Scheme Of Finite Type Research Articles

Frobenius split subvarieties pull back in almost all characteristics

Let 𝒳 and 𝒴 be schemes of finite type over Specℤ and let α:𝒴→𝒳 be a finite map. We show the following holds for all sufficiently large primes p: If ϕ and ψ are any splittings on 𝒳× Spec𝔽p and 𝒴× Spec𝔽p, such that the restriction of α is compatible with ϕ and ψ, and V is any compatibly split subvariety of (𝒳× Spec𝔽p,ϕ), then the reduction α−1(V)red is a compatibly split subvariety of (𝒴× Spec𝔽p,ψ). This is meant as a tool to aid in listing the compatibly split subvarieties of various classically split varieties.

On the fundamental groups of commutative algebraic groups

Consider the abelian category ${\mathcal C}$ of commutative group schemes of finite type over a field $k$, its full subcategory ${\mathcal F}$ of finite group schemes, and the associated pro category ${\rm Pro}({\mathcal C})$ (resp. ${\rm Pro}({\mathcal F})$) of pro-algebraic (resp. profinite) group schemes. When $k$ is perfect, we show that the profinite fundamental group $\varpi_1 : {\rm Pro}({\mathcal C}) \to {\rm Pro}({\mathcal F})$ is left exact and commutes with base change under algebraic field extensions; as a consequence, the higher profinite homotopy functors $\varpi_i$ vanish for $i \geq 2$. Along the way, we describe the indecomposable projective objects of ${\rm Pro}({\mathcal C})$ over an arbitrary field $k$.

Compatible systems and ramification

We show that compatible systems of$\ell$-adic sheaves on a scheme of finite type over the ring of integers of a local field are compatible along the boundary up to stratification. This extends a theorem of Deligne on curves over a finite field. As an application, we deduce the equicharacteristic case of classical conjectures on$\ell$-independence for proper smooth varieties over complete discrete valuation fields. Moreover, we show that compatible systems have compatible ramification. We also prove an analogue for integrality along the boundary.

The maximal dimension of formal fibers of local rings of an algebraic scheme of finite type

For a scheme [Formula: see text] of finite type over a Noetherian local ring [Formula: see text] with a closed point [Formula: see text] of the special fiber, we show that the maximal dimension of the formal fibers of the local algebra [Formula: see text] equals to [Formula: see text] provided that either [Formula: see text] is complete of dimension one or the dimensions of the formal fibers of [Formula: see text] are less than [Formula: see text]. This extends Matsumura’s theorem for algebraic varieties.

Białynicki-Birula decomposition for reductive groups

We generalize the Białynicki-Birula decomposition from actions of Gm on smooth varieties to actions of a linearly reductive group G on finite type schemes and algebraic spaces. We also provide a relative version and briefly discuss the case of algebraic stacks.We define the Białynicki-Birula decomposition functorially: for a fixed G-scheme X and a monoid G‾ which partially compactifies G, the BB decomposition parameterizes G-schemes over X for which the G-action extends to the G‾-action. The freedom of choice of G‾ makes the theory richer than the Gm-case.

The slice spectral sequence for singular schemes and applications

We examine the slice spectral sequence for the cohomology of singular schemes with respect to various motivic T-spectra, especially the motivic cobordism spectrum. When the base field k admits resolution of singularities and X is a scheme of finite type over k, we show that Voevodsky's slice filtration leads to a spectral sequence for MGL(X) whose terms are the motivic cohomology groups of X defined using the cdh-hypercohomology. As a consequence, we establish an isomorphism between certain geometric parts of the motivic cobordism and motivic cohomology of X. A similar spectral sequence for the connective K-theory leads to a cycle class map from the motivic cohomology to the homotopy invariant K-theory of X. We show that this cycle class map is injective for projective schemes. We also deduce applications to the torsion in the motivic cohomology of singular schemes.

New points on curves

Let K be a field and let L/K be a finite extension. Let X/K be a scheme of finite type. A point of X(L) is said to be new if it does not belong to the union of X(F), when F runs over all proper subextensions of L. Fix now an integer g>0 and a finite separable extension L/K of degree d. We investigate in this article whether there exists a smooth proper geometrically connected curve of genus g with a new point in X(L). We show for instance that if K is infinite of characteristic different from 2 and g is bigger or equal to [d/4], then there exist infinitely many hyperelliptic curves X/K of genus g, pairwise non-isomorphic over the algebraic closure of K, and with a new point in X(L). When d is between 1 and 10, we show that there exist infinitely many elliptic curves X/K with pairwise distinct j-invariants and with a new point in X(L).

On the Congruence Class Modulo Prime Numbers of the Number of Rational Points of a Variety

Let $X$ be a scheme of finite type over $\mathbf{Z}$. For $p \in \mathcal{P}$ the set of prime numbers, let $N_{X}(p)$ be the number of $\mathbf{F}_{p}$-points of $X/\mathbf{F}_{p}$. For fixed $n\geq 1$ and $a_{1}, \ldots, a_{n} \in \mathbf{Z}$, we study the set $\bigcap_{i=1}^{n}\lbrace p\in\mathcal{P}-\Sigma_{X}, N_{X}(p)\neq a_{i} [\bmod p]\rbrace$ where $\Sigma_{X}$ is the finite set of primes of bad reduction for $X$. In case $\dim X\leq 3$, we show the set is either empty or has positive lower-density. We also address the question of the size of the smallest prime in that set. Using sieve methods, we obtain for example an upper bound for the size of the least prime of $\lbrace p\in\mathcal{P}, p\nmid N_{X}(p)\rbrace$ on average in particular families of hyperelliptic curves.

On the degree of Fano schemes of linear subspaces on hypersurfaces

2. Equivariant intersectiontheoryEdidin and Graham [5, 6] gave an algebraic construction to equivari-ant intersection theory. In this section, we review the basic notionsand results of this theory. Let Gbe a linear algebraic group and letX be a scheme of finite type over C endowed with a G-action. Forany non-negative integer i, we can find a representation V of Gto-gether with a dense open subset U ⊂ V on which Gacts freely andwhose complement has codimension larger than dimX−isuch that theprincipal bundle quotient U→ U/Gexists in the category of schemes(see [5, Lemma 9]). The diagonal action on X× U is then also free,which implies that under mild assumption, a principal bundle quotientX×U→ (X×U)/Gexists in the category of schemes (see [5, Propo-sition 23]). In what follows, we will tacitly assume that the scheme(X× U)/Gexists and denote it by X

On the formal arc space of a reductive monoid

Let $X$ be a scheme of finite type over a finite field $k$, and let ${\cal L} X$ denote its arc space; in particular, ${\cal L} X(k)=X(k[[t]])$. Using the theory of Grinberg, Kazhdan, and Drinfeld on the finite-dimensionality of singularities of ${\cal L} X$ in the neighborhood of non-degenerate arcs, we show that a canonical ``basic function'' can be defined on the non-degenerate locus of ${\cal L} X(k)$, which corresponds to the trace of Frobenius on the stalks of the intersection complex of any finite-dimensional model. We then proceed to compute this function when $X$ is an affine toric variety or an ``$L$-monoid''. Our computation confirms the expectation that the basic function is a generating function for a local unramified $L$-function; in particular, in the case of an $L$-monoid we prove a conjecture formulated by the second author.

Computing zeta functions of arithmetic schemes

We present new algorithms for computing zeta functions of algebraic varieties over finite fields. In particular, let X be an arithmetic scheme (scheme of finite type over Z), and for a prime p let zeta_{X_p}(s) be the local factor of its zeta function. We present an algorithm that computes zeta_{X_p}(s) for a single prime p in time p^(1/2+o(1)), and another algorithm that computes zeta_{X_p}(s) for all primes p < N in time N (log N)^(3+o(1)). These generalise previous results of the author from hyperelliptic curves to completely arbitrary varieties.

Monomial principalization in the singular setting

We generalize an algorithm by Goward for principalization of monomial ideals in nonsingular varieties to work on any scheme of finite type over a field. The normal crossings condition considered by Goward is weakened to the condition that components of the generating divisors meet as complete intersections. This leads to a substantial generalization of the notion of monomial scheme; we call the resulting schemes `\textit{regular crossings} (r.c.) \textit{monomial}.' We prove that r.c.~monomial subschemes in arbitrarily singular varieties can be principalized by a sequence of blow-ups at codimension~2 r.c.~monomial centers.

Toric stacks II: Intrinsic characterization of toric stacks

The purpose of this paper and its prequel is to introduce and develop a theory of toric stacks which encompasses and extends several notions of toric stacks defined in the literature, as well as classical toric varieties. While the focus of the prequel is on how to work with toric stacks, the focus of this paper is how to show a stack is toric. For toric varieties, a classical result says that a finite type scheme with an action of a dense open torus arises from a fan if and only if it is normal and separated. In the same spirit, the main result of this paper is that any Artin stack with an action of a dense open torus arises from a stacky fan under reasonable hypotheses.

Comparison results for étale cohomology in rigid geometry

We give some comparison results between étale cohomology of schemes and that of associated rigid spaces. First we work in the framework of Fujiwara spaces, and prove a comparison theorem for the derived direct image of morphisms which are not necessarily proper. We need the assumption that the base of the comparison functor consists of schemes of finite type over a field. Next, we consider an analogous problem for adic spaces. In this case, we can prove the comparison theorem under the condition that the base of the comparison functor consists of schemes of finite type over a complete discrete valuation ring with equal characteristic. Finally we propose a definition of the nearby cycle functor for an adic space over a complete discrete valuation ring and give a comparison result on it.

Rigid Cohomology and de Rham-Witt Complexes

Let k be a perfect field of characteristic p &gt; 0, W_n =ˆ W_n(k) . For separated k -schemes of finite type, we explain how rigid cohomology with compact supports can be computed as the cohomology of certain de Rham-Witt complexes with coefficients. This result generalizes the classical comparison theorem of Bloch-Illusie for proper and smooth schemes. In the proof, the key step is an extension of the Bloch-Illusie theorem to the case of cohomologies relative to W_n with coefficients in a crystal that is only assumed to be flat over W_n .

Horizontal Factorizations of Certain Hasse-Weil Zeta Functions - a Remark on a Paper by Taniyama

The paper [T] by Taniyama contains several results in the arithmetic theory of CM abelian varieties which have become well known. Moreover the notion of a compatible system ρ = (ρl) of l-adic representations is introduced there for the first time. Under suitable conditions on ρ, Taniyama proves an interesting formula for the alternating product of the L-functions of the exterior powers of ρ: it is given by an infinite product of Artin L-functions each of which modified by a change of finitely many Euler factors. As an application, Taniyama obtains such a formula for the Hasse–Weil zeta function of an abelian scheme over a localized number ring. These infinite product formulas of Taniyama seem to be little known but they motivated the works [JY] and [JR]. In [JY] Joshi and Yogananda show that the Hasse–Weil zeta function ζGm(s) = ζ(s−1)/ζ(s) of Gm = specZ[T, T −1] is an infinite product of Dirichlet L-series. This is a special case of Taniyama’s formulas but they give an elementary direct argument. In [JR] Joshi and Raghunathan express quotients of very general Dirichlet series with Euler products as infinite products of “twisted Dirichlet series”. The method is purely local. It applies in particular to the Hasse–Weil zeta function of X×Gm where X is a scheme of finite type over Z. In this case the formula has a geometric proof which was suggested by Serre, c.f. [JR] § 3. Joshi and Raghunathan also obtain certain infinite product formulas for quotients of automorphic L-functions. Recently these have been related to canonical bases by Kim and Lee, [KL] Remark 2.21.

Odds and Ends on Finite Group Actions and Traces

In this article, we study several problems related to virtual traces for finite group actions on schemes of finite type over an algebraically closed field. We also discuss applications to fixed point sets. Our results generalize previous results obtained by Deligne, Laumon, Serre and others.

Perverse coherent sheaves and the geometry of special pieces in the unipotent variety

Let X be a scheme of finite type over a Noetherian base scheme S admitting a dualizing complex, and let U ⊂ X be an open set whose complement has codimension at least 2. We extend the Deligne–Bezrukavnikov theory of perverse coherent sheaves by showing that a coherent intermediate extension (or intersection cohomology) functor from perverse sheaves on U to perverse sheaves on X may be defined for a much broader class of perversities than has previously been known. We also introduce a derived category version of the coherent intermediate extension functor. Under suitable hypotheses, we introduce a construction (called “ S 2 -extension”) in terms of perverse coherent sheaves of algebras on X that takes a finite morphism to U and extends it in a canonical way to a finite morphism to X. In particular, this construction gives a canonical “ S 2 -ification” of appropriate X. The construction also has applications to the “Macaulayfication” problem, and it is particularly well-behaved when X is Gorenstein. Our main goal, however, is to address a conjecture of Lusztig on the geometry of special pieces (certain subvarieties of the unipotent variety of a reductive algebraic group). The conjecture asserts in part that each special piece is the quotient of some variety (previously unknown for the exceptional groups and in positive characteristic) by the action of a certain finite group. We use S 2 -extension to give a uniform construction of the desired variety.

Réalisation de Betti des motifs de Voevodsky

For any subfield k of the field of complex numbers C, we construct a Betti realization functor from the category DM − ( k ) of Voevodsky motivic complexes over k to the category of graded abelian groups. If X is a scheme of finite type over k, the image of the associated motive through this functor is the singular cohomology ⊕ p ⩾ 0 H p ( X ( C ) , Z ) of the variety X ( C ) of complex points. To cite this article: F. Lecomte, C. R. Acad. Sci. Paris, Ser. I 346 (2008).

Rigidity of Log Morphisms

In the paper [5], we proved Kato’s conjecture, that is, the finiteness of dominant rational maps in the category of log schemes as a generalization of Kobayashi-Ochiai theorem [4]. It guarantees the finiteness of K-rational points of a certain kind of log smooth schemes for a big function field K, which gives rise to an evidence for Lang’s conjecture. In the proof of the above theorem, the most essential part is the rigidity theorem of log morphisms. In this paper, we would like to generalize it to a semistable scheme over an arbitrary noetherian scheme. Let f : X → S be a scheme of finite type over a locally noetherian scheme S. We assume that f : X → S is a semistable scheme over S, namely, f is flat and, for any morphism Spec(Ω) → S with Ω an algebraic closed field, the completion of the local ring of X ×S Spec(Ω) at every closed point is isomorphic to a ring of the type