Complete Discrete Valuation Ring Research Articles

On symmetric quotients of symmetric algebras

We investigate symmetric quotient algebras of symmetric algebras, with an emphasis on finite group algebras over a complete discrete valuation ring O. Using elementary methods, we show that if an ordinary irreducible character χ of a finite group G gives rise to a symmetric quotient over O which is not a matrix algebra, then the decomposition numbers of the row labelled by χ are all divisible by the characteristic p of the residue field of O.

Parity sheaves

Given a stratified variety X X with strata satisfying a cohomological parity-vanishing condition, we define and show the uniqueness of “parity sheaves,” which are objects in the constructible derived category of sheaves with coefficients in an arbitrary field or complete discrete valuation ring. This construction depends on the choice of a parity function on the strata. If X X admits a resolution also satisfying a parity condition, then the direct image of the constant sheaf decomposes as a direct sum of parity sheaves, and the multiplicities of the indecomposable summands are encoded in certain refined intersection forms appearing in the work of de Cataldo and Migliorini. We give a criterion for the Decomposition Theorem to hold in the semi-small case. Our framework applies to many stratified varieties arising in representation theory such as generalised flag varieties, toric varieties, and nilpotent cones. Moreover, parity sheaves often correspond to interesting objects in representation theory. For example, on flag varieties we recover in a unified way several well-known complexes of sheaves. For one choice of parity function we obtain the indecomposable tilting perverse sheaves. For another, when using coefficients of characteristic zero, we recover the intersection cohomology sheaves and in arbitrary characteristic the special sheaves of Soergel, which are used by Fiebig in his proof of Lusztig’s conjecture.

Strong fusion control and stable equivalences

This article is dedicated to the proof of the following theorem. Let G be a finite group, p be a prime number, and e be a p-block of G. Assume that the centraliser CG(P) of an e-subpair (P,eP) “strongly” controls the fusion of the block e, and that a defect group of e is either abelian or (for odd p) has a non-cyclic centre. Then there exists a stable equivalence of Morita type between the block algebras OGe and OCG(P)eP, where O is a complete discrete valuation ring of residual characteristic p. This stable equivalence is constructed by gluing together a family of local Morita equivalences, which are induced by bimodules with fusion-stable endo-permutation sources.Broué had previously obtained a similar result for principal blocks, in relation with the search for a modular proof of the odd Zp⁎-theorem. Thus our theorem points towards a block-theoretic analogue of the Zp⁎-theorem, which we state in terms of fusion control and Morita equivalences.

Linear algebra over and related rings

AbstractLet $\mathfrak{R}$ be a complete discrete valuation ring, $S=\mathfrak{R}[[u]]$ and $d$ a positive integer. The aim of this paper is to explain how to efficiently compute usual operations such as sum and intersection of sub-$S$-modules of $S^d$. As $S$ is not principal, it is not possible to have a uniform bound on the number of generators of the modules resulting from these operations. We explain how to mitigate this problem, following an idea of Iwasawa, by computing an approximation of the result of these operations up to a quasi-isomorphism. In the course of the analysis of the $p$-adic and $u$-adic precisions of the computations, we have to introduce more general coefficient rings that may be interesting for their own sake. Being able to perform linear algebra operations modulo quasi-isomorphism with $S$-modules has applications in Iwasawa theory and $p$-adic Hodge theory. It is used in particular in Caruso and Lubicz (Preprint, 2013, arXiv:1309.4194) to compute the semi-simplified modulo $p$ of a semi-stable representation.

On the structure of Witt–Burnside rings attached to pro-$p$ groups

The p -typical Witt vectors are a ubiquitous object in algebra and number theory. They arise as a functorial construction that takes perfect fields k of prime characteristic p > 0 to p -adically complete discrete valuation rings of characteristic 0 with residue field k and are universal in that sense. A. Dress and C. Siebeneicher generalized this construction by producing a functor \mathbf W_G attached to any profinite group G . The p -typical Witt vectors arise as those attached to the p -adic integers. Here we examine the ring structure of \mathbf W_G(k) for several examples of pro- p groups G and fields k of characteristic p . We will show that the structure is surprisingly more complicated than the p -typical case.

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.

Cycle classes and the syntomic regulator

Let $V=Spec(R)$ and $R$ be a complete discrete valuation ring of mixed characteristic $(0,p)$. For any flat $R$-scheme $X$ we prove the compatibility of the de Rham fundamental class of the generic fiber and the rigid fundamental class of the special fiber. We use this result to construct a syntomic regulator map $r:CH^i(X/V,2i-n)\to H^n_{syn}(X,i)$, when $X$ is smooth over $V$, with values on the syntomic cohomology defined by A. Besser. Motivated by the previous result we also prove some of the Bloch-Ogus axioms for the syntomic cohomology theory, but viewed as an absolute cohomology theory.

Variants of formal nearby cycles

AbstractIn this paper, we introduce variants of formal nearby cycles for a locally noetherian formal scheme over a complete discrete valuation ring. If the formal scheme is locally algebraizable, then our nearby cycle gives a generalization of Berkovich’s formal nearby cycle. Our construction is entirely scheme theoretic, and does not require rigid geometry. Our theory is intended for applications to the local study of the cohomology of Rapoport–Zink spaces.

Constructible $$\nabla $$ ∇ -modules on curves

Let $$\mathcal{V }$$ be a complete discrete valuation ring of mixed characteristic with perfect residue field. Let $$X$$ be a geometrically connected smooth proper curve over $$\mathcal{V }$$ . We introduce the notion of constructible convergent $$\nabla $$ -module on the analytification $$X_{K}^{\mathrm{an}}$$ of the generic fiber of $$X$$ . A constructible module is an $$\mathcal{O }_{X_{K}^{\mathrm{an}}}$$ -module which is not necessarily coherent, but becomes coherent on a stratification by locally closed subsets of the special fiber $$X_{k}$$ of $$X$$ . The notions of connection, of (over-) convergence and of Frobenius structure carry over to this situation. We describe a specialization functor from the category of constructible convergent $$\nabla $$ -modules to the category of $$\mathcal{D }^\dagger _{\hat{X} \mathbf{Q }}$$ -modules. We show that specialization induces an equivalence between constructible $$F$$ - $$\nabla $$ -modules and perverse holonomic $$F$$ - $$\mathcal{D }^\dagger _{\hat{X} \mathbf{Q }}$$ -modules.

Extending Endo-monomial Modules

Let G be a finite group with a normal Sylow p-subgroup P. Let [Formula: see text] be a complete discrete valuation ring with residue field F of characteristic p. Let M be an indecomposable endo-monomial [Formula: see text]-module. In this paper we prove that M extends to an [Formula: see text]-module if and only if M is G-stable. A similar and well-known version for endo-permutation modules is due to Dade.

Weierstrass preparation and algebraic invariants

We prove a form of the Weierstrass Preparation Theorem for normal algebraic curves over complete discrete valuation rings. While the more traditional algebraic form of Weierstrass Preparation applies just to the projective line over a base, our version allows more general curves. This result is then used to obtain applications concerning the values of u-invariants, and on the period–index problem for division algebras, over fraction fields of complete two-dimensional rings. Our approach uses patching methods and matrix factorization results that can be viewed as analogs of Cartan’s Lemma.

Maximal wild monodromy in unequal characteristic

Let R be a complete discrete valuation ring of mixed characteristic ( 0 , p ) with fraction field K . We study stable models of p -cyclic covers of P K 1 . First, we determine the monodromy extension, the monodromy group, its filtration and the Swan conductor for special covers of arbitrarily high genus with potential good reduction. In the case p = 2 we consider hyperelliptic curves of genus 2.

Indecomposable and noncrossed product division algebras over curves over complete discrete valuation rings

Let T be a complete discrete valuation ring and Xˆ a smooth projective curve over S=spec(T) with closed fibre X. Denote by F the function field of Xˆ and by Fˆ the completion of F with respect to the discrete valuation defined by X, the closed fibre. In this paper, we construct indecomposable and noncrossed product division algebras over F. This is done by defining an index preserving group homomorphism s:Br(Fˆ)′→Br(F)′, and using it to lift indecomposable and noncrossed product division algebras over Fˆ.

Néron’s pairing and relative algebraic equivalence

Let R be a complete discrete valuation ring with algebraically closed residue field k and fraction field K. Let XK be a proper smooth and geometrically connected scheme over K. Neron defined a canonical pairing on XK between 0-cycles of degree zero and divisors which are algebraically equivalent to zero. When XK is an abelian variety, and if one restricts to those 0-cycles supported on K-rational points, Neron gave an expression of his pairing involving intersection multiplicities on the Neron model A of AK over R. When XK is a curve, Gross and Hriljac gave independently an analogous description of Neron’s pairing, but for arbitrary 0-cycles of degree zero, by means of intersection theory on a proper flat regular R-model X of XK. We show that these intersection computations are valid for an arbitrary scheme XK as above and arbitrary 0-cycles of degree zero, by using a proper flat normal and semifactorial model X of XK over R. When XK=AK is an abelian variety, and X=A¯ is a semifactorial compactification of its Neron model A, these computations can be used to study the relative algebraic equivalence on A¯∕R. We then obtain an interpretation of Grothendieck’s duality for the Neron model A, in terms of the Picard functor of A¯ over R. Finally, we give an explicit description of Grothendieck’s duality pairing when AK is the Jacobian of a curve of index one.

Pushout of quasi-finite and flat group schemes over a Dedekind ring

Let G, G1 and G2 be quasi-finite and flat group schemes over a complete discrete valuation ring R, φ1:G→G1 any morphism of R-group schemes and φ2:G→G2 a model map. We construct the pushout P of G1 and G2 over G in the category of R-affine group schemes. In particular when φ1 is a model map too we show that P is still a model of the generic fibre of G. We also provide a short proof for the existence of cokernels and quotients of finite and flat group schemes over any Dedekind ring.

An upper bound on the Abbes–Saito filtration for finite flat group schemes and applications

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

On Auslander–Reiten components and splitting trace lattices for integral group rings

Let G be a finite group and O a complete discrete valuation ring of characteristic zero with residue class field k=O/πO of characteristic p>0. Suppose that O is sufficiently large to satisfy certain conditions and the group ring OG is of infinite representation type. Let Θ be a connected component of the Auslander–Reiten quiver of OG. We show that if Θ contains an OG-lattice M such that M/πM is an indecomposable kG-module and rankOM is not divisible by p, then the tree class of Θ is A∞ and M lies at the end of Θ.

Local-global principle for quadratic forms over fraction fields of two-dimensional henselian domains

Let $R$ be a 2-dimensional normal excellent henselian local domain in which 2 is invertible and let $L$ and $k$ be respectively its fraction field and residue field. Let $\Omega_R$ be the set of rank 1 discrete valuations of $L$ corresponding to codimension 1 points of regular proper models of $\Spec R$. We prove that a quadratic form $q$ over $L$ satisfies the local-global principle with respect to $\Omega_R$ in the following two cases: (1) $q$ has rank 3 or 4; (2) $q$ has rank $\ge 5$ and $R=A[y]$, where $A$ is a complete discrete valuation ring with a not too restrictive condition on the residue field $k$, which is satisfied when $k$ is $C_1$.

The Ordinary Characters of the Involution Module of SL<sub>2</sub>(2<i><sup>f</sup></i>)

Let k be an algebraically closed field of characteristic two, and let G be a finite group. Furthermore let J be the set of involutions in G, that is, the set of elements in G of order two. Then J is a G-set under conjugation. In particular we get a fcG-permutation module fcJ, which is known as the Involution Module of G. The involution module has been studied in general by J. Murray in [5; 6; 7]. Also the author studied the involution module of the special linear group and general linear group in [10] and [11], respectively. The purpose of this paper is to investigate the ordinary characters of the involution module of the special linear group SL2(g), where q := 2-f, for some integer / > 1. We present our main result in Theorem 2.7. In the following we give a brief introduction to the idea of ordinary characters. Let (iž, F, k ) be a 2-modular system. That means R is a complete discrete valuation ring of characteristic 0, with field of fractions F, unique maximal ideal (7r) and residue field R/(7t) = k. Without loss of generality we assume that F is contained in the field C of complex numbers and furthermore that R contains all ( q l)-st roots of unity. Next let M be a &G-module. If there exists an i?G-module V such that Vo :=

Structure of nodal algebras

The structure of nodal algebras over a complete discrete valuation ring with algebraically closed residue field is described.