Complete Discrete Valuation Ring Research Articles

Specialization map between stratified bundles and pro-étale fundamental group

Given a projective family of semi-stable curves over a complete discrete valuation ring of characteristic p>0 with algebraically closed residue field, we construct a specialization functor between the category of continuous representations of the pro-étale fundamental group of the closed fibre and the category of stratified bundles on the geometric generic fibre. By Tannakian duality, this functor induces a morphism between the corresponding affine group schemes. We show that this morphism is a lifting of the specialization map, constructed by Grothendieck, between the étale fundamental groups.

Algèbres de distributions et $\mathcal D$-modules arithmétiques

Let p be a prime number, V a complete discrete valuation ring of unequal caracteristics (0,p) , G a smooth affine algebraic group over Spec V . Using partial divided powers techniques of Berthelot, we construct arithmetic distribution algebras, with level m , generalizing the classical construction of the distribution algebra. We also construct the weak completion of the classical distribution algebra over a finite extension K of \mathbf Q_p . We then show that these distribution algebras can be identified with invariant arithmetic differential operators over G , and prove a coherence result when the ramification index of K is < p-1 .

Galois descent of semi-affinoid spaces

We study the Galois descent of semi-affinoid non-archimedean analytic spaces. These are the non-archimedean analytic spaces which admit an affine special formal scheme as model over a complete discrete valuation ring, such as for example open or closed polydiscs or polyannuli. Using Weil restrictions and Galois fixed loci for semi-affinoid spaces and their formal models, we describe a formal model of a $K$-analytic space $X$, provided that $X\otimes_KL$ is semi-affinoid for some finite tamely ramified extension $L$ of $K$. As an application, we study the forms of analytic annuli that are trivialized by a wide class of Galois extensions that includes totally tamely ramified extensions. In order to do so, we first establish a Weierstrass preparation result for analytic functions on annuli, and use it to linearize finite order automorphisms of annuli. Finally, we explain how from these results one can deduce a non-archimedean analytic proof of the existence of resolutions of singularities of surfaces in characteristic zero.

Notes on the local p-adic Simpson correspondence

We complete the theory of the local p-adic Simpson correspondence developed by Faltings for rational coefficients. It asserts that there is an equivalence between the category of small \(\mathbb {Q}_p\)-generalized representations of the geometric fundamental group and that of small \(\mathbb {Q}_p\)-Higgs bundles for a certain kind of log smooth affine scheme over a complete discrete valuation ring of mixed characteristic (0, p) with perfect residue field. The difficulty lies in the construction of the latter from the former, and we give it via a generalized Sen’s theory for the log smooth affine scheme, which depends on Faltings’ almost purity theorem. Inspired by a recent work by R. Liu and X. Zhu, we also give a formulation of local p-adic Simpson correspondence for \(\mathbb {Q}_p\)-generalized representations of the arithmetic fundamental group, and a characterization of Hodge-Tate generalized representations in terms of the correspondence.

Weak completions, bornologies and rigid cohomology

Let $V$ be a complete discrete valuation ring with residue field $k$ of positive characteristic and with fraction field $K$ of characteristic 0. We clarify the analysis behind the Monsky--Washnitzer completion of a commutative $V$-algebra using completions of bornological $V$-algebras. This leads us to a functorial chain complex for a finitely generated commutative algebra over the residue field $k$ that computes its rigid cohomology in the sense of Berthelot.

Integrable Derivations and Stable Equivalences of Morita Type

AbstractUsing that integrable derivations of symmetric algebras can be interpreted in terms of Bockstein homomorphisms in Hochschild cohomology, we show that integrable derivations are invariant under the transfer maps in Hochschild cohomology of symmetric algebras induced by stable equivalences of Morita type. With applications in block theory in mind, we allow complete discrete valuation rings of unequal characteristic.

Effective Faithful Tropicalizations Associated to Adjoint Linear Systems

AbstractLet R be a complete discrete valuation ring of equi-characteristic zero with fraction field K. Let X be a connected smooth projective variety of dimension d over K, and let L be an ample line bundle over X. We assume that there exist a regular strictly semistable model ${\mathscr {X}}$ of X over R and a relatively ample line bundle ${\mathscr {L}}$ over ${\mathscr {X}}$ with $\left .{{\mathscr {L}}}\right \vert_{{X}} \cong L$. Let $S({\mathscr {X}})$ be the skeleton associated to ${\mathscr {X}}$ in the Berkovich analytification Xan of X. In this article, we study when $S({\mathscr {X}})$ is faithfully tropicalized into tropical projective space by the adjoint linear system |L⊗m ⊗ ωX|. Roughly speaking, our results show that if m is an integer such that the adjoint bundle is basepoint free, then the adjoint linear system admits a faithful tropicalization of $S({\mathscr {X}})$.

On Tate duality and a projective scalar property for symmetric algebras

We identify a class of symmetric algebras over a complete discrete valuation ring $\mathcal O$ of characteristic zero to which the characterisation of Kn\"orr lattices in terms of stable endomorphism rings in the case of finite group algebras, can be extended. This class includes finite group algebras, their blocks and source algebras and Hopf orders. We also show that certain arithmetic properties of finite group representations extend to this class of algebras. Our results are based on an explicit description of Tate duality for lattices over symmetric $\mathcal O$-algebras whose extension to the quotient field of $\mathcal O$ is separable.

Motivic realizations of singularity categories and vanishing cycles

In this paper we establish a precise comparison between vanishing cycles and the singularity category of Landau-Ginzburg models over an excellent Henselian discrete valuation ring. By using noncommutative motives, we first construct a motivic-adic realization functor for dg-categories. Our main result, then asserts that, given a Landau-Ginzburg model over a complete discrete valuation ring with potential induced by a uniformizer, the-adic realization of its singularity category is given by the inertia-invariant part of vanishing cohomology. We also prove a functorial and ∞-categorical lax symmetric monoidal version of Orlov's comparison theorem between the derived category of singularities and the derived category of matrix factorizations for a Landau-Ginzburg model over a noetherian regular local ring.

Regular characters of groups of type [formula omitted] over discrete valuation rings

Let o be a complete discrete valuation ring with finite residue field k of odd characteristic. Let G be a general or special linear group or a unitary group defined over o and let g denote its Lie algebra. For every positive integer ℓ, let Kℓ be the ℓ-th principal congruence subgroup of G(o). A continuous irreducible representation of G(o) is called regular of level ℓ if it is trivial on Kℓ+1 and its restriction to Kℓ/Kℓ+1≃g(k) consists of characters with G(k‾)-stabiliser of minimal dimension. In this paper we construct the regular characters of G(o), compute their degrees and show that the latter satisfy Ennola duality. We give explicit uniform formulae for the regular part of the representation zeta functions of these groups.

ON RAMIFIED TORSION POINTS ON A CURVE WITH STABLE REDUCTION OVER AN ABSOLUTELY UNRAMIFIED BASE

Let $p$ be an odd prime number, $W$ an {\it absolutely unramified} $p$-adically complete discrete valuation ring with algebraically closed residue field, and $X$ a curve of genus at least two over the field of fractions $K$ of $W$. In the present paper, we study, under the assumption that $X$ has {\it stable reduction} over $W$, {\it torsion points} on $X$, i.e., torsion points of the Jacobian variety $J$ of $X$ which lie on the image of the Albanese embedding $X\hookrightarrow J$ with respect to a $K$-rational point of $X$. A consequence of the main result of the present paper is that if, moreover, $J$ has good reduction over $W$, then every torsion point on $X$ is {\it $K$-rational} {\it after multiplying $p$}. This result is closely related to a conjecture of {\it R. Coleman} concerning the ramification of torsion points. For instance, this result leads us to a solution of the conjecture in the case where a given curve is hyperelliptic and of genus at least $p$.

Milnor K-theory of complete discrete valuation rings with finite residue fields

Consider a complete discrete valuation ring O with quotient field F and finite residue field. Then the inclusion map O↪F induces a map Kˆ⁎MO→Kˆ⁎MF on improved Milnor K-theory. We show that this map is an isomorphism in degrees bigger or equal to 3. This implies the Gersten conjecture for improved Milnor K-theory for O. This result is new in the p-adic case.

The third homology of $$\mathrm {SL}_2$$ of local rings

We describe the third homology of \(SL_2\) of local rings, over \(\mathbb {Z}\left[ \tfrac{1}{2}\right] \), in terms of a refined Bloch group. We use this result to elucidate the relationship of this homology group to the indecomposable part of \(K_3\) of the ring, extending and generalizing recent results in the case of fields. In particular, we prove that if A is a local domain with sufficiently large (possibly finite) residue field then the natural map \(\mathrm {H}_{3}\left( {\text {SL}}_{2}\left( A\right) ,{\mathbb {Z}}\left[ \tfrac{1}{2}\right] \right) \rightarrow {K^{\mathrm { ind}}_3(A)}\left[ \tfrac{1}{2}\right] \) induces an isomorphism \(\mathrm {H}_{3}\left( {\text {SL}}_{2}\left( A\right) ,{\mathbb {Z}}\left[ \tfrac{1}{2}\right] \right) _{A^\times }\cong {K^{\mathrm { ind}}_3(A)}\left[ \tfrac{1}{2}\right] \) on coinvariants for the natural action of units \(A^\times \). We prove that the action of \(A^\times \) on \(\mathrm {H}_{3}\left( {\text {SL}}_{2}\left( A\right) ,{\mathbb {Z}}\left[ \tfrac{1}{2}\right] \right) \) is trivial when A is a complete discrete valuation ring with finite residue field of odd characteristic, and we show by example that this action is non-trivial for certain complete discrete valuation rings with infinite residue field.

ON COMPONENTS OF STABLE AUSLANDER–REITEN QUIVERS THAT CONTAIN HELLER LATTICES: THE CASE OF TRUNCATED POLYNOMIAL RINGS

Let $A$ be a truncated polynomial ring over a complete discrete valuation ring ${\mathcal{O}}$, and we consider the additive category consisting of $A$-lattices $M$ with the property that $M\otimes {\mathcal{K}}$ is projective as an $A\otimes {\mathcal{K}}$-module, where ${\mathcal{K}}$ is the fraction field of ${\mathcal{O}}$. Then, we may define the stable Auslander–Reiten quiver of the category. We determine the shape of the components of the stable Auslander–Reiten quiver that contain Heller lattices.

Canonical bundle formula and base change

We study invariants of an elliptic fibration over a complete discrete valuation ring with algebraically closed residue field. The invariants are given by the relative dualizing sheaf and the first direct image sheaf of the structure sheaf. In the studies of an elliptic surface over an algebraically closed field, the invariants appear as local invariants that determine important global invariants such as its plurigenera. We determine the invariants by investigating the change of the invariants by base change.

Ramification theory for Artin–Schreier extensions of valuation rings

The goal of this paper is to generalize and refine the classical ramification theory of complete discrete valuation rings to more general valuation rings, in the case of Artin–Schreier extensions. We define refined versions of invariants of ramification in the classical ramification theory and compare them. Furthermore, we can treat the defect case.

On an isotropy criterion for quadratic forms over function fields of curves over non-dyadic complete discrete valuation rings

On an isotropy criterion for quadratic forms over function fields of curves over non-dyadic complete discrete valuation rings

Inductive system coherence for logarithmic arithmetic ${\mathcal D}$-modules, stability for cohomology operations

Let {\mathcal V} be a complete discrete valuation ring of unequal characteristic with perfect residue field, {\mathcal P} be a smooth, quasi-compact, separated formal scheme over {\mathcal V} , {\mathcal Z} be a strict normal crossing divisor of {\mathcal P} and {\mathcal P}^\sharp:=({\mathcal P},{\mathcal Z}) the induced smooth formal log-scheme over {\mathcal V} . In Berthelot's theory of arithmetic {\mathcal D} -modules, we work with the inductive system of sheaves of rings \widehat{{\mathcal D}}^{(\bullet)}_{{\mathcal P}^\sharp}:=(\widehat{{\mathcal D}}^{(m)}_{{\mathcal P}^\sharp})_{m\in\mathbb{N}} , where \widehat{{\mathcal D}}^{(m)}_{{\mathcal P}^\sharp} is the p -adic completion of the ring of differential operators of level m over {\mathcal P}^\sharp . Moreover, he introduced the sheaf {\mathcal D}^\dag_{{\mathcal P}^\sharp,\mathbb{Q}}:= \varinjlim_m\,\widehat{{\mathcal D}}^{(m)}_{{\mathcal P}^\sharp}\otimes_{\mathbb{Z}}\mathbb{Q} of differential operators over {\mathcal P}^\sharp of finite level. In this paper, we define the notion of (over)coherence for complexes of \widehat{{\mathcal D}}^{\bullet}_{{\mathcal P}^\sharp} -modules. In this inductive system context, we prove some classical properties including that of Berthelot–Kashiwara's theorem. Moreover, when {\mathcal Z} is empty, we check this notion is compatible to that already know of (over)coherence for complexes of {\mathcal D}^\dag_{{\mathcal P},\mathbb{Q}} -modules.

LIFTING TORSION GALOIS REPRESENTATIONS

Let$p\geqslant 5$be a prime, and let${\mathcal{O}}$be the ring of integers of a finite extension$K$of$\mathbb{Q}_{p}$with uniformizer${\it\pi}$. Let${\it\rho}_{n}:G_{\mathbb{Q}}\rightarrow \mathit{GL}_{2}\left({\mathcal{O}}/({\it\pi}^{n})\right)$have modular mod-${\it\pi}$reduction$\bar{{\it\rho}}$, be ordinary at$p$, and satisfy some mild technical conditions. We show that${\it\rho}_{n}$can be lifted to an${\mathcal{O}}$-valued characteristic-zero geometric representation which arises from a newform. This is new in the case when$K$is a ramified extension of$\mathbb{Q}_{p}$. We also show that a prescribed ramified complete discrete valuation ring${\mathcal{O}}$is the weight-$2$deformation ring for$\bar{{\it\rho}}$for a suitable choice of auxiliary level. This implies that the field of Fourier coefficients of newforms of weight 2, square-free level, and trivial nebentype that give rise to semistable$\bar{{\it\rho}}$of weight 2 can have arbitrarily large ramification index at$p$.

ON IMAGES OF REAL REPRESENTATIONS OF SPECIAL LINEAR GROUPS OVER COMPLETE DISCRETE VALUATION RINGS

AbstractIn this paper, we investigate the abstract homomorphisms of the special linear group SLn($\mathfrak{O}$) over complete discrete valuation rings with finite residue field into the general linear group GLm($\mathbb{R}$) over the field of real numbers. We show that for m < 2n, every such homomorphism factors through a finite index subgroup of SLn($\mathfrak{O}$). For $\mathfrak{O}$ with positive characteristic, this result holds for all m ∈ ${\mathbb N}$.