Galois modules arising from Faltings's strict modules

This work concerns representations of a finite flat group scheme G defined over a noetherian commutative ring R . The focus is on lattices, namely, finitely generated G -modules that are projective as R -modules, and on the full subcategory of all G -modules projective over R generated by the lattices. The stable category of such G -modules is a rigidly-compactly generated, tensor triangulated category. The main result is that this stable category is stratified and costratified by the natural action of the cohomology ring of G . Applications include formulas for computing the support and cosupport of tensor products and the module of homomorphisms, and a classification of the thick ideals in the stable category of lattices.

Let $k$ be a complete discretely valued field of equal characteristic $p > 0$ with possibly imperfect residue field and let $G_k$ be its Galois group. We prove that the conductors computed by the arithmetic ramification filtrations on $G_k$ coincide with the differential Artin conductors and Swan conductors of Galois representations of $G_k$. As a consequence, we give a Hasse-Arf theorem for arithmetic ramification filtrations in this case. As applications, we obtain a Hasse-Arf theorem for finite flat group schemes; we also give a comparison theorem between the differential Artin conductors and Borger's conductors.

Torsors under some group schemes of order [formula omitted

Given a finite group scheme $\cG$ over an algebraically closed field $k$ of characteristic $\Char(k)=p>0$, we introduce new invariants for a $\cG$-module $M$ by associating certain morphisms $\deg^j_M : U_M \lra \Gr_d(M) \ \ (1\!\le\!j\!\le\! p\!-\!1)$ to $M$ that take values in Grassmannians of $M$. These maps are studied for two classes of finite algebraic groups, infinitesimal group schemes and elementary abelian group schemes. The maps associated to the so-called modules of constant $j$-rank have a well-defined degree ranging between $0$ and $j\rk^j(M)$, where $\rk^j(M)$ is the generic $j$-rank of $M$. The extreme values are attained when the module $M$ has the equal images property or the equal kernels property. We establish a formula linking the $j$-degrees of $M$ and its dual $M^\ast$. For a self-dual module $M$ of constant Jordan type this provides information concerning the indecomposable constituents of the pull-back $\alpha^\ast(M)$ of $M$ along a $p$-point $\alpha : k[X]/(X^p) \lra k\cG$.

We give a definition of polynomial identities for affine group schemes; over a field F of characteristic 0, the identities of an affine group scheme turn out to coincide with the identities of a certain matrix algebra over F. We prove the identities of an algebraic affine group scheme are those of a matrix algebra whose size is related to the rank of the representative algebra of the group scheme. Moreover, we relate the existence of a polynomial identity of an affine group scheme with the existence of an abelian subgroupscheme of finite index, giving a Passman type result for affine group schemes.

In the first part of this work [ 12], we studied affine group schemes over a discrete valuation ring (DVR) by means of Neron blowups. We also showed how to apply these findings to throw light on the group schemes coming from Tannakian categories of $\mathcal{D}$-modules. In the present work, we follow up this theme. We show that a certain class of affine group schemes of “infinite type,” Neron blowups of formal subgroups, are quite typical. We also explain how these group schemes appear naturally in Tannakian categories of $\mathcal{D}$-modules. To conclude, we isolate a Tannakian property of affine group schemes, named prudence, which allows one to verify if the underlying ring of functions is a free module over the base ring. This is then successfully applied to obtain a general result on the structure of differential Galois groups over complete DVRs.

Let be a finite group scheme over an algebraically closed field k of characteristic char(k) = p ≥ 3. In generalisation of the familiar notion from the modular representation theory of finite groups, we define the p-rank rkp() of and determine the structure of those group schemes of p-rank 1, whose linearly reductive radical is trivial. The most difficult case concerns infinitesimal groups of height 1, which correspond to restricted Lie algebras. Our results show that group schemes of p-rank ≤ 1 are closely related to those being of finite or domestic representation type.

Algebraic groups play much the same role for algebraists as Lie groups play for analysts. This book is the first comprehensive introduction to the theory of algebraic group schemes over fields that includes the structure theory of semisimple algebraic groups, and is written in the language of modern algebraic geometry. The first eight chapters study general algebraic group schemes over a field and culminate in a proof of the Barsotti–Chevalley theorem, realizing every algebraic group as an extension of an abelian variety by an affine group. After a review of the Tannakian philosophy, the author provides short accounts of Lie algebras and finite group schemes. The later chapters treat reductive algebraic groups over arbitrary fields, including the Borel–Chevalley structure theory. Solvable algebraic groups are studied in detail. Prerequisites have also been kept to a minimum so that the book is accessible to non-specialists in algebraic geometry.

The so-called class-invariant homomorphism $\psi$ measures the Galois module structure of torsors--under a finite flat group scheme $G$--which lie in the image of a coboundary map associated to an isogeny between (N\'eron models of) abelian varieties with kernel $G$. When the varieties are elliptic curves with semi-stable reduction and the order of $G$ is coprime to 6, is is known that the homomorphism $\psi$ vanishes on torsion points. In this paper, using Weil restrictions of elliptic curves, we give the construction, for any prime number $p>2$, of an abelian variety $A$ of dimension $p$ endowed with an isogeny (with kernel $\mu_p$) whose coboundary map is surjective. In the case when $A$ has rank zero and the $p$-part of the Picard group of the base is non-trivial, we obtain examples where $\psi$ does not vanishes on torsion points.

In our previous study of duality for complete discrete valuation fields with perfect residue field, we treated coefficients in finite flat group schemes. In this paper, we treat abelian varieties. This, in particular, implies Grothendieck’s conjecture on the perfectness of his pairing between the Néron component groups of an abelian variety and its dual. The point is that our formulation is well suited to Galois descent. From the known case of semistable abelian varieties, we deduce the perfectness in full generality. We also treat coefficients in tori and, more generally, 1-motives.

Consider the algebraic function $\Phi _{g,n}$ that assigns to a general $g$-dimensional abelian variety an $n$-torsion point. A question first posed by Klein asks: What is the minimal $d$ such that, after a rational change of variables, the function $\Phi _{g,n}$ can be written as an algebraic function of $d$ variables? Using techniques from the deformation theory of $p$-divisible groups and finite flat group schemes, we answer this question by computing the essential dimension and $p$-dimension of congruence covers of the moduli space of principally polarized abelian varieties. We apply this result to compute the essential $p$-dimension of congruence covers of the moduli space of genus $g$ curves, as well as its hyperelliptic locus, and of certain locally symmetric varieties. These results include cases where the locally symmetric variety $M$ is proper. As far as we know, these are the first examples of nontrivial lower bounds on the essential dimension of an unramified, nonabelian covering of a proper algebraic variety.

We introduce the space Π(G) of equivalence classes of π-points of a finite group scheme G and associate a subspace Π(G)M to any G-module M. Our results extend to arbitrary finite group schemes G over arbitrary fields k of positive characteristic and to arbitrarily large G-modules, the basic results about “cohomological support varieties” and their interpretation in terms of representation theory. In particular, we prove that the projectivity of any (possibly infinite-dimensional) G-module can be detected by its restriction along π-points of G. Unlike the cohomological support variety of a G-module M, the invariant M↦Π(G)M satisfies good properties for all modules, thereby enabling us to determine the thick, tensor-ideal subcategories of the stable module category of finite-dimensional G-modules. Finally, using the stable module category of G, we provide Π(G) with the structure of a ringed space which we show to be isomorphic to the scheme ProjH•(G,k)

Finite group scheme actions on the affine plane

On the structure of commutative pointed Hopf algebras

Let S be a non-empty scheme with 2 invertible.In this paper, we present a functor F : AZ n * GS n * , where AZ n * and GS n * are fibered categories over Sch S given, respectively, by degree n Azumaya algebras with an involution of type * and rank n adjoint group schemes of classical type * with absolutely simple fibers.Here n is a function of n.We show that this functor is an equivalence of fibered categories using tale descent, thus giving a classification of adjoint (as well as simply connected) group schemes over S, generalizing the well-known case when the base scheme is the spectrum of a field.In particular, this implies that every adjoint group scheme of classical type with absolutely simple fibers is isomorphic to the neutral component of the automorphism group scheme of a unique (up to isomorphism) Azumaya algebra with involution.We also show interesting applications of this classification such as a specialization theorem for isomorphism classes of Azumaya algebra with involution over Henselian local rings and the uniqueness of an integral model for groups with good reduction over discrete valued fields, and we discuss its implications for the Grothendieck-Serre conjecture over certain domains.