Lagrange’s theorem for a class of finite flat group schemes over local Artin rings

Let p be an odd prime. We show that the classification of p -divisible groups by Breuil windows and the classification of commutative finite flat group schemes of p -power order by Breuil modules hold over every complete regular local ring with perfect residue field of characteristic p . We set up a formalism of frames and windows with an abstract deformation theory that applies to Breuil windows.

Let A′ be a complete characteristic (0,p) discrete valuation ring with absolute ramification degree e and a perfect residue field. We are interested in studying the category FFA' of finite flat commutative group schemes over A′ withp-power order. When e= 1, Fontaine formulated the purely ‘linear algebra’ notion of a finite Honda system over A′ and constructed an anti-equivalence of categories betweenineFFA'> and the category of finite Honda systems over A′ when p> 2. We generalize this theory to the case e≤ − 1.

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.

Given a flat, finite group scheme G finitely presented over a base scheme we introduce the notion of ramified Galois cover of group G (or simply G-cover), which generalizes the notion of G-torsor. We study the stack of G-covers, denoted with G, mainly in the abelian case, precisely when G is a finite diagonalizable group scheme over . In this case, we prove that G is connected, but it is irreducible or smooth only in few finitely many cases. On the other hand, it contains a “special” irreducible component , which is the closure of G and this reflects the deep connection we establish between G and the equivariant Hilbert schemes. We introduce “parametrization” maps from smooth stacks, whose objects are collections of invertible sheaves with additional data, to and we establish sufficient conditions for a G-cover in order to be obtained (uniquely) through those constructions. Moreover, a toric description of the smooth locus of is provided.

HTML view is not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button. Let R be a local Artin ring with maximal ideal $\frak m$ and residue class field of characteristic p > 0. We show that every finite flat group scheme over R is annihilated by its rank, whenever $\frak m$p = p$\frak m$ = 0. This implies that any finite flat group scheme over an Artin ring the square of whose maximal ideal is zero, is annihilated by its rank.

Let G be a finite group scheme over a field k, that is, an affine group scheme whose coordinate ring k[G] is finite dimensional. The dual algebra k[G]* ≡ Homk(k[G], k) is then a finite dimensional cocommutative Hopf algebra. Indeed, there is an equivalence of categories between finite group schemes and finite dimensional cocommutative Hopf algebras (cf. [19]). Further the representation theory of G is equivalent to that of k[G]*. Many familiar objects can be considered in this context. For example, any finite group G can be considered as a finite group scheme. In this case, the algebra k[G]* is simply the group algebra kG. Over a field of characteristic p > 0, the restricted enveloping algebra u([gfr ]) of a p-restricted Lie algebra [gfr ] is a finite dimensional cocommutative Hopf algebra. Also, the mod-p Steenrod algebra is graded cocommutative so that some finite dimensional Hopf subalgebras are such algebras.Over the past thirty years, there has been extensive study of the modular representation theory (i.e. over a field of positive characteristic p > 0) of such algebras, particularly in regards to understanding cohomology and determining projectivity of modules. This paper is primarily interested in the following two questions:Questions1·1. Let G be a finite group scheme G over a field k of characteristic p > 0, and let M be a rational G-module.(a) Does there exist a family of subgroup schemes of G which detects whether M is projective?(b) Does there exist a family of subgroup schemes of G which detects whether a cohomology class z ∈ ExtnG(M, M) (for M finite dimensional) is nilpotent?

Let R be a discrete valuation ring with field of fractions K and residue field k of characteristic $$p>0$$ . Given a commutative finite group scheme G over K and a smooth projective curve C over K with a rational point, we study the extension of pointed fppf G-torsors over C to pointed torsors over some R-regular model $${\mathcal {C}}$$ of C. We first study this problem in the category of log schemes: given a finite flat R-group scheme $${\mathcal {G}}$$ , we prove that the data of a pointed $${\mathcal {G}}$$ -log torsor over $${\mathcal {C}}$$ is equivalent to that of a morphism $${\mathcal {G}}^D \rightarrow {{\,\mathrm{Pic}\,}}^{log}_{{\mathcal {C}}/R}$$ , where $${\mathcal {G}}^D$$ is the Cartier dual of $${\mathcal {G}}$$ and $${{\,\mathrm{Pic}\,}}^{log}_{{\mathcal {C}}/R}$$ the log Picard functor. After that, we give a sufficient condition for such a log extension to exist, and then we compute the obstruction for the existence of an extension in the category of usual schemes. In a second part, we generalize a result of Chiodo (Manuscr Math 129(3):337–368, 2009) which gives a criterion for the r-torsion subgroup of the Néron model of J to be a finite flat group scheme, and we combine it with the results of the first part. Finally, we give a detailed example of extension of torsors when C is a hyperelliptic curve defined over $${\mathbb {Q}}$$ , which illustrates our techniques.

Let K be a local field of characteristi c 0 which is complete relative to a discrete valuation with a perfect residue field k of characteristic ρ > 0. If Ko is the quotient field of the ring of Witt vectors OQ = W(k), then K is a totally ramified extension of KQ of degree e, and we assume that e < ρ - 1 and that k is algebraically closed for e = ρ - 1. We denote by O the ring of integers of K, by π a prime element in O, and by σ a Frobenius automorphism on k, W(k), and KQ. The objects of the category ©g are finite flat commutative group schemes over O which can be annihilated by multiplicatio n by p. Morphisms in ®Q are morphisms of group schemes Gj -*· G^ to within morphisms of the form Gx -*• G\et^ -*• G^ -*• Gt, where the extreme morphisms are the projection on the largest etale quotient-object and the embedding of the largest multiplicative subobject, respectively. We note that for e < ρ - 1 the category ©^ is identified with a complete subcategory in the category of all group schemes over O. We introduce the category HQ which generalizes a complete subcategory of finite Honda systems (1) consisting of objects annihilated by multiplicatio n by p. Objects in HQ are collections M = (M°, M1, φ, ψ), where Λ/° is an 00-module of finite rank, pM = 0, M1 is an Og-submodule m

In this paper, it is shown that the projectivity of a rational module for an infinitesimal unipotent group scheme over an algebraically closed field of positive characteristic can be detected on a family of closed subgroups. Let k be an algebraically closed field of characteristic p > 0 and G be an infinitesimal group scheme over k, that is, an affine group scheme G over k whose coordinate (Hopf) algebra k[G] is a finite-dimensional local k-algebra. A rational G-module is equivalent to a k[G]-comodule and further equivalent to a module for the finitedimensional cocommutative Hopf algebra k[G]∗ ≡ Homk(k[G], k). Since k[G]∗ is a Frobenius algebra (cf. [Jan]), a rational G-module (even infinite-dimensional) is in fact projective if and only if it is injective (cf. [FW]). Further, for any rational G-module M and any closed subgroup scheme H ⊂ G, if M is projective over G, then it remains projective upon restriction to H (cf. [Jan]). We consider the question of whether there is a “nice” collection of closed subgroups of G upon which projectivity (over G) can be detected. For an example of what we mean by a “nice” collection, consider the situation of modules over a finite group. Over a field of characteristic p > 0, a module over a finite group is projective if and only if it is projective upon restriction to a p-Sylow subgroup (cf. [Rim]). For a p-group (and hence for any finite group), L. Chouinard [Ch] showed that a module is projective if and only if it is projective upon restriction to every elementary abelian subgroup. If the module is assumed to be finite-dimensional, this result follows from the theory of varieties for finite groups (cf. [Ca] or [Ben]). Indeed, elementary abelian subgroups play an essential role in this theory. In work of A. Suslin, E. Friedlander, and the author [SFB1], [SFB2], a theory of varieties for infinitesimal group schemes was developed. In this setting, subgroups of the form Ga(r) (the rth Frobenius kernel of the additive group scheme Ga) play the role analogous to that of elementary abelian subgroups in the case of finite groups. Not surprisingly then, for finite-dimensional modules, one obtains the following analogue of Chouinard’s Theorem. Proposition 1 ([SFB2, Proposition 7.6]). Let k be an algebraically closed field of characteristic p > 0, r > 0 be an integer, G be an infinitesimal group scheme over k of height ≤ r, and M be a finite-dimensional rational G-module. Then M is Received by the editors January 27, 1998 and, in revised form, March 24, 1998 and May 19, 1999. 2000 Mathematics Subject Classification. Primary 14L15, 20G05; Secondary 17B50. c ©2000 American Mathematical Society

Boundedness results for finite flat group schemes over discrete valuation rings of mixed characteristic

We define a motivic conductor for any presheaf with transfersFusing the categorical framework developed for the theory of motives with modulus by Kahn, Miyazaki, Saito and Yamazaki. IfFis a reciprocity sheaf, this conductor yields an increasing and exhaustive filtration on$F(L)$, whereLis any henselian discrete valuation field of geometric type over the perfect ground field. We show that ifFis a smooth group scheme, then the motivic conductor extends the Rosenlicht–Serre conductor; ifFassigns toXthe group of finite characters on the abelianised étale fundamental group ofX, then the motivic conductor agrees with the Artin conductor defined by Kato and Matsuda; and ifFassigns toXthe group of integrable rank$1$connections (in characteristic$0$), then it agrees with the irregularity. We also show that this machinery gives rise to a conductor for torsors under finite flat group schemes over the base field, which we believe to be new. We introduce a general notion of conductors on presheaves with transfers and show that on a reciprocity sheaf, the motivic conductor is minimal and any conductor which is defined only for henselian discrete valuation fields of geometric type withperfectresidue field can be uniquely extended to all such fields without any restriction on the residue field. For example, the Kato–Matsuda Artin conductor is characterised as the canonical extension of the classical Artin conductor defined in the case of a perfect residue field.

Ramification correspondence of finite flat group schemes over equal and mixed characteristic local fields

Negative cohomology and the endomorphism ring of the trivial module

Continuous-time quantum walk (CTQW) over finite group schemes is investigated, where it is shown that some properties of a CTQW over a group scheme defined on a finite group G induces a CTQW over group scheme defined on G/H, where H is a normal subgroup of G with prime index. This reduction can be helpful in analyzing CTQW on underlying graphs of group schemes. Even though this claim is proved for normal subgroups with prime index (using the Clifford's theorem from representation theory), it is checked in some examples that for other normal subgroups or even non-normal subgroups, the result is also true! Moreover, it is shown that the Bose-Mesner (BM) algebra associated with the group scheme over G is isomorphic to the corresponding BM algebra of the association scheme defined over the coset space G/H up to the scale factor |H|. In fact, we show that the underlying graph defined over group G is a covering space for the quotient graph defined over G/H, so that CTQW over the graph on G, starting from any arbitrary vertex, is isomorphic to the CTQW over the quotient graph on G/H if we take the sum of the amplitudes corresponding to the vertices belonging to the same cosets.

Let OK be a complete discrete valuation ring of residue characteristic p > 0, and G be a finite flat group scheme over OK of order a power of p. We prove in this paper that the Abbes-Saito filtration of G is bounded by a linear function of the degree of G. Assume OK has generic characteristic 0 and the residue field of OK is perfect. Fargues constructed the higher level canonical subgroups for a not too supersingular Barsotti- Tate group G over OK. As an application of our bound, we prove that the canonical subgroup of G of level n � 2 constructed by Fargues appears in the Abbes-Saito filtration of the p n -torsion subgroup of G. Let OK be a complete discrete valuation ring with residue field k of characteristic p > 0 and fraction field K. We denote by vthe valuation on K normalized by v�(K × ) = Z. Let G be a finite and flat group scheme over OK of order a power of p such that G K is etale. We denote by (G a ;a 2 Q�0) the Abbes-Saito filtration of G. This is a decreasing and separated filtration of G by finite and flat closed subgroup schemes. We refer the readers to (AS02, AS03, AM04) for a full discussion, and to section 1 for a brief review of this filtration. Let !G be the module of invariant differentials of G. The generic etaleness of G implies that !G is a torsion OK-module of finite type. There exist thus nonzero elements a1;���;ad 2 O K such that !G ' d