Complete Discrete Valuation Ring Research Articles

The torsion in the cohomology of wild elliptic fibers

Given an elliptic fibration f:X→S over the spectrum of a complete discrete valuation ring with algebraically closed residue field, we use a Hochschild–Serre spectral sequence to express the torsion in R1f⁎OX as the first group cohomology H1(G,H0(S′,OS′)). Here, G is the Galois group of the maximal extension K′/K such that the normalization of X×SS′ induces an étale covering of X, where S′ is the normalization of S in K′. The case where S is a Dedekind scheme is easily reduced to the local case. Moreover, we generalize to higher-dimensional fibrations.

2-Blocks whose defect group is homocyclic and whose inertial quotient contains a Singer cycle

We consider a block B of a finite group with defect group D≅(C2m)n and inertial quotient E containing a Singer cycle (an element of order 2n−1). This implies E=E⋊F, where E≅C2n−1, F≤Cn, and E acts transitively on the elements in D of order 2. We classify the basic Morita equivalence classes of B over a complete discrete valuation ring O: when m=1, B is basic Morita equivalent to the principal block of one of SL2(2n)⋊F, D⋊E, or J1 (where J1 occurs only when n=3). When m>1, B is basic Morita equivalent to D⋊E.

On relative projectivity of lattices in Auslander-Reiten components for group rings

Let G be a finite group, Q a p-subgroup of G and P a Sylow p-subgroup of Let be a complete discrete valuation ring of characteristic zero with residue class field of characteristic p > 0. Suppose that the group ring is of infinite representation type and is sufficiently large. Let be a stable Auslander-Reiten component containing a Scott -lattice S(Q) with vertex Q. Then all the -lattices in have P as their vertices. Also, we show that there exists an indecomposable -lattice L with vertex P such that a kG-module is Q-projective.Communicated by J. Zhang

Nonarchimedean Analytic Cyclic Homology

Let V be a complete discrete valuation ring with fraction field F of characteristic zero and with residue field \mathbb{F} . We introduce analytic cyclic homology of complete torsion-free bornological algebras over V . We prove that it is homotopy invariant, stable, invariant under certain nilpotent extensions, and satisfies excision. We use these properties to compute it for tensor products with dagger completions of Leavitt path algebras. If R is a smooth commutative V -algebra of relative dimension 1 , then we identify the analytic cyclic homology of its dagger completion with Berthelot's rigid cohomology of R\otimes_V\mathbb{F} .

A natural Morita equivalence reduction for some blocks of G-crossed product -lattices and applications to the Clifford theory of finite group modular representation theory

Let be a complete discrete valuation ring with an algebraically closed residue field of characteristic p. Let G be a finite group and let A be a G-crossed product finitely generated -lattice.Section 1 presents a “natural Morita equivalence” reduction for some blocks of A. This setting was pioneered by E. C. Dade and extends some results of A. Hida and S. Koshitani, for example.In Section 2, we apply results in Section 1 and a basic result of Knorr to Finite Group Block Theory. In particular, we obtain a “lift from k to ” of fundamental results of Külshammer. This latter result has recently also been obtained by F. Eisele. We also present an “axiomitization” of the proof of an important result of B. Külshammer and some examples.

ARITHMETIC STRUCTURES FOR DIFFERENTIAL OPERATORS ON FORMAL SCHEMES

Let $\mathfrak{o}$ be a complete discrete valuation ring of mixed characteristic $(0,p)$ and $\mathfrak{X}_{0}$ a smooth formal $\mathfrak{o}$-scheme. Let $\mathfrak{X}\rightarrow \mathfrak{X}_{0}$ be an admissible blow-up. In the first part, we introduce sheaves of differential operators $\mathscr{D}_{\mathfrak{X},k}^{\dagger }$ on $\mathfrak{X}$, for every sufficiently large positive integer $k$, generalizing Berthelot’s arithmetic differential operators on the smooth formal scheme $\mathfrak{X}_{0}$. The coherence of these sheaves and several other basic properties are proven. In the second part, we study the projective limit sheaf $\mathscr{D}_{\mathfrak{X},\infty }=\mathop{\varprojlim }\nolimits_{k}\mathscr{D}_{\mathfrak{X},k}^{\dagger }$ and introduce its abelian category of coadmissible modules. The inductive limit of the sheaves $\mathscr{D}_{\mathfrak{X},\infty }$, over all admissible blow-ups $\mathfrak{X}$, is a sheaf $\mathscr{D}_{\langle \mathfrak{X}_{0}\rangle }$ on the Zariski–Riemann space of $\mathfrak{X}_{0}$, which gives rise to an abelian category of coadmissible modules. Analogues of Theorems A and B are shown to hold in each of these settings, that is, for $\mathscr{D}_{\mathfrak{X},k}^{\dagger }$, $\mathscr{D}_{\mathfrak{X},\infty }$, and $\mathscr{D}_{\langle \mathfrak{X}_{0}\rangle }$.

Infinitely generated pseudocompact modules for finite groups and Weiss' Theorem

One of the most beautiful results in the integral representation theory of finite groups is a theorem of A. Weiss that detects a permutation R-lattice for the finite p-group G in terms of the restriction to a normal subgroup N and the N-fixed points of the lattice, where R is a finite extension of the p-adic integers. Using techniques from relative homological algebra, we generalize Weiss' Theorem to the class of infinitely generated pseudocompact lattices for a finite p-group, allowing R to be any complete discrete valuation ring in mixed characteristic. A related theorem of Cliff and Weiss is also generalized to this class of modules. The existence of the permutation cover of a pseudocompact module is proved as a special case of a more general result. The permutation cover is explicitly described.

A hermitian analog of a quadratic form theorem of Springer

We show that the second residue map for hermitian Witt groups of an Azumaya algebra A with involution $$\tau $$ of first- or second kind over a semilocal Dedekind domain R is surjective. This proves a generalization to hermitian Witt groups of an exact sequence for Witt groups of quadratic forms due to Springer. If R is a complete discrete valuation ring and $$\tau $$ is of the first kind we show that our short exact sequence of hermitian Witt groups is split. As a corollary we prove a purity theorem for hermitian Witt groups of Azumaya algebras with involutions over a regular semilocal domain of dimension two.

Some examples of Picard groups of blocks

We calculate examples of Picard groups for 2-blocks with abelian defect groups with respect to a complete discrete valuation ring. These include all blocks with abelian 2-groups of 2-rank at most three with the exception of the principal block of J1. In particular this shows directly that all such Picard groups are finite and Picent, the group of Morita auto-equivalences fixing the centre, is trivial. These are amongst the first calculations of this kind. Further we prove some general results concerning Picard groups of blocks with normal defect groups as well as some other cases.

On the non-periodic stable Auslander-Reiten Heller component for the Kronecker algebra over a complete discrete valuation ring

We consider the Kronecker algebra $A=\mathcal{O}[X,Y]/(X^2,Y^2)$, where $\mathcal{O}$ is a complete discrete valuation ring. Since $A\otimes\kappa$ is a special biserial algebra, where $\kappa$ is the residue field of $\mathcal{O}$, one can compute a complete list of indecomposable $A\otimes \kappa$-modules. For each indecomposable $A\otimes \kappa$-module, we obtain a special kind of $A$-lattices called ``Heller lattices''. In this paper, we determine the non-periodic component of a variant of the stable Auslander--Reiten quiver for the category of $A$-lattices that contains ``Heller lattices''.

The asymptotic estimates and Hasse principle for multidimensional Waring's problem

Motivated by the asymptotic estimates and Hasse principle for multidimensional Waring's problem via the circle method, we prove for the first time that the corresponding singular series is bounded below by an absolute positive constant without any nonsingular local solubility assumption. The number of variables we need is near-optimal. By proving a more general uniform density result over certain complete discrete valuation rings with finite residue fields, we also establish uniform lower bounds for both singular series and singular integral in Fq[t]. We thus obtain asymptotic formulas and the Hasse principle for multidimensional Waring's problem in Fq[t] via a variant of the circle method.

Notes on a $p$-adic Exponential Map for the Picard Group

For proper flat schemes over complete discrete valuation rings of mixed characteristic, we construct an isomorphism of certain subgroups of the Picard group and the first cohomology group of the structure sheaf. When the Picard functor is representable and smooth, our construction recovers and gives finer information to the isomorphism coming from its formal completion. An alternative proof of an old theorem of Mattuck is given.

Isocrystals associated to arithmetic jet spaces of abelian schemes

Using Buium's theory of arithmetic differential characters, we construct a filtered F-isocrystal H(A)K associated to an abelian scheme A over a p-adically complete discrete valuation ring with perfect residue field. As a filtered vector space, H(A)K admits a natural map to the usual de Rham cohomology of A, but the Frobenius operator comes from arithmetic differential theory and is not the same as the usual crystalline one. When A is an elliptic curve, we show that H(A)K has a natural integral model H(A), which implies an integral refinement of a result of Buium's on arithmetic differential characters. The weak admissibility of H(A)K depends on the invertibility of an arithmetic-differential modular parameter. Thus the Fontaine functor associates to suitably generic A a local Galois representation of an apparently new kind.

Common hyperbolic bases for chains of alternating or quadratic lattices

We give a short and purely bilinear proof of the fact that two chains of $p$-elementary lattices with quadratic form or alternating bilinear form over the $p$-adic integers ore more generally over a complete discrete valuation ring have common hyperbolic bases. This fact, which is useful for the study of Bruhat-Tits buildings, has been proven before with different methods by Abramenko and Nebe and by Frisch.

On Picard groups of blocks of finite groups

We show that the subgroup of the Picard group of a p-block of a finite group given by bimodules with endopermutation sources modulo the automorphism group of a source algebra is determined locally in terms of the fusion system on a defect group. We show that the Picard group of a block over a complete discrete valuation ring O of characteristic zero with an algebraic closure k of Fp as residue field is a colimit of finite Picard groups of blocks over p-adic subrings of O. We apply the results to blocks with an abelian defect group and Frobenius inertial quotient, and specialise this further to blocks with cyclic or Klein four defect groups.

Regular characters of classical groups over complete discrete valuation rings

Let o be a complete discrete valuation ring with finite residue field k of odd characteristic, and let G be a symplectic or special orthogonal group scheme over o. For any ℓ∈N let Gℓ denote the ℓ-th principal congruence subgroup of G(o). An irreducible character of the group G(o) is said to be regular if it is trivial on a subgroup Gℓ+1 for some ℓ, and if its restriction to Gℓ/Gℓ+1≃Lie(G)(k) consists of characters of minimal G(kalg)-stabilizer dimension. In the present paper we consider the regular characters of such classical groups over o, and construct and enumerate all regular characters of G(o), when the characteristic of k is greater than two. As a result, we compute the regular part of their representation zeta function.

On the lifting of the Dade group

Abstract For the group of endo-permutation modules of a finite p-group, there is a surjective reduction homomorphism from a complete discrete valuation ring of characteristic 0 to its residue field of characteristic p. We prove that this reduction map always has a section which is a group homomorphism.

Equivariant motivic integration on formal schemes and the motivic zeta function

For a formal scheme over a complete discrete valuation ring with a good action of a finite group, we define equivariant motivic integration, and we prove a change of variable formula for that. To do so, we construct and examine an induced group action on the Greenberg scheme of such a formal scheme. Using this equivariant motivic integration, we define an equivariant volume Poincaré series, from which we deduce Denef and Loeser’s motivic zeta function including the action of the profinite group of roots of unity.

REDUCTIVE SUBGROUP SCHEMES OF A PARAHORIC GROUP SCHEME

Let K be the field of fractions of a complete discrete valuation ring $$ \mathcal{A} $$ with residue field k, and let G be a connected reductive algebraic group over K. Suppose $$ \mathcal{P} $$ is a parahoric group scheme attached to G. In particular, $$ \mathcal{P} $$ is a smooth affine $$ \mathcal{A} $$-group scheme having generic fiber $$ \mathcal{P} $$K = G; the group scheme $$ \mathcal{P} $$ is in general not reductive over $$ \mathcal{A} $$. If G splits over an unramified extension of K, we find in this paper a closed and reductive $$ \mathcal{A} $$-subgroup scheme $$ \mathcal{M}\subset \mathcal{P} $$ for which the special fiber $$ \mathcal{M} $$k is a Levi factor of $$ \mathcal{P} $$k. Moreover, we show that the generic fiber $$ M={\mathcal{M}}_{\mathrm{K}} $$ is a subgroup of G which is geometrically of type C(μ) – i.e., after a separable field extension, M is the identity component $$ M={C}_G^o\left(\phi \right) $$ of the centralizer of the image of a homomorphism ϕ: μn → H, where μn is the group scheme of n-th roots of unity for some n ≥ 2. For a connected and split reductive group H over any field $$ \mathcal{F} $$, the paper describes those subgroups of H which are of type C(μ).

Universal deformation rings, endo-trivial modules, and semidihedral and generalized quaternion 2-groups

Let k be a field of characteristic p>0, and let W be a complete discrete valuation ring of characteristic 0 that has k as its residue field. Suppose G is a finite group and Gab,p is its maximal abelian p-quotient group. We prove that every endo-trivial kG-module V has a universal deformation ring that is isomorphic to the group ring WGab,p. In particular, this gives a positive answer to a question raised by Bleher and Chinburg for all endo-trivial modules. Moreover, we show that the universal deformation of V over WGab,p is uniquely determined by any lift of V over W. In the case when p=2 and G is a 2-group that is either semidihedral or (generalized) quaternion, we give an explicit description of the universal deformation of every indecomposable endo-trivial kG-module V.