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We prove that every filtered fiber functor on the category of dualizable representations of a smooth affine group scheme with enough dualizable representations comes from a graded fiber functor.

The aim of the paper is to study the group scheme $G:=\operatorname{SL}(2, A)$, it's comodules and a virtual Clebsch-Gordan formula. Here $A$ is a field or a Dedekind domain. If $V:=A\{e_1,e_2\}$ is the free rank $2$ module on $A$ and if we give $V$ the "standard" structure as comodule on $G$, we may form the symmetric powers $\operatorname{Sym}^n(V)$ for $n \geq 1$ an integer. If $A$ is a field of characteristic zero, there is a direct sum decomposition of the tensor product $\operatorname{Sym}^n(V) \otimes \operatorname{Sym}^m(V)$ into irreducible $G$-comodules and the main aim of the paper is to investigate if similar results hold over the ring of integers or a more general commutative ring such as a Dedekind domain. For $A:=\mathbb{Z}$ we will find that there is for any pair of integers $1 \leq n \leq m$ a finite filtration $F_i \subseteq \operatorname{Sym}^n(V) \otimes \operatorname{Sym}^m(V)$ with $F_i/F_{i+1} \cong \operatorname{Sym}^{n+m-2i}(V)$ for $i=0,..,n$. This implies there is a version of the Clebsch-Gordan formula valid in the Grothendieck group of coherent comodules on $G$. We moreover prove that the group scheme $G$ is not "completely reducible" in the sense that there are surjections $\phi: V \rightarrow W$ of finite rank comodules on $G$ that do not split. I also discuss the notion "good filtration" for torsion free comodules and give an explicit construction of an infinte set of non trival comodules with a good filtration. I give a functorial definition of the dual comodule of any comodule $(V, \Delta)$, where $V$ is a free and finite rank $A$-module. This construction has the property that the double dual $V^{**}$ is canonically isomorphic to $V$ as comodule. I use the construction to study the dual of the symmetric powers and symmetric tensors of the standard module and give explicit examples of comodules without a good filtration., Comment: Oct-06-2023: Extended version with a functorial construction of the dual, new examples using the symmetric tensors and new references. 11.10.2023: Minor corrections. 18.10.2023: New introduction. 01.02.2024: Minor corrections

Given a reductive group scheme $G$, we give a linear algebraic description of reduced \'etale $4$-cocycles on its classifying stack $\mathrm B(G)$. These cocycles form a $2$-groupoid, which we interpret as parameters of metaplectic covers of $G$. We use our linear algebraic description to define the Langlands dual of a metaplectic cover., Comment: 78 pages, a substantial revision of "metaplectic group schemes"

Let $\mathcal{G}^{(\lambda)}$ be a group scheme which deforms $\mathbb{G}_a$ to $\mathbb{G}_m$. We explicitly describe the Cartier dual of the $l$-th Frobenius type kernel $N_l$ of the group scheme $\mathcal{E}^{(\lambda,\mu;D)}$ which is an extension of $\mathcal{G}^{(\lambda)}$ by $\mathcal{G}^{(\mu)}$. Here we assume that the base ring $A$ is a $\mathbb{Z}_{(p)}$-algebra containing some nilpotent elements. The obtained result generalizes a previous result by N. Aki and M. Amano (Tsukuba J.Math. $\textbf{34}$ (2010)) which assumes that $A$ is an $\mathbb{F}_p$-algebra., Comment: 13pages

We use tools of representation theory to get a better understanding of the cohomology of graded group schemes. For that, we focus our attention on the case in which the base field is of characteristic $p > 0$. Using as inspiration the work of Friedlander, et al, we build the theory of graded $1$-parameter subgroups denoted by $V_r^*(G)$. We give a natural homomorphism of bigraded $\boldsymbol{\rm k}$-algebras $\psi: H^{*, *}(G, \boldsymbol{\rm k}) \to \boldsymbol{\rm k}[V_r^* (G)],$ where $\boldsymbol{\rm k}[V_r^* (G)]$ is the bigraded coordinate ring for $V_r^*(G)$. We show that $\psi$ is an $F$-monomorphism for a class of graded group schemes. This provides evidence that with the appropriate detection property, a Quillen-type result could exist for graded group schemes.

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 \'{e}tale graded group schemes, among others. We give a generalization of the result that connected graded bialgebras are graded Hopf algebra. Our result is given in for a broader class of graded Hopf algebras; the coordinate rings of graded group varieties, In addition, this result can also be interpreted as the algebraic analogue of a well known geometric result regarding projective group schemes. We also give a classification for graded group algebras and graded group varieties.

It is well known that all torsors under an affine algebraic group over an algebraically closed field are trivial. We note that under suitable conditions this also holds if the the group is not necessarily of finite type. This has an application to isomorphisms of fibre functors on neutral Tannakian categories.

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 $\mu_p$ by $\Z/p\Z$ split over the ring of $S$-integers $O_S$ of $K$.

Let $k$ be an algebraically closed field of characteristic $p>0$. For a loop $\circlearrowleft$, denote its path coalgebra by $k\circlearrowleft$. In this paper, all the finite-dimensional commutative Hopf algebras over the sub coalgebras of $k\circlearrowleft$ are given. As a direct consequence, all the commutative infinitesimal groups $\mathcal{G}$ with dim$_{k}$Lie$(\mathcal{G})=1$ are classified., Comment: 16 pages

For $K$ an extension of $\mathbb{Q}_{p}$ with ring of integers $R$ we show how Breuil-Kisin modules can be used to determine Hopf orders in $K$-Hopf algebras of $p$-power dimension. We find all cyclic Breuil-Kisin modules, and use them to compute all of the Hopf orders in the group ring $K\Gamma$ where $\Gamma$ is cyclic of order $p$ or $p^{2}.$ We also give a Laurent series interpretation of the Breuil-Kisin modules that give these Hopf orders., Comment: 17 pages

Let R be a discrete valuation ring with residue field of characteristic p>0. Let K be its fraction field. We prove that any finite and flat R-group scheme, isomorphic to \mu_{p^2,K} on the generic fiber, is the kernel in a short exact sequence which generically coincides with the Kummer sequence. We will explicitly describe and classify such models. In the appendix X. Caruso shows how to classify models of \mu_{p^2,K}, in the case of unequal characteristic, using the Breuil-Kisin theory., Comment: 38 pages. This paper replaces the previous preprint "Models over a d.v.r. of unequal caracteristic" arXiv:0803.3702. To appear on Journal of Algebra

Let X be a smooth projective variety defined over an algebraically closed field k. Nori constructed a category of vector bundles on X, called essentially finite vector bundles, which is reminiscent of the category of representations of the fundamental group (in characteristic zero). In fact, this category is equivalent to the category of representations of a pro--finite group scheme which controls all finite torsors. We show that essentially finite vector bundles coincide with those which become trivial after being pulled back by some proper and surjective morphism to X., Comment: Final version

Let A be the N\'eron model of an abelian variety A_K over the fraction field K of a discrete valuation ring R. Due to work of Mazur-Messing, there is a functorial way to prolong the universal extension of A_K by a vector group to a smooth and separated group scheme over R, called the canonical extension of A. In this paper, we study the canonical extension when A_K=J_K is the Jacobian of a smooth proper and geometrically connected curve X_K over K. Assuming that X_K admits a proper flat regular model X over R that has generically smooth closed fiber, our main result identifies the identity component of the canonical extension with a certain functor Pic^{\natural,0}_{X/R} classifying line bundles on X that have partial degree zero on all components of geometric fibers and are equipped with a regular connection. This result is a natural extension of a theorem of Raynaud, which identifies the identity component of the N\'eron model J of J_K with the functor Pic^0_{X/R}. As an application of our result, we prove a comparison isomorphism between two canonical integral structures on the de Rham cohomology of X_K., Comment: To appear in Algebra & Number Theory

Let M be a geometrically irreducible smooth projective variety, defined over a finite field k, such that M admits a k-rational point x_0. Let \varpi(M,x_0) denote the corresponding fundamental group--scheme introduced by Nori. Let E_G be a principal G-bundle over M, where G is a reduced reductive linear algebraic group defined over the field k. Fix a polarization \xi on M. We prove that the following three statements are equivalent: The principal G-bundle E_G over M is given by a homomorphism \varpi(M,x_0) --> G. There are integers b > a > 0 such that the principal G-bundle (F^b_M)^*E_G is isomorphic to (F^a_M)^*E_G, where F_M is the absolute Frobenius morphism of M. The principal G-bundle E_G is strongly semistable, degree(c_2(ad(E_G))c_1(\xi)^{d-2}) = 0, where d = \dim M, and degree(c_1(E_G(\chi))c_1(\xi)^{d-1}) = 0 for every character \chi of G, where E_G(\chi) is the line bundle over $M$ associated to $E_G$ for \chi. The equivalence between the first statement and the third statement was proved by S. Subramanian under the extra assumption that dim(M) = 1 and $G$ is semisimple., Comment: Final version of preprint (2007)

For a reductive group scheme over a regular semi-local ring, we prove an equivarinat version of the Gersten conjecture. We draw some interesting consequences for the representation rings of such reductive group schemes. We also prove the rigidity for the equivariant K-theory of reductive group schemes over a henselian local ring. This is then used to compute the equivariant K-theory of algebraically closed fields.

Let G be a connected linear algebraic group over a number field k. We establish an exact sequence describing the closure of the group G(k) of rational points of G in the group of adelic points of G. This exact sequence describes the defect of strong approximation on G in terms of the algebraic Brauer group of G. In particular, we deduce from those results that the integral Brauer-Manin obstruction on a torsor under the group G is the only obstruction to the existence of an integral point on this torsor. We also obtain a non-abelian Poitou-Tate exact sequence for the Galois cohomology of the linear group G. The main ingredients in the proof of those results are the local and global duality theorems for complexes of k-tori of length two and the abelianization maps in Galois cohomology introduced by Borovoi., Comment: 33 pages

We refine the notion of variety over the "field with one element" developed by C. Soul\'e by introducing a grading in the associated functor to the category of sets, and show that this notion becomes compatible with the geometric viewpoint developed by J. Tits. We then solve an open question of C. Soul\'e by proving, using results of J. Tits and C. Chevalley, that Chevalley group schemes are examples of varieties over a quadratic extension of the above "field"., Comment: 26 pages

The stratified vector bundles on a smooth variety defined over an algebraically closed field $k$ form a neutral Tannakian category over $k$. We investigate the affine group--scheme corresponding to this neutral Tannakian category., Comment: 17 pages

We classify principal bundles over anti-affine schemes with affine and commutative structural group. We show that this yields the classification of quasi-abelian varieties over a field k (i.e., group k-schemes with no non constant global functions). The interest of this result is given by the fact that the classification of smooth group k-schemes is reduced to the classification of quasi-abelian varieties and of certain affine group schemes., Comment: 26 pages

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., Comment: technical corrections were made and new applications in Section 8 were added

Let R be a discrete valuation ring of unequal characteristic with fraction field K which contains a primitive p^2-th root of unity. Let X be a faithfully flat R-scheme and G be a finite abstract group. Let us consider a G-torsor Y_K\to X_K and let Y be the normalization of X_K in Y. If G=Z/p^n Z, n<3, under some hypothesis on X, we attach some invariants to Y_K \to X_K. If p>2, we determine, through these invariants, when Y\to X has a structure of torsor which extends that of Y_K\to X_K. Moreover we explicitly calculate the effective model (defined by Romagny) of the action of G on Y., Comment: 40 pages, final version, corrected and improved some statements in the sections 4,5,6 and 7

Let R be a discrete valuation ring of unequal characteristic which contains a primitive p^2-th root of unity. If K is the fraction field of R, it is well known that (Z/p^2 Z)_K is isomorphic to \mu_{p^2,K}. We prove that any finite and flat R-group scheme of order p^2 isomorphic to (Z/p^2 Z)_K on the generic fiber (i.e. a model of (Z/p^2 Z)_K), is the kernel in a short exact sequence which generically coincides with the Kummer sequence. We will explicitly describe and classify such models., Comment: 40 pages

In this paper we obtain the Cartier duality for k-schemes of commutative monoids functorially without providing the vector spaces of functions with a topology, generalizing a result for finite commutative algebraic groups by M. Demazure and P. Gabriel., Comment: 12 pages

This note is a correction of (statement and proof of) proposition 3.3.1 of Toby Gee's preprint intitled *On the weights of mod p Hilbert modular forms*. The aim is to compare Galois representations arising from extensions of some group schemes (over the ring of integers of a p-adic field) endowed with a descent data, and extensions of some crystalline representations with given Hodge-Tate weights. The main tool of the proof is the theory of Breuil., Comment: 18 pages

We introduce and study a class of algebraic stacks with finite inertia in positive and mixed characteristic, which we call tame algebraic stacks. They include tame Deligne-Mumford stacks, and are arguably better behaved than general Deligne-Mumford stacks. We also give a complete characterization of finite flat linearly reductive schemes over an arbitrary base. Our main result is that tame algebraic stacks are \'etale locally quotient by actions of linearly reductive finite group schemes. In a subsequent paper we will show that tame algebraic stacks admit a good theory of stable maps., Comment: 31 pages, 3 sections and 1 appendix

We define the local fundamental group scheme, defined by the F-trivial vector bundles, and give necessary and sufficient conditions for it to base change., Comment: submitted to the Proceedings of the Indian Academy of Sciences, Mathematical Sciences on the 3rd of April, 2006

Let O\_K be a complete discrete valuation ring. Denote by K its fractions field and by k its residue field. Assume that k is of characteristic p>0 and perfect. Breuil gives an anti-equivalence between the category of finite flat O\_K-group schemes killed by a power of p and a category of linear algebra objects which is called (Mod/S). The aim of this article is to make explicit the Cartier duality on the category (Mod/S).

We show that the pairing on de Rham realizations of 1-motives in "Theorie di Hodge III", IHES 44, can be defined over any base scheme and we prove that it gives rise to a perfect duality if one is working with a 1-motive and its Cartier dual. Furthermore we study universal extensions of 1-motives and their relation with $\natural$-extensions., Comment: 21 pages. Last section rewritten. New proofs

In this paper we investigate the image of the $l$-adic representation attached to the Tate module of an abelian variety over a number field with endomorphism algebra of type I or II in the Albert classification. We compute the image explicitly and verify the classical conjectures of Mumford-Tate, Hodge, Lang and Tate, for a large family of abelian varieties of type I and II. In addition, for this family, we prove an analogue of the open image theorem of Serre., Comment: 34 pages

Suppose O is a complete discrete valuation ring of positive characteristic with perfect residue field. The category of finite flat strict modules was introduced recently by Faltings and appears as an equal characteristic analogue of the classical category of finite flat group schemes. In this paper we obtain a classification of these modules and apply it to prove analogues of properties, which were known earlier for group schemes., Comment: 19 pages

Let T be a Tannakian category over a field k of characteristic 0 and \pi(T) its fundamental group. In this paper we prove that there is a bijection between the otimes-equivalence classes of Tannakian subcategories of T and the normal affine group sub-T-schemes of \pi(T). We apply this result to the Tannakian category T_1(k) generated by motives of niveau \leq 1 defined over k, whose fundamental group is called the motivic Galois group G_mot(T_1(k)) of motives of niveau \leq 1. We find four short exact sequences of affine group sub-T_1(k)-schemes of G_mot(T_1(k)), correlated one to each other by inclusions and projections. Moreover, given a 1-motive M, we compute explicitly the biggest Tannakian subcategory of the one generated by M, whose fundamental group is commutative., Comment: Version with more details