Quasi-projective Schemes Research Articles

Equivariant algebraic K-theory, G-theory and derived completions

In the mid 1980s, while working on establishing completion theorems for equivariant Algebraic K-Theory similar to the well-known Atiyah-Segal completion theorem for equivariant topological K-theory, the late Robert Thomason found the strong finiteness conditions that are required in such theorems to be too restrictive. Then he made a conjecture on the existence of a completion theorem in the sense of Atiyah and Segal for equivariant algebraic G-theory, for actions of linear algebraic groups on schemes that holds without any of the strong finiteness conditions that are required in such theorems proven by him, and also appearing in the original Atiyah-Segal theorem. The main goal of the present paper is to provide a proof of this conjecture in as broad a context as possible, making use of the technique of derived completion, and to consider several of the applications.Our solution is broad enough to allow actions by all linear algebraic groups, irrespective of whether they are connected or not, and acting on any quasi-projective scheme of finite type over a field, irrespective of whether they are regular or projective. This allows us therefore to consider the equivariant algebraic G-Theory of large classes of varieties like all toric varieties (for the action of a torus) and all spherical varieties (for the action of a reductive group). Restricting to actions by split tori, we are also able to consider actions on algebraic spaces. Moreover, the restriction that the base scheme be a field is also not required often, but is put in mainly to simplify some of our exposition. These enable us to obtain a wide range of applications, some of which are briefly sketched and which we plan to explore in detail in the future. In fact, we discuss an extension of our results to equivariant homotopy K-theory along with various Riemann-Roch theorems in a sequel. A comparison of our results with previously known results, none of which made use of derived completions, shows that without the use of derived completions one can only obtain results which are indeed very restrictive.

The étale symmetric Künneth theorem

Let k be an algebraically closed field, lne {{,textrm{char},}}k a prime number, and X a quasi-projective scheme over k. We show that the étale homotopy type of the dth symmetric power of X is textbf{Z}/l-homologically equivalent to the dth strict symmetric power of the étale homotopy type of X. We deduce that the textbf{Z}/l-local étale homotopy type of a motivic Eilenberg–Mac Lane space is an ordinary Eilenberg–Mac Lane space.

Experiments on the Brauer map in high codimension

Using twisted and formal-local methods, we prove that every separated algebraic space which is the (open/flat) pushout of affine schemes has enough Azumaya algebras. As a corollary we show that, under mild hypothesis, every cohomological Brauer class is representable by an Azumaya algebra away from a closed subset of codimension $\geq 3$, generalizing an early result of Grothendieck. Next, we show that $\mathrm{Br}(X)=\mathrm{Br}'(X)$ when $X$ is an algebraic space obtained from a quasi-projective scheme by contracting a curve. This is the first method of descending an Azumaya algebra along a Chow cover which works in all dimensions. As a corollary, we prove that cohomological Brauer classes are geometric on arbitrary separated surfaces. In higher dimensions, we obtain the first examples of non-quasi-projective algebraic spaces with $\mathrm{Br}(X)=\mathrm{Br}'(X)$.

Rank 2 local systems and abelian varieties II

Let $X/\mathbb {F}_{q}$ be a smooth, geometrically connected, quasi-projective scheme. Let $\mathcal {E}$ be a semi-simple overconvergent $F$-isocrystal on $X$. Suppose that irreducible summands $\mathcal {E}_i$ of $\mathcal {E}$ have rank 2, determinant $\bar {\mathbb {Q}}_p(-1)$, and infinite monodromy at $\infty$. Suppose further that for each closed point $x$ of $X$, the characteristic polynomial of $\mathcal {E}$ at $x$ is in $\mathbb {Q}[t]\subset \mathbb {Q}_p[t]$. Then there exists a dense open subset $U\subset X$ such that $\mathcal {E}|_U$ comes from a family of abelian varieties on $U$. As an application, let $L_1$ be an irreducible lisse $\bar {\mathbb {Q}}_l$ sheaf on $X$ that has rank 2, determinant $\bar {\mathbb {Q}}_l(-1)$, and infinite monodromy at $\infty$. Then all crystalline companions to $L_1$ exist (as predicted by Deligne's crystalline companions conjecture) if and only if there exist a dense open subset $U\subset X$ and an abelian scheme $\pi _U\colon A_U\rightarrow U$ such that $L_1|_U$ is a summand of $R^{1}(\pi _U)_*\bar {\mathbb {Q}}_l$.

ZERO-CYCLES ON NORMAL PROJECTIVE VARIETIES

AbstractWe prove an extension of the Kato–Saito unramified class field theory for smooth projective schemes over a finite field to a class of normal projective schemes. As an application, we obtain Bloch’s formula for the Chow groups of $0$ -cycles on such schemes. We identify the Chow group of $0$ -cycles on a normal projective scheme over an algebraically closed field to the Suslin homology of its regular locus. Our final result is a Roitman torsion theorem for smooth quasiprojective schemes over algebraically closed fields. This completes the missing p-part in the torsion theorem of Spieß and Szamuely.

Zero-cycles with modulus and relative K-theory

Let D be an effective Cartier divisor on a regular quasiprojective scheme X of dimension d≥1 over a field. For an integer n≥0, we construct a cycle class map from the higher Chow groups with modulus {CHn+d(X|mD,n)}m≥1 to the relative K-groups {Kn(X,mD)}m≥1 in the category of pro-abelian groups. We show that this induces a proisomorphism between the additive higher Chow groups of relative 0-cycles and the reduced algebraic K-groups of truncated polynomial rings over a regular semilocal ring which is essentially of finite type over a characteristic zero field.

Projective bundle formula for Heller's relative $K_{0}$

In this article, we study the Heller relative $K_{0}$ group of the map $\mathbf{P}_{X}^{r} \to \mathbf{P}_{S}^{r}$, where $X$ and $S$ are quasi-projective schemes over a commutative ring. More precisely, we prove that the projective bundle formula holds for Heller's relative $K_{0}$, provided $X$ is flat over $S$. As a corollary, we get a description of the relative group $K_{0}(\mathbf{P}_{X}^{r} \to \mathbf{P}_{S}^{r})$ in terms of generators and relations, provided $X$ is affine and flat over $S$.

Bivariant derived algebraic cobordism

We extend the derived algebraic bordism of Lowrey and Schürg to a bivariant theory in the sense of Fulton and MacPherson and establish some of its basic properties. As a special case, we obtain a completely new theory of cobordism rings of singular quasi-projective schemes. The extended cobordism is shown to specialize to algebraicK0K^0analogously to the Conner-Floyd theorem in topology. We also give a candidate for the correct definition of Chow rings of singular schemes.

Geometrical symmetric powers in the motivic homotopy category

Symmetric powers of quasi-projective schemes can be extended via Kan extensions to geometrical symmetric powers of motivic spaces. In this work, we study geometrical symmetric powers and compare them with various symmetric powers in the unstable and stable A1-homotopy category of schemes over a field.

Mixed motives and geometric representation theory in equal characteristic

Let $\mathbb{k}$ be a field of characteristic $p$. We introduce a formalism of mixed sheaves with coefficients in $\mathbb{k}$ and showcase its use in representation theory. More precisely, we construct for all quasi-projective schemes $X$ over an algebraic closure of $\mathbb{F}_p$ a $\mathbb{k}$-linear triangulated category of motives on $X$. Using work of Ayoub (2007), Cisinski-Deglise (2012) and Geisser-Levine (2000), we show that this system of categories has a six functors formalism and computes higher Chow groups. Indeed, it behaves similarly to other categories of sheaves that one is used to. We attempt to make its construction also accessible to non-experts. We then consider the subcategory of stratified mixed Tate motives defined for affinely stratified varieties $X$, discuss perverse and parity motives and prove formality results. As an example, we combine these results and Soergel (2000) to construct a geometric and graded version of Soergel's modular category $\mathscr O(G)$, consisting of rational representations of a split semisimple group $G/\mathbb{k}$, and thereby equip it with a full six functor formalism (see Riche-Soergel-Williamson (2014) and Achar-Riche (2016) for other approaches). The main idea of using motives in geometric representation theory in this way as well as many results about stratified mixed Tate motives are directly borrowed from Soergel and Wendt, who tell the story in characteristic zero.

Multigraded Hilbert schemes parametrizing ideals in the Weyl algebra

Results of Haiman and Sturmfels \cite {HS04} on multigraded Hilbert schemes are used to establish a quasi-projective scheme which parametrizes all left homogeneous ideals in the Weyl algebra having a fixed Hilbert function with respect to a given grading by an abelian group.

Equivariant K-theory and higher Chow groups of schemes

For a smooth quasi-projective scheme $X$ over a field $k$ with an action of a reductive group, we establish a spectral sequence connecting the equivariant and the ordinary higher Chow groups of $X$. For $X$ smooth and projective, we show that this spectral sequence degenerates, leading to an explicit relation between the equivariant and the ordinary higher Chow groups. We obtain several applications to algebraic $K$-theory. We show that for a reductive group $G$ acting on a smooth projective scheme $X$, the forgetful map $K^G_i(X) \to K_i(X)$ induces an isomorphism $K^G_i(X)/{I_G K^G_i(X)} \cong K_i(X)$ with rational coefficients. This generalizes a result of Graham to higher $K$-theory of such schemes. We prove an equivariant Riemann-Roch theorem, leading to a generalization of a result of Edidin and Graham to higher $K$-theory. Similar techniques are used to prove the equivariant Quillen-Lichtenbaum conjecture.

A module structure and a vanishing theorem for cycles with modulus

We show that the higher Chow groups with modulus of Binda-Kerz-Saito for a smooth quasi-projective scheme $X$ is a module over the Chow ring of $X$. From this, we deduce certain pull-backs, the projective bundle formula, and the blow-up formula for higher Chow groups with modulus. We prove vanishing of $0$-cycles of higher Chow groups with modulus on various affine varieties of dimension at least two. This shows in particular that the multivariate analogue of Bloch-Esnault--Rulling computations of additive higher Chow groups of 0-cycles vanishes.

Fully faithfulness criteria for quasi-projective schemes

We extend to quasi-projective schemes that kind of Bondal-Orlov’s criterion for a Fourier-Mukai functor to be fully faithful and we prove a variant of it valid in arbitrary characteristic. As a first application of these criteria we prove the fully faithfulness of the Fourier-Mukai functor used to show the autoduality of the compactified Jacobian for reduced projective curves with locally planar singularities.

Cobordism invariants of the moduli space of stable pairs

For a quasi-projective scheme M which carries a perfect obstruction theory, we construct the virtual cobordism class of M. If M is projective, we prove that the corresponding Chern numbers of the virtual cobordism class are given by integrals of the Chern classes of the virtual tangent bundle. Further, we study cobordism invariants of the moduli space of stable pairs introduced by Pandharipande-Thomas. Rationality of the partition function is conjectured together with a functional equation, which can be regarded as a generalization of the rationality and 1/q <-> q symmetry of the Calabi-Yau case. We prove rationality for nonsingular projective toric 3-folds by the theory of descendents.

Witt, GW, K-theory of quasi-projective schemes

In this article, we prove some results on Witt, Grothendieck–Witt (GW) and K-theory of noetherian quasi-projective schemes X, over affine schemes Spec(A). For integers k≥0, let CMk(X) denote the category of coherent OX-modules F, with locally free dimension dimV(X)⁡(F)=k=grade(F). We prove that there is an equivalence Db(CMk(X))→Dk(V(X)) of the derived categories. It follows that there is a sequence of zig-zag maps K(CMk+1(X))⟶K(CMk(X))⟶∐x∈X(k)K(CMk(Xx)) of the K-theory spectra that is a homotopy fibration. In fact, this is analogous to the homotopy fiber sequence of the G-theory spaces of Quillen (see proof of [16, Theorem 5.4]). We also establish similar homotopy fibrations of GW-spectra and GW-bispectra, by application of the same equivalence theorem.

Motivic cohomology, localized Chern classes, and local terms

Let \({c : C \rightarrow X \times X}\) be a correspondence with C and X quasi-projective schemes over an algebraically closed field k. We show that if \({u_\ell : c_1^*\mathbb{Q}_\ell \rightarrow c_2^!\mathbb{Q}_\ell}\) is an action defined by the localized Chern classes of a c2-perfect complex of vector bundles on C, where l is a prime invertible in k, then the local terms of ul are given by the class of an algebraic cycle independent of l. We also prove some related results for quasi-finite correspondences. The proofs are based on the work of Cisinski and Deglise on triangulated categories of motives.

Foxby-morphism and derived equivalences

For quasi-projective schemes X over affine schemes Spec(A), resolving subcategories A of Coh(X) were considered. The equivalences ▪ of derived categories were established, where Mgk(A)={F∈Coh(X):dimA⁡(F)<∞,grade(F)≥k} and Dk denote the corresponding filtration of the derived category.

Realization of Levine's motives and p-adic Hodge cohomology

We construct a realization functor from Levine's triangulated category of motives, for any cohomology theory that takes values in a tensor category that is not necessarily a category of vector spaces, on smooth quasi-projective schemes over a reduced separated noetherian scheme. As a sample application, we construct a realization functor associated with the triple that consists of the rigid cohomology, the de Rham cohomology and the specialization map between them, whose domain is Levine's motivic category for the case in which the base scheme is the spectrum of a ring of p-adic integers.

Berkovich spaces embed in Euclidean spaces

Let K be a field that is complete with respect to a nonarchimedean absolute value such that K has a countable dense subset. We prove that the Berkovich analytification V^{\mathrm {an}} of any d -dimensional quasi-projective scheme V over K embeds in \mathbb R^{2d+1} . If, moreover, the value group of K is dense in \mathbb R_{&gt;0} and V is a curve, then we describe the homeomorphism type of V^{\mathrm {an}} by using the theory of local dendrites.