aa r X i v : . [ m a t h . K T ] S e p Weibel’s conjecture for twisted K-theory
Joel StapletonSeptember 29, 2020
Abstract
We prove Weibel’s conjecture for twisted K -theory when twisting by a smooth proper connective dg-algebra. Our main contribution is showing we can kill a negative twisted K -theory class using a projectivebirational morphism (in the same twisted setting). We extend the vanishing result to relative twisted K -theory of a smooth affine morphism and describe counter examples to some similar extensions. The so-called fundamental theorem for K and K states that for any ring R there is an exact sequence0 → K ( R ) → K ( R [ t ]) ⊕ K ( R [ t − ]) → K ( R [ t ± ]) → K ( R ) → . We see K can be defined using K . There is an analogous exact sequence, truncated on the right, for K .Bass defines K − ( X ) as the cokernel of the final morphism. He then iterates the construction to define atheory of negative K-groups [Bas68, Sections XII.7 and XII.8].Weibel’s conjecture, originally posed in [Wei80], asks if K − i ( R ) = 0 for i > dim R when R has finiteKrull dimension. Kerz–Strunk–Tamme [KST18] have proven Weibel’s conjecture for any Noetherian schemeof finite Krull dimension (see the introduction for a historical summary of progress) by establishing pro cdh-descent for algebraic K -theory. Land–Tamme [LT18] have shown that a general class of localizing invariantssatisfy pro cdh-descent. With this improvement, we extend Weibel’s vanishing to some cases of twisted K -theory. Theorem 1.1.
Let X be a Noetherian d -dimensional scheme and A a sheaf of smooth proper connectivequasi-coherent differential graded algebras over X , then K − i (Perf( A )) vanishes for i > d . The original goal of this paper was to extend Weibel’s conjecture to an Azumaya algebra over a scheme.To an Azumaya algebra A of rank r on X we can associate a Severi-Brauer variety P of relative dimension r − X . Such a variety is ´etale-locally isomorphic over X to P r − X . In Quillen’s work [Qui73], hegeneralizes the projective bundle formula to Severi-Brauer varities showing (for i ≥ K i ( P ) ∼ = r − M n =0 K i ( A ⊗ n ) . At the root of this computation is a semi-orthogonal decomposition of Perf( P ). Consequently, the compu-tation lifts to the level of nonconnective K -theory spectra. Statements about the K -theory of Azumayaalgebras can generally be extracted through this decomposition. In our case, the dimension of the Severi-Brauer variety jumps and so Weibel’s conjecture (for our noncommutative dg-algebra) does not follow fromthe commutative setting. 1e could remedy this by characterizing a class of morphisms to X , which should include Severi-Brauervarieties, and then show the relative K -theory vanishes under − d −
1. In Remark 4.4, we show that smoothand proper morphisms (in fact, smooth and projective) are not sufficient. We warn the reader that we willuse the overloaded words “smooth and proper” in both the scheme and dg-algebra settings.For dg-algebras and dg-categories, properness and smoothness are module and algebraic finiteness condi-tions, see To¨en–Vaqui´e [TV07, Definition 2.4]. Together, the two conditions characterize the dualizable ob-jects in Mod
Mod R (Pr Lst ,ω ), whose objects are ω -compactly generated R -linear stable presentable ∞ -categories.More surprisingly, the invertible objects of Mod Mod R (Pr Lst ,ω ) are exactly the module categories over derivedAzumaya algebras, see Antieau–Gepner [AG14, Theorem 3.15]. So Theorem 1.1 recovers the discrete Azu-maya algebra case.However, any connective derived Azumaya algebras is discrete. After base-changing to a field k , A k ∼ = H ∗ A k is a connective graded k -algebra and H ∗ A k ⊗ k ( H ∗ A k ) op is Morita equivalent to k . So H ∗ A k is discrete.The scope of Theorem 1.1 is not wasted as smooth proper connective dg-algebras can be nondiscrete, seeRaedschelders–Stevenson [RS19, Section 4.3].The proof of Theorem 1.1 follows Kerz [Ker18]. In Section 2, we define and study twisted K -theory. Wekill a negative twisted K -theory class using a projective birational morphism in Section 3. Lastly, Section 4holds the main theorem and we consider some extensions. Conventions:
We make very little use of the language of ∞ -categories. For a commutative ring R , thereis an equivalence of ∞ -categories between the E -ring spectra over HR and differential graded algebras over R localized at the quasi-isomorphisms (see [Lur, 7.1.4.6]). For a dg-algebra (or E -ring) A , we can considerthe ∞ -category RMod( A ) of spectra which have a right A -module structure. We will refer to this ∞ -categoryas the derived category of A and denote it by D ( A ). The subcategory Perf( A ) consists of all compact objectsof RMod( A ), or the right A -modules which corepresent a functor that commutes with filtered colimits. Weshall refer to objects of Perf( A ) as perfect complexes over A .We use K ( − ) undecorated as non-connective algebraic K -theory and consider it as a localizing invariantin the sense of Blumberg–Gepner–Tabuada [BGT13]. In particular, it is an ∞ -functor from Cat perf ∞ , the ∞ -category of idempotent complete small stable infinity categories with exact functors, taking values inSp, the ∞ -category of spectra. For X a quasi-compact quasi-separated scheme, K (Perf( X )) is equivalentto the non-connective K -theory spectrum of Thomason–Trobaugh [TT90]. The ∞ -category Cat perf ∞ has asymmetric monoidal structure which we will denote by b ⊗ . For R an E ∞ -ring spectrum, Perf( R ) is an E ∞ algebra in Cat perf ∞ . We will restrict the domain of algebraic K -theory to Mod Perf( R ) (Cat perf ∞ ). Acknowledgements:
The author is thankful to his advisor, Benjamin Antieau, for the suggestedproject, patience, and guidance. He also thanks Maximilien P´eroux for helpful comments on an earlierdraft. The author was partially supported by NSF Grant DMS-1552766 and NSF RTG grant DMS-1246844. K -theory In Grothendieck’s original papers [Gro68a] [Gro68b] [Gro68c], he globalizes the notion of a central simplealgebra over a field.
Definition 2.1.
A locally free sheaf of O X -algebras A is a sheaf of Azumaya algebras if it is ´etale-locallyisomorphic to M n ( O X ) for some n .An Azumaya algebra is then a P GL n -torsor over the ´etale topos of X and so, by Giraud, isomorphismclasses are in bijection with H ( X, P GL n ). The central extension of sheaves of groups in the ´etale topology1 → G m → GL n → P GL n → · · · H ( X, G m ) H ( X, GL n ) H ( X, P GL n ) H ( X, G m ) . For d | n we have a morphism of exact sequences1 G m GL n P GL n G m GL d P GL d n/d times. The Brauergroup is the filtered colimit of cofibers Br ( X ) := colim −−−−−→ (cofib( H ( X, GL n ) → H ( X, P GL n )))along the partially-ordered set of the natural numbers under division. This is the group of Azumaya algebrasmodulo Morita equivalence with group operation given by tensor product (see [Gro68a]). We have an injection Br ( X ) ֒ → H ( X, G m ) and when X is quasi-compact this injection factors through the torsion subgroup.We will call Br ′ ( X ) := H ( X, G m ) tor the cohomological Brauer group. Grothendieck asked if the injection Br ( X ) ֒ → Br ′ ( X ) is an isomorphism.This map is not generally surjective. Edidin–Hassett–Kresch–Vistoli [EHKV01] give a non-separatedcounter example by connecting the image of the Brauer group to quotient stacks. There are two ways toproceed in addressing the question. The first is to provide a class of schemes for when this holds. In [dJ],de Jong publishes a proof of O. Gabber that Br ( X ) ∼ = Br ′ ( X ) when X is equipped with an ample linebundle. Along with reproving Gabber’s result for affines, Lieblich [Lie04] shows that for a regular schemewith dimension less than or equal to 2 there are isomorphisms Br ( X ) ∼ = Br ′ ( X ) ∼ = H ( X, G m ).The second perspective is to enlarge the class of objects considered. The Morita equivalence classes of G m -gerbes over the ´etale topos of a scheme X are in bijection with H ( X, G m ). In [Lie04], Lieblich associatesto any Azumaya algebra A a G m -gerbe of Morita-theoretic trivializations. Over an ´etale open U → X , thegerbe gives a groupoid of Morita equivalences from A to O X . The gerbe of trivializations represents theboundary class δ ([ A ]) = α ∈ H ( X, G m ).Any class α ∈ H ( X, G m ) is realizable on a ˇCech cover. We can use this data to build a well-definedcategory of sheaves of O X -modules which “glue up to α ”, see C˘ald˘araru [C˘al00, Chapter 1]. Let Mod αX denote the corresponding derived ∞ -category and Perf αX the full subcategory of compact objects. K (Perf αX )is the classical definition of α -twisted algebraic K -theory. Determining when the cohomology class α isrepresented by an Azumaya algebra reduces to finding a twisted locally-free sheaf with trivial determinanton a G m -gerbe associated to α [Lie04, Section 2.2.2]. The endomorphism algebra of the twisted locally-freesheaf gives the Azumaya algebra and the twisted module represents the tilt Mod αX ≃ Mod A .Lieblich also compactifies the moduli of Azumaya algebras. This necessarily includes developing a defi-nition of a derived Azumaya algebra. Definition 2.2. A derived Azumaya algebra over a commutative ring R is a proper dg-algebra A such thatthe natural map of R -algebras A ⊗ L R A op ≃ −→ R Hom D ( R ) ( A , A )is a quasi-isomorphism.After Lieblich, To¨en [To¨e12] and (later) Antieau–Gepner [AG14] consider the analogous problem posedby Grothendieck in the dg-algebra and E ∞ -algebra settings, respectively. Antieau–Gepner construct an ´etale3heaf Br in the ∞ -topos Shv ´et R . For any ´etale sheaf X , we can now associate a Brauer space Br ( X ). For X a quasi-compact quasi-separated scheme, they show π ( Br ( X )) ∼ = H ( X, Z ) × H ( X, G m ) and every suchBrauer class is algebraic. Now for any (possibly nontorsion) α ∈ H et ( X, G m ) there is a derived Azumayaalgebra A and an equivalence Mod αX ≃ Mod A of stable ∞ -categories.This reframes classical twisted K -theory as K -theory with coefficients in a particularly special dg-algebrain D ( X ). For our purposes, we work with a generalized definition of twisted K -theory which allows “twisting”by any dg-algebra. Definition 2.3.
Let R be a commutative ring. For a dg-algebra A over R , we define the A - twisted K -theory K A : Mod Perf( R ) (Cat perf ∞ ) → Sp by K A ( C ) := K ( C b ⊗ Perf( R ) Perf( A )).When the dg-algebra “ A ” is clear, we just write twisted K-theory. If our input to K A is an R -algebra S then K A ( S ) = K (Perf( S ) b ⊗ Perf( R ) Perf( A )) ≃ K (Perf( S ⊗ R A )) ≃ K ( S ⊗ R A ) . Our definition recovers the historical definition of twisted K-theory when A is a derived Azumaya algebraand we evaluate on the base ring R . The same definition works for a scheme X and A ∈
Alg E ( D qc ( X )).We will refer to such an A as a sheaf of quasi-coherent dg-algebras over X . By Theorem 9.36 of Blumberg–Gepner–Tabuada [BGT13], twisted K -theory is a localizing invariant. When X is a quasi-compact quasi-separated scheme, Proposition A.15 of Clausen–Mathew–Naumann–Noel [CMNN16] establishes Nisnevichdescent when X is quasi-compact quasi-separated. Definition 2.4.
A dg-algebra A over a ring R is proper if it is perfect as a complex over R and smooth ifit is perfect over A op ⊗ R A .The following is Lemma 2.8 of [TV07] and is an essential property for our proof in Section 3. Lemma 2.5.
Let A be a smooth proper dg-algebra over a ring R . Then a complex of D ( A ) is perfect over A if and only if it is perfect as an object of D ( R ) . The previous definition and lemma both generalize to a sheaf of quasi-coherent dg-algebras over a schemeas perfection is a local property. For the remainder of the section, we prove some basic properties of A -twisted K-theory, often assuming A is connective. We will not use smooth and properness until the latersections. Proposition 2.6.
Let A , S be connective dg-algebras over R . Then the natural maps induce isomorphisms K A i ( S ) ∼ = K A i ( π ( S )) ∼ = K π ( A ) i ( S ) ∼ = K π ( A ) i ( π ( S )) for i ≤ .Proof. We have the following isomorphisms of discrete rings π ( A ⊗ R S ) ∼ = π ( A ⊗ R π ( S )) ∼ = π ( π ( A ) ⊗ R S ) ∼ = π ( π ( A ) ⊗ R π ( S )) . The lemma follows since K i ( R ) ∼ = K i ( π ( R )) for i ≤ π of a connective dg-algebra does not preservesmoothness, which is a necessary property for our proof of Proposition 3.2. We will also need reductioninvariance for low dimensional K-groups. 4 roposition 2.7. Let R be a commutative ring and A a connective dg-algebra over R . Let S be a commu-tative ring under R and Let I be a nilpotent ideal of S . Then the induced morphism K A i ( S ) ∼ = −→ K A i ( S/I ) isan isomorphism for i ≤ .Proof. By naturality of the fundamental exact sequence of twisted K -theory (see ( † ) and the surroundingdiscussion at the beginning of Section 3), we can restrict the proof to K A . By Proposition 2.6, we canassume A is a discrete algebra. Let ϕ : S ։ S/I be the surjection. After − ⊗ R A we have a surjection(ker ϕ ) ⊗ R A ։ ker( ϕ ⊗ R A ). The nonunital ring (ker ϕ ) ⊗ R A is nilpotent. So ker( ϕ ⊗ R A ) is nilpotent aswell. The proposition follows from nil-invariance of K .A Zariski descent spectral sequence argument gives us a global result. Corollary 2.8.
Let X be a quasi-compact quasi-separated scheme of finite Krull dimension d and A a sheafof connective quasi-coherent dg-algebras over X . The natural morphism f : X red → X induces isomorphisms K f ∗ A− i ( X red ) ∼ = K A− i ( X ) for i ≥ d .Proof. We have descent spectral sequences E p,q = H pZar ( X, ( π q K A ) ∼ ) ⇒ π q − p K A ( X ) and E p,q = H pZar ( X, f ∗ ( π q K f ∗ ( A ) ) ∼ ) ⇒ π q − p K f ∗ A ( X red )both with differential d = (2 , F ∼ denote the Zariski sheafification of the presheaf F . The spectralsequences agree for q ≤
0. By Corollary 3.27 of [CM19], the spectral sequences vanishes for p > d .In Theorem 4.3, we extend our main theorem across smooth affine morphisms. We will need reductioninvariance in this setting.
Definition 2.9.
For f : S → X a morphism of quasi-compact quasi-separated schemes and A a sheaf ofquasi-coherent dg-algebras over X , the relative A -twisted K -theory of f is K A ( f ) := fib( K A ( X ) f ∗ −→ K A ( S )) . As defined, K A ( f ) is a spectrum. There is an associated presheaf of spectra on the base scheme X givenby U K A ( f | U ). This presheaf sits in a fiber sequence K A ( f ) → K A → K A S where the presheaf K A S is also defined by pullback along f . Both presheaves K A and K A S satisfy Nisnevichdescent and so K A ( f ) does as well. Corollary 2.10.
Let f : S → X be an affine morphism of quasi-compact quasi-separated schemes. Suppose X has Krull dimension d and let A be a sheaf of connective quasi-coherent dg-algebras over X . Then thecommutative diagram S red X red S X f red gf induces an isomorphism of relative twisted K -theory groups K g ∗ A− i ( f red ) ∼ = K A− i ( f ) for i ≥ d + 1 . roof. We have two descent spectral sequences E p,q = H pZar ( X, ( π q K A ( f )) ∼ ) ⇒ π q − p K A ( f )( X ) and E p,q = H pZar ( X, g ∗ ( π q K g ∗ A ( f red )) ∼ ) ⇒ π q − p K g ∗ A ( f red )( X red )with differential of degree d = (2 ,
1) and F ∼ the sheafification of the presheaf F . For an open affineSpec R → X with pullback Spec A → S we examine the morphism of long exact sequences when q ≤ · · · π q K A ( R ) π q K A ( A ) π q − K A ( f ) π q − K A ( R ) π q − K A ( A ) · · ·· · · π q K A ( R red ) π q K A ( A red ) π q − K A ( f red ) π q − K A ( R red ) π q − K A ( A red ) · · · ∼ = ∼ = ∼ = ∼ = By the 5-lemma, this induces sheaf isomorphisms g ∗ ( π q K g ∗ A ( f red )) ∼ ∼ = ( π q K A ( f )) ∼ for q < p > d .We will need pro-excision for abstract blow-up squares. Recall that an abstract blow-up square is apullback square D ˜ XY X ( ∗ )with Y → X a closed immersion and ˜ X → X a proper morphism which restricts to an isomorphism of opensubschemes ˜ X \ D → X \ Y . The theorem is stated using the ∞ -category of pro-spectra Pro (Sp), where anobject is a small cofiltered diagram, E : Λ → Sp, valued in spectra. We write { E n } for the correspondingpro-spectrum. If the brackets and index are omitted, then the pro-spectrum is considered constant. Afteradjusting equivalence class representatives, we may assume the cofiltered diagram is fixed when working witha finite set of pro-spectra. Any morphism can then be represented by a natural transformation of diagrams(also known as a level map). We will need no knowledge of the ∞ -category beyond the following definition. Definition 2.11.
A square of pro-spectra { E n } { F n }{ X n } { Y n } is pro-cartesian if and only if the induced map on the level-wise fiber pro-spectra is a weak equivalence (seeDefinition 2.27 of [LT18]).The following is Theorem A.8 of Land–Tamme [LT18]. The theorem holds much more generally for any k -connective localizing invariant (see Definition 2.5 of [LT18]). Twisted K -theory is 1-connective. Theorem 2.12 (Land–Tamme [LT18]) . Given an abstract blow-up square ( ∗ ) of schemes and a sheaf ofdg-algebras A on X then the square of pro-spectra K A ( X ) K A ( ˜ X ) { K A ( Y n ) } { K A ( D n ) } s pro-cartesian (where Y n is the infinitesimal thickening of Y ). The pro-cartesian square of pro-spectra gives a long exact sequence of pro-groups · · · { K A− i +1 ( E n ) } K A− i ( X ) K A− i ( ˜ X ) ⊕ { K A− i ( Y n ) } { K A− i ( E n ) } · · · which is the key to our induction argument. K -theory classes We turn to our main contribution of the existence of a projective birational morphism which kills a givennegative twisted K -theory class (when twisting by a smooth proper connective dg-algebra). Let X be aquasi-compact quasi-separated scheme and A a sheaf of quasi-coherent dg-algebras on X . We first constructgeometric cycles for negative twisted K-theory classes on X using a classical argument of Bass (see XII.7of [Bas68]) which works for a general additive invariant. We have an open cover X [ t ± ] X [ t − ] X [ t ] P X . fg jk Since twisted K -theory satisfies Zariski descent, there is an associated Mayer-Vietoris sequence of homotopygroups · · · K A− n ( P X ) K A− n ( X [ t ]) ⊕ K A− n ( X [ t − ]) K A− n ( X [ t ± ]) K A− n − ( P X ) · · · ( j ∗ k ∗ ) f ∗ − g ∗ ∂ . As an additive invariant, K A ( P X ) ≃ K A ( X ) ⊕ K A ( X ) splits as a K A ( X )-module with generators[ O ⊗ O X A ] = [ A ] and [ O (1) ⊗ O X A ] = [ A (1)]corresponding to the Beilinson semiorthogonal decomposition. Adjusting the generators to [ A ] and [ A ] − [ A (1)], we can identify the map ( j ∗ , k ∗ ) as it is a map of K A ( X )-modules. The second generator vanishesunder each restriction. This identifies the map as K A ( P X ) ≃ K A ( X )[ A ] ⊕ K A ( X )([ A ] − [ A (1)]) ∆ ⊕ −−−→ K A ( X [ t ]) ⊕ K A ( X [ t − ])with ∆ the diagonal map corresponding to pulling back along the projections X [ t ] → X and X [ t − ] → X .As ∆ is an embedding the long exact sequence splits as0 K A− n ( X ) K A− n ( X [ t ]) ⊕ K A− n ( X [ t − ]) K A− n ( X [ t ± ]) K A− n − ( X ) 0 ∆ ± ∂ . ( † )After iterating the complex K A− n ( X [ t ]) → K A− n ( X [ t ± ]) ։ K A− n − ( X ) , we can piece together a complex K A ( A n +1 X ) → K A ( G n +1 m,X ) ։ K A− n − ( X ) . Negative twisted K -theory classes have geometric representations as twisted perfect complexes on G im,X .There is even a sufficient geometric criterion implying a given representative is 0; it is the restriction of atwisted perfect complex on A iX . Our proof of the main proposition of this section will use these representa-tives. We first need a lemma about extending finitely-generated discrete modules in a twisted setting.7 emma 3.1. Let j : U → X be an open immersion of quasi-compact quasi-separated schemes. Let A bea sheaf of proper connective quasi-coherent dg-algebras on X and j ∗ A its restriction. Let N be a discrete j ∗ A -module which is finitely generated as an O U -module. Then there exists a discrete A -module M , finitelygenerated over O X , such that j ∗ M ∼ = N .Proof. Note that H ≥ ( j ∗ A ) necessarily acts trivially on N . So the j ∗ A -module structure on N comes fromforgetting along the map j ∗ A → H ( j ∗ A ) and the natural H ( j ∗ A )-module structure. Under restriction, j ∗ H ( A ) ∼ = H ( j ∗ A ) . We reduce to when A is a quasi-coherent sheaf of discrete O X -algebras, finite over the structure sheaf. Wehave an isomorphism N ∼ = j ∗ j ∗ N . Write j ∗ N as a filtered colimit of its finitely generated A -submodules j ∗ N ∼ = colim −−−−−→ λ M λ . The pullback is exact, so we can write N ∼ = colim −−−−−→ λ j ∗ M λ as a filtered colimit of finitelygenerated submodules. As N is finitely generated itself, this isomorphism factors at some stage and N ∼ = j ∗ M λ . Proposition 3.2.
Let X be a reduced scheme which is quasi-projective over a Noetherian affine scheme.Let A be a sheaf of smooth proper connective quasi-coherent dg-algebras on X . Let γ ∈ K A− i ( X ) for i > .Then there is a projective birational morphism ρ : ˜ X → X so that ρ ∗ γ = 0 ∈ K A− i ( ˜ X ) .Proof. We fix a diagram of schemes over X G im,X A iX X π j π . For any morphism f : Y → Y , we let ˜ f : G im,Y → G im,Y denote the pullback. Lift γ to a K A ( G im,X )-class[ P • ], with P • some π ∗ A -twisted perfect complexes on G im,X . The Induction Step :We induct on the range of homology of P • . As π ∗ A is a sheaf of proper quasi-coherent dg-algebras, P • isperfect on G im,X by Lemma 2.5. Since G im,X has an ample family of line bundles, we may choose P • to bestrict perfect without changing the quasi-isomorphism class. After some (de)suspension, we may assume P • is connective as this only alters the K -class by ±
1. For the lowest nontrivial differential of P • , d , we utilizepart (iv) of Lemma 6.5 of [KST18] (with the morphism G im,X → X ) to construct a projective birationalmorphism ρ : X → X so that coker (˜ ρ ∗ d ) (= H (˜ ρ ∗ P • )) has tor-dimension ≤ X . Consider thefollowing distinguished triangle of ˜ ρ ∗ π ∗ A -complexes on G im,X F • → ˜ ρ ∗ P • → H (˜ ρ ∗ P • ) ∼ = coker ˜ ρ ∗ d . In Lemma 3.3 below, we cover the base induction step, when the homology is concentrated in a single degree.Using this, construct a projective birational morphism φ : X → X such that L ˜ φ ∗ H (˜ ρ ∗ P • ) is a perfectcomplex and is the restriction of a perfect complex from A iX . By two out of three, L ˜ φ ∗ F • is perfect and[ ˜ φ ∗ ˜ ρ ∗ P • ] = [ L ˜ φ ∗ F • ] + [ L ˜ φ ∗ H (˜ ρ ∗ P • )] in K A ( G im,X ). We then repeat the entire induction step with L ˜ φ ∗ F • .We need the induction will terminate, which is the purpose of the first projective birational morphismof each step. Since coker (˜ ρ ∗ d ) has tor-dimension ≤ X , by [KST18][Lemma 6.5], L ˜ φ ∗ coker (˜ ρ ∗ d ) ∼ =˜ φ ∗ coker (˜ ρ ∗ d ). This implies L ˜ φ ∗ F • will have no homology outside the original range of homology of P • .Since ˜ φ ∗ coker (˜ ρ ∗ d ) ∼ = coker ( ˜ φ ∗ ˜ ρ ∗ d ), this guarantees H ( L ˜ φ ∗ F • ) = 0, so the homology of L ˜ φ ∗ F • lies in astrictly smaller range than ˜ φ ∗ ˜ ρ ∗ P • . Proposition 3.2 follows from the next lemma.8 emma 3.3. Let X be a reduced scheme which is quasi-projective over a Noetherian affine scheme. Let A be a sheaf of smooth proper connective quasi-coherent dg-algebras on X . Let N be a discrete π ∗ A -modulewhich is coherent on G im,X . Then there exists a birational blow-up φ : ˜ X → X so that ˜ φ ∗ N is perfect over ˜ φ ∗ π ∗ A on G m, ˜ X and is the restriction of a perfect complex over the pullback of A to A i ˜ X .Proof. Using Lemma 3.1, extend N from G im,X to a coherent π ∗ A -module M on A iX . Using the amplefamily, choose a resolution in O A iX -modules of the form0 → K → F → M → F is a vector bundle and K is the coherent kernel. As X is reduced, K is flat over some dense openset U of X . By platification par ´eclatement (see Theorem 5.2.2 of Raynaud–Gruson [RG71]), there is a U -admissable blow-up φ : ˜ X → X so that the strict transform of K along the pullback morphism p : A i ˜ X → A iX is flat over ˜ X .We now show the pullback p ∗ M is perfect as a p ∗ π ∗ A -module. Let j : A iU → A i ˜ X be the inclusion of theopen set and Z the closed complement. For any sheaf of modules G on A i ˜ X , we let G Z denote the subsheafof sections supported on Z . We have a short exact sequence natural in G → G Z → G → j st G → . We also obtain the following exact sequence of sheaves of abelian groups via pullback0 → T or p − O A iX ( p − M , O A i ˜ X ) → p ∗ K → p ∗ F → p ∗ M → . To make our notation clearer, we set T = T or p − O A iX ( p − M , O A i ˜ X ). We flesh both these exact sequencesout into a (nonexact) commutative diagram of p − O A iX -modules0 0 00 T Z T j st T
00 ( p ∗ K ) Z p ∗ K j st p ∗ K
00 ( p ∗ F ) Z p ∗ F j st p ∗ F
00 ( p ∗ M ) Z p ∗ M j st p ∗ M
00 0 0 . We observe that every row and the middle column is exact. The first map in the left column is an injectionand the last map in the right column is a surjection. Since p ∗ F is flat, we have ( p ∗ F ) Z = 0. This induces alifting of the injection T Z T ( p ∗ K ) Z p ∗ K .
9e finish the proof by showing j ∗ T or p − O A iX ( p − M , O A i ˜ X ) = 0. Since j : A iU → A i ˜ X is flat, the sheaf isisomorphic to T or A iU ( j ∗ p − M , j ∗ O A i ˜ X ) and j ∗ O A i ˜ X ∼ = O A iU . Our big diagram can be rewritten as0 0 00 T Z T p ∗ K ) Z p ∗ K j st p ∗ K
00 0 p ∗ F j st p ∗ F
00 ( p ∗ M ) Z p ∗ M j st p ∗ M
00 0 0 ∼ = ∼ = and we can glue together to get a flat resolution of p ∗ M as an O A i ˜ X -module0 → j st p ∗ K → p ∗ F → p ∗ M → A i ˜ X is Noetherian and p ∗ M is coherent. Since p ∗ π ∗ A is a sheaf of smooth quasi-coherent dg-algebrasover O A i ˜ X , the complex p ∗ M is perfect over p ∗ π ∗ A by Lemma 2.5. By commutativity, p ∗ M restricts to ˜ φ ∗ N on G im, ˜ X . This completes the proof of Proposition 3.2.We will need a relative version of Proposition 3.2. Corollary 3.4.
Let f : S → X be a smooth quasi-projective morphism of Noetherian schemes with X reducedand quasi-projective over a Noetherian base ring. Let A be a sheaf of smooth proper connective quasi-coherentdg-algebras over X and consider a negative twisted K -theory class γ ∈ K A i ( S ) for i < . Then there exists aprojective birational morphism ρ : ˜ X → X such that, under the pullback of the pullback morphism, ρ ∗ S γ = 0 .Proof. We will briefly check that we can run the induction argument in the proof of Proposition 3.2. Theassumptions of this corollary are invariant under pullback along projective birational morphisms ˜ X → X .We need to ensure we can select projective birational morphisms to our base X . Lemma 6.5 of Kerz–Strunk–Tamme [KST18] is stated in a relative setting. The proof also relies on platification par ´eclatement. Thiscan still be applied in our relative setting as X is reduced (see Proposition 5 of Kerz–Strunk [KS17]). We now prove Theorem 1.1 and an extension across a smooth affine morphism. We begin with the baseinduction step for both theorems. Kerz–Strunk [KS17] use a sheaf cohomology result of Grothendieck alongwith a spectral sequence argument to show vanishing for a Zariski sheaf of spectra can be reduced to thesetting of local ring. 10 roposition 4.1.
Let R be a regular Noetherian ring of Krull dimension d over a local Artinian ring k . Let A be a smooth proper connective dg-algebra over R , then K A i ( R ) = 0 for i < .Proof. By Proposition 2.10, we may assume k is a field. Proposition 5.4 of [RS19] shows that the t-structureon D ( A ) restricts to a t-structure on Perf( A ), which is observably bounded. The heart is the category offinitely-generated modules over H ( A ). As H ( A ) is finite-dimensional over k , this is a Noetherian abeliancategory. By Theorem 1.2 of Antieau–Gepner–Heller [AGH19]), the negative K -theory vanishes. Theorem 1.1.
Let X be a Noetherian scheme of Krull dimension d and A a sheaf of smooth proper con-nective quasi-coherent dg-algebras on X , then K A− i ( X ) vanishes for i > d .Proof. Proposition 4.1 covers the base case so assume d >
0. By the Kerz–Strunk spectral sequence argumentand Corollary 2.8, we may assume X is a Noetherian reduced affine scheme.Choose a negative K A -theory class γ ∈ K A− i ( X ) for i ≥ dim X + 1. Using Proposition 3.2, construct aprojective birational morphism that kills γ and extend it to an abstract blow-up square E ˜ XY X .
By [LT18, Theorem A.8], there is a Mayer-Vietoris exact sequence of pro-groups · · · { K A− i +1 ( E n ) } K A− i ( X ) K A− i ( ˜ X ) ⊕ { K A− i ( Y n ) } { K A− i ( E n ) } · · · . When i ≥ dim X + 1, by induction every nonconstant pro-group vanishes and K A− i ( X ) ∼ = K A− i ( ˜ X ) showing γ = 0.By [AG14, Theorem 3.15], we recover Weibel’s vanishing for discrete Azumaya algebras. Corollary 4.2.
For X a Noetherian d -dimensional scheme and A a quasi-coherent sheaf of discrete Azumayaalgebras, then K A− i ( X ) = 0 for i > d . The next result nearly covers the K-regularity portion of Weibel’s conjecture, but we are missing theboundary case K A− d ( X ) ∼ = K A− d ( A nX ). Theorem 4.3.
Let f : S → X be a smooth affine morphism of Noetherian schemes and A a sheaf of smoothproper connective quasi-coherent dg-algebras on X . Then K A− i ( f ) = 0 for i > dim X + 1 .Proof. The base case is covered by Proposition 4.1 and our reductions are analagous to those in the proof ofTheorem 1.1. So assume X is a Noetherian reduced affine scheme of dimension d . Choose γ ∈ K A− i ( S ) with i > d . Using Corollary 3.4, construct a projective birational morphism ρ : ˜ X → X that kills γ . We thenbuild a morphism of abstract blow-up squares D ˜ SE ˜ XV SY X
11y Theorem 2.12, we again get a long exact sequence of pro-groups corresponding to the back square · · · { K A− i +1 ( D n ) } K A− i ( S ) K A− i ( ˜ S ) ⊕ { K A− i ( V n ) } { K A− i ( D n ) } · · · . When i ≥ dim X + 1, every nonconstant pro-group vanishes by induction and we have an isomorphism K A− i ( S ) ∼ = K A− i ( ˜ S ) implying γ = 0. Remark 4.4.
The conditions on the morphism in Corollary 3.4 are more general than those of Theorem4.3. We might hope to generalize Theorem 4.3 to a smooth quasi-projective or smooth projective map ofNoetherian schemes. Although the induction step is present, both base cases fail. Consider the descentspectral sequence E p,q := H p ( X, ˜ K q ) ⇒ K q − p ( X ) with d = (2 , X ≤
3, then E , = E , ∞ = coker ( H ( X, Z ) d −→ H ( X, O ∗ X ))contributes to K − ( X ). The differential is zero as the edge morphism K ( X ) E , ∞ rank identifies E , ∞ with the rank component of K , implying E , = E , ∞ . We now construct a family of examplesfor schemes X with nontrivial H ( X, O ∗ X ). Let X red be quasi-projective smooth over a field k and form thecartesian diagram X X red
Spec ( k [ t ] / ( t )) Spec k f . The pullback X will be our counter-example. We have an isomorphism O ∗ X ∼ = g ∗ ( O ∗ X red ) ⊕ g ∗ ( O X red )of sheaves of abelian groups on X with g : X red → X the pullback of the reduction morphism Spec k → Spec k [ t ] / ( t ). Locally, ( R [ t ] / ( t )) × consists of all elements of the form u + v · t where u ∈ R × and v ∈ R .Sheaf cohomology commutes with coproducts so this turns into an isomorphism H ( X, O ∗ X ) ∼ = H ( X, g ∗ ( O ∗ X red )) ⊕ H ( X, g ∗ ( O X red )) ∼ = H ( X red , O ∗ X red ) ⊕ H ( X red , O X red ) . Now the problem reduces to finding a surface or 3-fold X red with nontrivial degree 2 sheaf cohomology. Takea smooth quartic in P k for a counter-example which is smooth and proper. Here is a counter-example whichis smooth and quasi-affine. Let ( A, m ) be a 3-dimensional local ring which is smooth over a field k . Take X = Spec A \ { m } to be the punctured spectrum. Then H ( X, O X ) ∼ = H m ( A ), which is the injective hull ofthe residue field A/ m . References [AG14] Benjamin Antieau and David Gepner. Brauer groups and ´etale cohomology in derived algebraicgeometry.
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