Victor Abrashkin
For a prime number p>2, we give a direct proof of Breuil's classification of killed by p finite flat group schemes over the valuation ring of a p-adic field with perfect residue field. As application we prove that the Galois modules of geometric points of such group schemes and of their characteristic p analogues coming from Faltings's strict modules can be identified via the Fontaine-Wintenberger field-of-norms functor.
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We give an explicit construction of the antiequivalence of the category of finite flat commutative group schemes of period 2 defined over the valuation ring of a 2-adic field with algebraically closed residue field. This result extends the earlier author's approach to group schemes of period p>2 from Proceedings of LMS, 101, 2010, 207-259.
For a number field $K$, a finite set of primes $S$ not containing a fixed prime $p$, we explain when extensions of group schemes of $μ_p$ by $\Z/p\Z$ split over the ring of $S$-integers $O_S$ of $K$.
We give an explicit description of the category of central extensions of a group scheme by a sheaf of Abelian groups. Based on this, we describe a framework for computing with central extensions of finite commutative group schemes, torsors under such group schemes and groups of isomorphism classes of these objects.
We prove equality of the various $p$-adic period morphisms for smooth, not necessarily proper, schemes. We start with showing that the $K$-theoretical uniqueness criterium we had found for proper smooth schemes extends to proper finite simplicial schemes in the good reduction case and to cohomology with compact support in the semistable reduction case. It yields the equality of the period morphisms for cohomology with compact support defined using the syntomic, almost étale, and motivic constructions. We continue with showing that the $h$-cohomology period morphism agrees with the syntomic and…
We give a definition of full level structure on group schemes of the form $G\times G$, where $G$ is a finite flat commutative group scheme of rank $p$ over a $\mathbb{Z}_p$-scheme $S$ or, more generally, a truncated $p$-divisible group of height $1$. We show that there is no natural notion of full level structure over the stack of all finite flat commutative group schemes.
In their book Rapoport and Zink constructed rigid analytic period spaces for Fontaine's filtered isocrystals, and period morphisms from moduli spaces of p-divisible groups to some of these period spaces. We determine the image of these period morphisms, thereby contributing to a question of Grothendieck. We give examples showing that only in rare cases the image is all of the Rapoport-Zink period space.
A semicommutative finite group scheme is a finite group scheme which can be obtained from commutative finite group schemes by iterated performing semidirect products with commutative kernels and taking quotients by normal subgroups. In this article, for an étale tame semicommutative finite group scheme $G$, we give a lower bound on the number of connected $G$-torsors of bounded height (such as discriminant).
In this paper we develop the theory of essential dimension of group schemes over an integral base. Shortly we concentrate over a local base. As a consequence of our theory we give a result of invariance of the essential dimension over a field. The case of group schemes over a discrete valuation ring is discussed. Moreover we propose a generalization of Ledet conjecture, which predicts the essential dimension of cyclic $p$-groups in positive characteristic, for finite commutative unipotent group schemes. And we show some results and some consequences of this new conjecture.
Let S be a locally noetherian regular scheme. We compute the units-Picard complex of a reductive S-group scheme G in terms of the dual algebraic fundamental complex of G. To do so, we establish a units-Picard-Brauer exact sequence for a torsor under a smooth S-group scheme.
We generalize J.-L. Colliot-Thelene's construction of flasque resolutions of reductive group schemes over a field to a broad class of base schemes.
Log $p$-divisible groups associated with semi-abelian degeneration
本文研究阿贝尔概形沿正规交叉除子半阿贝尔退化时挠点群概形的退化行为，证明其退化为对数有限群概形，并说明该对象比半阿贝尔概形挠点的拟有限平坦群概形保留更多信息。
We prove several basic extension theorems for reductive group schemes. We also prove that each Lie algebra with a perfect Killing form over a commutative $\dbZ$-algebra, is the Lie algebra of an adjoint group scheme.
Let $p$ be a prime number and let $k$ be an algebraically closed field of characteristic $p$. A $BT_1$ group scheme over $k$ is a finite commutative group scheme which arises as the kernel of $p$ on a $p$-divisible (Barsotti--Tate) group. Our main result is that every $BT_1$ scheme group over $k$ occurs as a direct factor of the $p$-torsion group scheme of the Jacobian of an explicit curve defined over $\mathbb{F}_p$. We also treat a variant with polarizations. Our main tools are the Kraft classification of $BT_1$ group schemes, a theorem of Oda, and a combinatorial description of the de Rham …
Let $p$ be a prime number and let $k$ be an algebraically closed field of characteristic $p$. A $BT_1$ group scheme over $k$ is a finite commutative group scheme which arises as the kernel of $p$ on a $p$-divisible (Barsotti--Tate) group. We compare three classifications of $BT_1$ group schemes, due in large part to Kraft, Ekedahl, and Oort, and defined using words, canonical filtrations, and permutations. Using this comparison, we determine the Ekedahl--Oort types of Fermat quotient curves and we compute four invariants of the $p$-torsion group schemes of these curves.
We introduce the abelian class group C_{ab}(G) of a reductive group scheme G over a ring A of arithmetical interest and study some of its properties. In particular, we show that if the fraction field of A is a global field without real primes, then there exists a surjection C(G)-->> C_{ab}(G), where C(G) is the class set of G.
In this paper we study multilinear morphisms between commutative group schemes and the associated tensor constructions. We will also do some explicit calculations and give examples that show that this theory behaves in a way that one would naturally expect.
In this paper we will give the computation of the etale fundamental group of an arithmetic scheme.
We determine the fundamental group of period domains over finite fields.
We define the algebraic fundamental group functor of a reductive group scheme over an arbitrary (non-empty) base scheme and prove that this functor is exact.
In this article, we study the log-scheme theoretic version of the Nori fundamental group scheme. Similar to the classical Nori fundamental group scheme, the log Nori fundamental group scheme classifies torsors on log flat topology. We also compute the nilpotent pro-p quotient of the log Nori fundamental group scheme with log at $\{0, 1, \infty\}$.
We extend the definition of fundamental group scheme to non reduced schemes over any connected Dedekind scheme. Then we compare the fundamental group scheme of an affine scheme with that of its reduced part.
The paper was motivated by a question of Vilonen, and the main results have been used by Mirkovic and Vilonen to give a geometric interpretation of the dual group (as a Chevalley group over Z) of a reductive group. We define a quasi-reducitve group over a discrete valuation ring R to be an affine flat group scheme over R such that (i) the fibers are of finite type and of the same dimension; (ii) the generic fiber is smooth and connected, and (iii) the netural component of the reduced special fiber is a reductive group. We show that such a group scheme is of finite type over R, the generic fibe…
Every finite flat finitely presented group scheme G of square free order over a scheme S can be written as an extension of a finite etale S-group scheme G" by a commutative finite flat finitely presented S-group scheme G' that is a direct sum of group schemes of prime order. Such an extension splits over a finite etale surjective base change.
Finite group schemes as fundamental group schemes of smooth projective varieties
本文研究将有限群概形实现为光滑射影簇基本群概形的问题，通过贝尔蒂尼型结果、戈多-塞雷构造等方法，证明在无限完美域k上任意有限群概形G/k均可作为对应类型的基本群概形实现于特定维数的连通光滑射影簇。
In this note, we prove that the F-fundamental group scheme is birational invariant for smooth projective varieties. We prove that the F-fundamental group scheme is naturally a quotient of the Nori fundamental group scheme. For elliptic curves, it turns out that the F-fundamental group scheme and the Nori fundamental group scheme coincides. We also consider an extension of the Nori fundamental group scheme in positive characteristic using semi-essentially finite vector bundles and prove that in this way, we do not get a non-trivial extension of the Nori fundamental group scheme for elliptic cur…
For a linear algebraic group $G$ over $\mathbf Q$, we consider the period domains $D$ classifying $G$-mixed Hodge structures, and construct the extended period domains $D_Σ$. In particular, we give toroidal partial compactifications of mixed Mumford--Tate domains, mixed Shimura varieties over $\mathbf C$, and higher Albanese manifolds.
We define and study the Picard group of a monoid scheme and the class group of a normal monoid scheme. To do so, we develop some ideal theory for (pointed abelian) noetherian monoids, including primary decomposition and discrete valuations. The normalization of a monoid turns out to be a monoid scheme, but not always a monoid.
The goal of this paper is to give an explicit formula for the l-adic cohomology of period domains over finite fields for arbitrary reductive groups. The result is a generalisation of the computation in math.AG/9907098 which treats the case of the general linear group.
For a linear algebraic group $G$ over $\bf Q$, we consider the period domains $D$ classifying $G$-mixed Hodge structures, and construct the extended period domains $D_{\mathrm{BS}}$, $D_{\mathrm{SL}(2)}$, and $Γ\backslash D_Σ$. In particular, we give toroidal partial compactifications of mixed Mumford--Tate domains.
Flasque resolutions of linear reductive group schemes
本文推广了Colliot-Thelene的flasque分解理论到任意基上的线性约化群概形，利用Borovoi基本群和Chevalley群自同构的下陷理论构造分解，并应用于环和低维域上的R等价性。
We prove that any commutative group scheme separated over a noetherian normal scheme admits a canonical structure of a presheaf with transfers, which is characterized by a simple condition on radicial transfers.
The tame fundamental group schemes of curves in positive characteristic
本文研究正特征代数闭域上光滑曲线的驯服基本群概形，发展余特化映射理论，证明其依赖曲线且可重构数值不变量。
Let $k$ be an algebraically closed field of characteristic $p > 3$. Let $X$ be an irreducible smooth projective surface over $k$. Fix an integer $n \geq 1$ and let ${\mathcal{H}{\it ilb}}_X^n$ be the Hilbert scheme parameterizing effective $0$-cycles of length $n$ on $X$. The aim of the present article is to find the $S$-fundamental group scheme and Nori's fundamental group scheme of the Hilbert scheme $\mathcal{H}{\it ilb}_X^n$.
On $X$ projective smooth over an algebraically closed field, we show that if Nori's fundamental group scheme is trivial, then there are no nontrivial Nori semistable bundles of degree 0, that is the group scheme $π^S(X)$ studied in particular by Langer is trivial.
Let $X$ be a smooth projective curve over the complex numbers. We compute the Brauer group of the moduli stack of Bruhat-Tits group scheme $\mathcal{G}$-torsors on $X$. When $g(X) \geq 3$ we compute the Brauer group of the regularly stable locus of the coarse moduli space of semi-stable $\mathcal{G}$-torsors.
In this paper we describe the fundamental group-scheme of a proper variety fibered over an abelian variety with rationally connected fibers over an algebraically closed field. We use old and recent results for the Nori fundamental group-scheme, and of finite group-schemes in general to prove that the kernel of such a fibration is finite, and that the homotopy exact sequence holds in this case. As an application, we describe the fundamental group-schemes of certain proper varieties that are connected by curves or are related to them.
For a linear algebraic group G over the field of rational numbers, we consider the period domains D classifying G-mixed Hodge structures, and construct the extended period domains as toroidal partial compactifications. We give an interpretation of higher Albanese manifolds by Hain and Zucker by using the above D for some G, and extend them via the toroidal partial compactifications.
We investigate the algebras of invariants and the properties of the quotient morphism by an action of a finite group scheme in terms of stabilizers of points.
For $X$ a complete, reduced, geometrically connected scheme over a perfect field of characteristic $p>0$, we analyze the decomposition of Nori's fundamental group scheme into its local and étale parts and raise the question of the relation between the geometry and the splitting of the group scheme. We also describe in categorial terms the functor which corresponds to the inclusion of the maximal reduced subgroup scheme.
Let $k$ be an algebraically closed field. Let $C$ be an irreducible smooth projective curve over $k$. Let $E$ be a locally free sheaf on $C$ of rank $\geq 2$. Fix an integer $d \geq 2$. Let $\mathcal{Q}$ denote the Quot scheme parameterizing torsion quotients of $E$ of degree $d$. In this article we compute the $S$-fundamental group scheme of $\mathcal{Q}$.
The aim of the present paper is to study the (abstract) Picard group and the Picard group scheme of the moduli stack of stable pointed curves over an arbitrary scheme. As a byproduct, we compute the Picard groups of the moduli stack of stable or smooth pointed curves over a field of characteristic different from two.
We prove that if a finite group scheme $G$ over a field $k$ has essential dimension one, then it embeds in $PGL_{2/k}$. We use this to give an explicit classification of all infinitesimal group schemes of essential dimension one over any field and a characterisation of all finite group schemes of essential dimension one over algebraically closed fields.
Let $X$ and $Y$ be nonsingular projective varieties over an algebraically closed field $k$ of positive characteristic. If $X$ and $Y$ are birational, we show their $S$-fundamental group schemes are isomorphic.
The S-fundamental group scheme is the group scheme corresponding to the Tannaka category of numerically flat vector bundles. We use determinant line bundles to prove that the S-fundamental group of a product of two complete varieties is a product of their S-fundamental groups as conjectured by V. Mehta and the author. We also compute the abelian part of the S-fundamental group scheme and the S-fundamental group scheme of an abelian variety or a variety with trivial etale fundamental group.
We prove that the Local Fundamental Group Scheme satisfies the Lefschetz - Bott theorems in characteristic p. The proofs are standard applications of the Enriques-Severi -Zariski-Serre vanishing theorems and known facts about the p-curvature
We generalize the classical Chevalley-Shephard-Todd theorem to the case of finite linearly reductive group schemes. As an application, we prove that every scheme X which is etale locally the quotient of a smooth scheme by a finite linearly reductive group scheme is the coarse space of a smooth tame Artin stack (as defined by Abramovich, Olsson, and Vistoli) whose stacky structure is supported on the singular locus of X.
We present an elementary proof of the fact that every torsor for an affine group scheme over an algebraically closed field is trivial. This is related to the uniqueness of fibre functors on neutral tannakian categories.
We define the notion of inseparable coverings of schemes and we propose a ramification formalism for them, along the lines of the classical one. Using this formalism we prove a formula analogous to the classical Riemann-Hurwitz formula for generic torsors under infinitesimal diagonalizable group schemes.
Let $k$ be an algebraically closed field. Chambert-Loir proved that the étale fundamental group of a normal rationally chain connected variety over $k$ is finite. We prove that the fundamental group scheme of a normal rationally chain connected variety over $k$ is finite and étale. In particular, the fundamental group scheme of a Fano variety is finite and étale.
We prove a non-integrability result concerning iterative derivations on projective line, where the iterative rule is given by a non-algebraic formal group.
We give sharp criteria for when a reductive group scheme satisfies Tannakian reconstruction. When the base scheme is Noetherian, we explicitly identify its Tannaka group scheme.
We define graded group schemes and graded group varieties and develop their theory. Graded group schemes are the graded analogue of group schemes and are in correspondence with graded Hopf algebra. In this setting, graded group varieties take the place of infinitesimal group schemes; an important and well studied class of group schemes. We give some examples, derive their properties, and define some special type of graded group schemes, like connected and étale graded group schemes, among others. We give a generalization of the result that connected graded bialgebras are graded Hopf algebra. O…
This article is concerned with Galois theory for iterative differential fields (ID-fields) in positive characteristic. More precisely, we consider purely inseparable Picard-Vessiot extensions, because these are the ones having an infinitesimal group scheme as iterative differential Galois group. In this article we prove a necessary and sufficient condition to decide whether an infinitesimal group scheme occurs as Galois group scheme of a Picard-Vessiot extension over a given ID-field or not. In particular, this solves the inverse ID-Galois problem for infinitesimal group schemes.
We are considering iterative derivations on the function field L of abelian schemes in positive characteristic p>0, and give conditions when the torsion group schemes of this abelian scheme occur as ID-automorphism groups, i.e. are the ID-Galois groups of L over certain ID-subfields. For an explicit example, we even give a construction of (a family of) such iterative derivations.
In this short note, we prove that there is a well behaved notion of normal hull for smooth algebraic group schemes over a field and that the commutator group $(G,H)$ is well defined for $H\subset G$ smooth, even when both of them are not connected.
We establish some structural results for the Witt and Grothendieck-Witt groups of schemes over $\mathbb{Z}[1/2]$, including homotopy invariance for Witt groups and a formula for the Witt and Grothendieck-Witt groups of punctured affine spaces over a scheme. All these results hold for singular schemes and at the level of spectra.
We show that a reductive group scheme over a base scheme S admits a faithful linear representation if and only if its radical torus is isotrivial, that is, it splits after a finite {é}tale cover.
We compare two notions of tameness: one introduced in [CEPT96] for actions of affine group schemes and one introduced in [AOV08] for stacks. From this comparison, we deduce results on the structure of inertia groups in these tame situations.
In this paper we determine the scheme-theoretic fixed points of pinned reductive group schemes acted upon by a group of pinning-preserving automorphisms. The results are used in a companion paper to establish a ramified geometric Satake equivalence with integral or modular coefficients.
Let S be a scheme. We compute explicitly the group of homomorphisms, the S-sheaf of homomorphisms, the group of extensions, and the S-sheaf of extensions involving locally constant S-group schemes, abelian S-schemes, and S-tori. Using the obtained results, we study the categories of biextensions involving these geometrical objets. In particular, we prove that if G_i (for i=1,2,3) is an extension of an abelian S-scheme A_i by an S-torus T_i, the category of biextensions of (G_1,G_2) by G_3 is equivalent to the category of biextensions of the underlying abelian S-schemes (A_1,A_2) by the underly…
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