Quasi-projective Schemes Research Articles

Framed sheaves on projective stacks

Given a normal projective irreducible stack X over an algebraically closed field of characteristic zero we consider framed sheaves on X, i.e., pairs (E,ϕE), where E is a coherent sheaf on X and ϕE is a morphism from E to a fixed coherent sheaf F. After introducing a suitable notion of (semi)stability, we construct a projective scheme, which is a moduli space for semistable framed sheaves with fixed Hilbert polynomial, and an open subset of it, which is a fine moduli space for stable framed sheaves. If X is a projective irreducible orbifold of dimension two and F a locally free sheaf on a smooth divisor D⊂X satisfying certain conditions, we consider (D,F)-framed sheaves, i.e., framed sheaves (E,ϕE) with E a torsion-free sheaf which is locally free in a neighbourhood of D, and ϕE|D an isomorphism. These pairs are μ-stable for a suitable choice of a parameter entering the (semi)stability condition, and of the polarization of X. This implies the existence of a fine moduli space parameterizing isomorphism classes of (D,F)-framed sheaves on X with fixed Hilbert polynomial, which is a quasi-projective scheme. In an appendix we develop the example of stacky Hirzebruch surfaces. This is the first paper of a project aimed to provide an algebro-geometric approach to the study of gauge theories on a wide class of 4-dimensional Riemannian manifolds by means of framed sheaves on “stacky” compactifications of them. In particular, in a subsequent paper [20] these results are used to study gauge theories on ALE spaces of type Ak.

Riemann–Roch for equivariant K-theory

We show that for a linear algebraic group G acting on a smooth quasi-projective scheme X over a field, there is a Chern character map KiG(X)⊗R(G)R(G)ˆ→chXGCHG⁎(X,i)⊗S(G)S(G)ˆ with rational coefficients, which is an isomorphism. This establishes the equivariant version of the Riemann–Roch isomorphism between the higher algebraic K-theory and the higher Chow groups of smooth quasi-projective schemes.

Characteristic classes for GO(2n) in étale cohomology

Let $GO(2n)$ be the general orthogonal group (the group of similitudes) over any algebraically closed field of characteristic $\ne 2$ . We determine the smooth-étale cohomology ring with $\mathbb F_2$ coefficients of the algebraic stack $BGO(2n)$ . In the topological category, Holla and Nitsure determined the singular cohomology ring of the classifying space $BGO(2n)$ of the complex Lie group $GO(2n)$ in terms of explicit generators and relations. We extend their results to the algebraic category. The chief ingredients in this are: (i) an extension to étale cohomology of an idea of Totaro, originally used in the context of Chow groups, which allows us to approximate the classifying stack by quasi projective schemes; and (ii) construction of a Gysin sequence for the ${\mathbb G_m}$ -fibration $BO(2n)\to BGO(2n)$ of algebraic stacks.

Spectrum of some triangulated categories

In this note we announce the computation of the triangular spectrum (as defined by P. Balmer) of two classes of tensor triangulated categories which are quite common in algebraic geometry. One of them is the derived category of $G$-equivariant sheaves on a smooth quasi projective scheme $X$ for a finite group $G$ which acts on $X$. The other class is the derived category of split super-schemes.

Derived categories of sheaves on singular schemes with an application to reconstruction

We prove that the bounded derived category of coherent sheaves with proper support is equivalent to the category of locally-finite, cohomological functors on the perfect derived category of a quasi-projective scheme over a field. We introduce the notions of pseudo-adjoints and Rouquier functors and study them. As an application of these ideas and results, we extend the reconstruction result of Bondal and Orlov to Gorenstein projective varieties.

Moduli of framed sheaves on projective surfaces

We show that there exists a fine moduli space for torsion-free sheaves on a projective surface which have a “good framing” on a big and nef divisor. This moduli space is a quasi-projective scheme. This is accomplished by showing that such framed sheaves may be considered as stable pairs in the sense of Huybrechts and Lehn. We characterize the obstruction to the smoothness of the moduli space and discuss some examples on rational surfaces.

Motivic Eilenberg-MacLane spaces

In this paper we construct symmetric powers in the motivic homotopy categories of morphisms and finite correspondences associated with f-admissible subcategories in the categories of schemes of finite type over a field. Using this construction we provide a description of the motivic Eilenberg-MacLane spaces representing motivic cohomology on some f-admissible categories including the category of semi-normal quasi-projective schemes and, over fields which admit resolution of singularities, on some admissible subcategories including the category of smooth schemes. This description is then used to give a complete computation of the algebra of bistable motivic cohomological operations on smooth schemes over fields of characteristic zero and to obtain partial results on unstable operations which are required for the proof of the Bloch-Kato conjecture.

Uniqueness of enhancement for triangulated categories

The paper contains general results on the uniqueness of a DG enhancement for triangulated categories. As a consequence we obtain such uniqueness for the unbounded derived categories of quasi-coherent sheaves, for the triangulated categories of perfect complexes, and for the bounded derived categories of coherent sheaves on quasi-projective schemes. If a scheme is projective, then we also prove a strong uniqueness for the triangulated category of perfect complexes and for the bounded derived categories of coherent sheaves. These results directly imply that fully faithful functors from the bounded derived categories of coherent sheaves and the triangulated categories of perfect complexes on projective schemes can be represented by objects on the product.

Algebraic connective [formula omitted]-theory and the niveau filtration

Algebraic connective K -theory is a universal homology theory that connects K -homology groups (in particular Chow groups) and algebraic K -theories in some intimate manner. In this report we investigate how certain groups in the Brown–Gersten–Quillen spectral sequence for quasi-projective schemes give rise to a definition of connective K -theory groups.

A mod- [formula omitted] vanishing theorem of Beilinson–Soulé type

Let L be a field containing an algebraically closed field and X an equidimensional quasiprojective scheme over L . We prove that CH i ( X , n ; Z / ℓ ) = 0 when n > 2 i and ℓ ≠ 0 ; this was known previously when i ≥ dim X and L is itself algebraically closed. This “mod- ℓ ” version of the Beilinson–Soulé conjecture implies the equivalence of the rational and integral versions of the conjecture for varieties over fields of this type and can be used to prove the vanishing of the (integral) groups CH i ( X , n ) (for n > 2 i ) in certain cases.

On proper coverings of Artin stacks

We prove that every separated Artin stack of finite type over a noetherian base scheme admits a proper covering by a quasi-projective scheme. An application of this result is a version of the Grothendieck existence theorem for Artin stacks.

Bertini theorems over finite fields

Let X be a smooth quasiprojective subscheme of P^n of dimension m >= 0 over F_q. Then there exist homogeneous polynomials f over F_q for which the intersection of X and the hypersurface f=0 is smooth. In fact, the set of such f has a positive density, equal to zeta_X(m+1)^{-1}, where zeta_X(s)=Z_X(q^{-s}) is the zeta function of X. An analogue for regular quasiprojective schemes over Z is proved, assuming the abc conjecture and another conjecture.

Deformations of quasicoherent sheaves of algebras

Gerstenhaber and Schack [NATO Adv. Sci. Inst. Ser. C, Vol. 247, 1986] developed a deformation theory of presheaves of algebras on small categories. We translate their cohomological description to sheaf cohomology. More precisely, we describe the deformation space of (admissible) quasicoherent sheaves of algebras on a quasiprojective scheme X in terms of sheaf cohomology on X and X × X . These results are applied to the study of deformations of the sheaf D X of differential operators on X . In particular, in case X is a flag variety we show that any deformation of D X , which is induced by a deformation of O X , must be trivial. This result is used in [Lunts, Rosenberg, manuscript], where we study the localization construction for quantum groups.

Algebraic Homology For Real Hyperelliptic and Real Projective Ruled Surfaces

AbstractLet X be a reduced nonsingular quasiprojective scheme over such that the set of real rational points X() is dense in X and compact. Then X() is a real algebraic variety. Denote by (X(), ) the group of homology classes represented by Zariski closed k-dimensional subvarieties of X(). In this note we show that (X(), ) is a proper subgroup of H1(X(), ) for a nonorientable hyperelliptic surface X. We also determine all possible groups (X(), ) for a real ruled surface X in connection with the previously known description of all possible topological configurations of X.

On Macaulayfication of Noetherian schemes

The Macaulayfication of a Noetherian scheme X is a birational proper morphism from a Cohen-Macaulay scheme to X. In 1978 Faltings gave a Macaulayfication of a quasi-projective scheme if its non-Cohen-Macaulay locus is of dimension 0 or 1. In the present article, we construct a Macaulayfication of Noetherian schemes without any assumption on the non-Cohen-Macaulay locus. Of course, a desingularization is a Macaulayfication and, in 1964, Hironaka already gave a desingularization of an algebraic variety over a field of characteristic 0. Our method, however, to construct a Macaulayfication is independent of the characteristic.

Galois Module Structure and the γ-Filtration

We describe a general method for calculating equivariant Euler characteristics. The method exploits the fact that the γ-filtration on the Grothendieck group of vector bundles on a Noetherian quasi-projective scheme has finite length; it allows us to capture torsion information which is usually ignored by equivariant Riemann–Roch theorems. As applications, we study the G-module structure of the coherent cohomology of schemes with a free action by a finite group G and, under certain assumptions, we give an explicit formula for the equivariant Euler characteristic \(\chi (\mathcal{O}_X ) = {\text{H}}^{\text{0}} (X,\mathcal{O}_X ) - {\text{H}}^{\text{1}} (X,\mathcal{O}_X )\) in the Grothendieck group of finitely generated Z[G]-modules, when X is a curve over Z and G has prime order.

On Macaulayfication of sheaves

Let X be a quasi-projective scheme and ℱ a coherent sheaf of modules over X such that its non-Cohen–Macaulay locus is at most one dimensional. We use and extend the techniques of Brodmann to construct proper birational morphisms of quasi-projective schemes f:Y→X and Cohen–Macaulay coherent sheaves of modules over Y that are isomorphic to the pull-back of ℱ away from the exceptional locus of f. Certain blow-ups of X at locally complete intersections subschemes which contain non-reduced scheme structures on the non-Cohen–Macaulay locus of ℱ are the main part of the construction.

On Macaulayfication of certain quasi-projective schemes

The notion of Macaulayfication, which is analogous of the desingularization, was introduced by Faltings in 1978 and he construct a Macaulayfication of quasi-projective scheme whose non-Cohen-Macaulay locus is of dimension 0 or 1 by a characteristic free method. In this paper, we gave a Macaulayfication of a quasi-projective scheme whose non-Cohen-Macaulay locus is of dimension 2. Of course out method is independent of the characteristic.

Hochschild cohomology of quasiprojective schemes

Three definitions for the Hochschild cohomology of schemes are considered and shown to coincide for quasiprojective schemes. In the smooth case, the associated Hodge spectral sequences are also shown to be isomorphic.

Quasi-projective quotients by compact equivalence relations

In [9, III], we tried to construct quasi-projective moduli schemes M h for certain moduli functors .4/h of polarized compact complex manifolds X with Hilbert polynomial h. However, being unable to handle the compact part of the equivalence relation, we could achieve our aim only after changing the definition of the moduli functor [9, III, 1.10]. Instead of considering the functor of polarized manifolds up to Q-numerical equivalence, as it was done in [6] and [7], we allowed only isomorphisms of the invertible sheaves giving the polarization. In this article we want to show that our "cheating" was unnecessary and we will prove: