible.Then F satisfies closed descent. This follows immediately from the closedbase change formula (cf. [Kha19, Prop. 3.3.2]). Remark 5.10.
Let E be a localizing invariant and suppose that it is more-over truncating in the sense of [LT18]. That is, if R is a connective E -ring spectrum and Mod perfR denotes the stable ∞ -category of left R-modules,then the canonical map E ( Mod perfR ) → E ( Mod perf π ( R ) ) is invertible. ThenLand–Tamme have recently proven that E has closed descent, at least ifwe restrict to noetherian algebraic spaces (see Step 1 in the proof of [LT18,Thm. A.2]). Remark 5.11.
There are a few variants of Theorem 5.6 with the same proof.For example:(i) On the category of (qcqs) schemes, descent with respect to the rh topology(generated by Zariski squares and proper cdh squares) can be checked withthe same criteria, except that Nisnevich squares are replaced by Zariskisquares in condition (ii).(ii) If we do not assume either Nisnevich or Zariski descent, descent for theproper cdh topology is still equivalent to conditions (i), (iii), and (iv), aslong as we restrict to a full subcategory of algebraic spaces or schemes whichsatisfy Thomason’s resolution property . For example, this holds on the cat-egory of quasi-projective schemes. LGEBRAIC K-THEORY OF QUASI-SMOOTH BLOW-UPS AND CDH DESCENT 19 (iii) One can extend the criterion to qcqs Artin stacks as follows. The defi-nition of Nisnevich square extends without modification (cf. [HK17, Sub-sect. 2.3]). In the definition of proper cdh square, we add the requirementthat the proper morphism p is representable (cf. op. cit. ). Then the cri-terion of Theorem 5.6 holds for stacks which admit the resolution propertyNisnevich-locally , see (5.3.4). This condition is relatively mild. For exam-ple, many quotient stacks have the resolution property ([Tho87, Lem. 2.4],[HR17, Exam. 7.5]). By the Nisnevich-local structure theorem of Alper–Hall–Rydh [HK17, Thm. 2.9], any stack with linearly reductive and almostmultiplicative stabilizers satisfies the resolution property Nisnevich-locally.5.3. Proof of Theorem 5.6.
Since Nisnevich squares, closed squares, andquasi-smooth blow-up squares are all cdh squares, the conditions are clearlynecessary. Conversely suppose that F is a presheaf satisfying the conditionsand let Q be a proper cdh square of algebraic spaces of the form(5.2) E YZ X . pi It will suffice to show that the induced square Γ ( Q , F ) is homotopy cartesian.5.3.1. Assume first that Q is a blow-up square, i.e., that Y = Bl Z / X is theblow-up of X centred in Z (and E = P ( C Z / X ) is the projectivized normalcone). By Nisnevich descent we may assume that X satisfies the resolutionproperty (e.g. X is affine). Since i ∶ Z → X is of finite presentation, theideal of definition I ⊂ O X is of finite type. Thus by the resolution propertythere exists a surjection u ∶ E → I with E a locally free O X -module of finiterank. Denote by V = V X ( E ) = Spec X ( Sym O X ( E )) the associated vectorbundle and 0 ∶ X → V the zero section. The O X -module homomorphism u ∶ E → I ⊂ O X induces a section of V, whose derived zero-locus ̃ Z fits in thehomotopy cartesian square ̃ Z XX V . ̃ i u By construction, ̃ i ∶ ̃ Z → X is a quasi-smooth closed immersion and there is acanonical morphism Z → ̃ Z which induces an isomorphism Z ≃ ̃ Z cl . Regarding F as a presheaf on derived algebraic spaces as in Remark 5.7, the squareΓ ( Q , F ) now factors as follows:Γ ( X , F ) Γ (̃ Z , F ) Γ ( Z , F ) Γ ( Bl ̃ Z / X , F ) Γ ( P ( N ̃ Z / X ) , F ) Γ ( Bl Z / X , F ) Γ ( P ( C Z / X ) , F ) The upper square is induced by a quasi-smooth blow-up square, hence iscartesian. The lower square is induced by a closed square, hence is alsocartesian. Therefore it follows that the outer composite square is also carte-sian. This shows that F satisfies descent for blow-up squares.5.3.2. Slightly more generally, suppose that Y = Bl Z ′ / X is a blow-up centredin some closed immersion Z ′ → X with ∣ Z ′ ∣ ⊆ ∣ Z ∣ on underlying topologicalspaces, and let E ′ → Y denote the exceptional divisor. Since F is invariantunder nilpotent extensions (Example 5.5) we may assume that i ′ ∶ Z ′ → X actually factors through a closed immersion Z ′ → Z (see Example 5.5).Applying descent for blow-up squares (5.3.1), it will suffice to show that F satisfies descent for the square E ′ EZ ′ Z . Note that the blow-up Bl Z ′ / Z is equipped with a canonical closed immersioninto E so that E ′ → E and Bl Z ′ / Z → Z form a closed covering. Applying closeddescent and descent for the blow-up square associated to Z ′ → Z (5.3.1), weconclude.5.3.3. For the case of an arbitrary proper cdh square, we recall the standardargument using Raynaud–Gruson’s technique of platification par ´eclatements [RG71, I, Cor. 5.7.12] to reduce to the case considered above (see e.g. [KST18,Subsect. 5.2]).
Construction 5.12.
Let Q be a proper cdh square of the form (5.2). As-sume that X is quasi-compact and quasi-separated and that the open sub-space X ∖ Z is quasi-compact and dense in X. Then there exists a propercdh square Q ′ sitting above Q such that the composite Q ′ ○ QE ′ Y ′ E YZ X Q ′ Q pi is of the form considered in (5.3.2). That is, Y ′ is a blow-up of X centred insome closed immersion Z ′ → X with ∣ Z ′ ∣ ⊆ ∣ Z ∣ . This follows from Raynaud–Gruson just as in the proof of [HK17, Cor. 2.4].Let Q be a proper cdh square of the form (5.2). Since F is a Nisnevichsheaf, we may assume that X is quasi-compact and quasi-separated. Since i ∶ Z → X is of finite presentation, its open complement U = X ∖ Z is quasi-compact. Using closed descent, we can ensure that U is dense in X. Nowapply the construction above to get a proper cdh square Q ′ such that Q ′ ○ Qis of the form considered in (5.3.2). Applying the construction again, thistime to Q ′ , we end up with a third square Q ′′ such that the composite Q ′′ ○ Q ′ LGEBRAIC K-THEORY OF QUASI-SMOOTH BLOW-UPS AND CDH DESCENT 21 is also of the form considered in (5.3.2). Then we know that Γ ( Q ′ ○ Q , F ) and Γ ( Q ′′ ○ Q ′ , F ) are both homotopy cartesian. It follows that the squareΓ ( Q ′ , F ) is also homotopy cartesian (since F takes values in a stable ∞ -category, it suffices to check that the induced map on homotopy fibres isinvertible), and hence so is Γ ( Q , F ) .5.3.4. We now discuss the extension to stacks mentioned in Remark 5.11(iii).The precise statement is as follows. Let C be a category of qcqs Artin stackssuch that (a) every stack X ∈ C admits a Nisnevich atlas by stacks with theresolution property; (b) for every stack X ∈ C and every blow-up Y → X, theqcqs Artin stack Y also belongs to C . Then the statement of Theorem 5.6holds for presheaves on C .The proof for the case of a blow-up square (5.3.1) has been presented insuch a way that it holds mutatis mutandis under the above assumptions.The argument of [KST18, Claim 5.3] also goes through, using descent forclosed squares and blow-up squares, to deal with the slightly more generalcase where Y = Bl Z ′ / X is a blow-up centred in some closed immersion Z ′ → Xthat factors through Z. To reduce a general proper cdh square to thatcase, we use Rydh’s extension of Raynaud–Gruson [HK17, Thm. 2.2]. First,closed descent allows us to assume that X ∖ Z is dense in X. Then we applyRydh–Raynaud–Gruson just as in the proof of [HK17, Cor. 2.4]. The onlydifference with the case of schemes or algebraic spaces is that in general weget a sequence of ( X ∖ Z ) -admissible blow-ups ˜X → X which factors through p ∶ Y → X. The addition of a simple induction is then the only modificationrequired to run the same argument.5.4.
Homotopy invariant K-theory. ( X , KH ) = lim Ð→ [ n ] ∈ ∆ op K ( X × A n ) . In other words, Γ ( X , KH ) is the geometric realization of the simplicial di-agram K ( X × A ● ) , where A ● is regarded as a cosimplicial scheme in theusual way (see e.g. [MV99, p. 45]). This extends the usual definition[Wei89, TT90], and is a way to formally impose the property of A -homotopyinvariance: for any qcqs algebraic space X, the projection p ∶ X × A → Xinduces an isomorphism of spectra p ∗ ∶ Γ ( X , KH ) → Γ ( X × A , KH ) . homotopy cartesian squares of spectraK ( X × A n ) K ( A × A n ) K ( Y × A n ) K ( B × A n ) for every [ n ] ∈ ∆ op . Passing to the colimit over n , we deduce that KH alsosatisfies Nisnevich descent. We have: Theorem 5.13.
For every qcqs derived algebraic space S , the canonicalmorphism of spectra Γ ( S , KH ) → Γ ( S cl , KH ) is invertible.Proof. By Nisnevich descent, we may as well assume S is an affine derivedscheme. Let KH S denote the restriction of KH to the site of affine derivedschemes that are smooth and of finite presentation over S. This is still an A -homotopy invariant Nisnevich sheaf, and it is equipped with a canonicalmorphism K cnS → K S → KH S , where K cnS and K S are the respective restrictions of connective and non-connective K-theory. By Cisinski, this morphism exhibits KH S as the Bottperiodization of the A -localization of K cnS , i.e., the periodization with re-spect to the Bott element b ∈ K ( G m, S ) (the proof is the same as in the casewhere S is classical [Cis13, Cor. 2.12]). It follows from this description thatfor any morphism of affine derived schemes f ∶ T → S, there is a canonical iso-morphism f ∗ ( KH S ) ≃ KH T , where f ∗ denotes the functor of inverse imageof A -invariant Nisnevich sheaves. Indeed, we reduce to checking the sameproperty for K cnS , which is clear as this is identified up to Zariski localizationwith the group completion of the presheaf ∐ n BGL n, S .In particular, we get a canonical isomorphism i ∗ ( KH S ) ≃ KH S cl , where i ∶ S cl → S is the inclusion of the underlying classical scheme. Moreover, i ∗ induces an equivalence between the ∞ -categories of A -invariant Nisnevichsheaves on S and S cl , respectively, by [Kha16, Cor. 1.3.5]. We deduce thatthe canonical morphismΓ ( S , KH ) ≃ Γ ( S , KH S ) → Γ ( S cl , KH S cl ) ≃ Γ ( S cl , KH ) is invertible. (cid:3) Proof of Theorem D.
We use the criterion of Theorem 5.6. Condi-tion (i) is obvious. Nisnevich descent (condition (ii)) was verified above(5.4.2). For condition (iv), it will suffice by Theorem 5.13 and Remark 5.7to show that KH sends quasi-smooth blow-up squares of derived algebraicspaces to homotopy cartesian squares. This follows from the same propertyfor K-theory (Theorem A) using the formula (5.3) (just as in the proof ofNisnevich descent). For closed descent (condition (iii)), we may restrict ourattention to closed squares of affine schemes (by Nisnevich descent). Thisis classical, see [TT90, Exer. 9.11(f)] or [Wei89, Cor. 4.10]. Alternatively, itfollows from the criterion of Example 5.9.
LGEBRAIC K-THEORY OF QUASI-SMOOTH BLOW-UPS AND CDH DESCENT 23
Remark 5.14.
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