Complete Discrete Valuation Ring Research Articles

Completed prismatic F-crystals and crystalline Zp-local systems

We introduce the notion of completed $F$-crystals on the absolute prismatic site of a smooth $p$-adic formal scheme. We define a functor from the category of completed prismatic $F$-crystals to that of crystalline étale $\mathbf {Z}_p$-local systems on the generic fiber of the formal scheme and show that it gives an equivalence of categories. This generalizes the work of Bhatt and Scholze, which treats the case of a mixed characteristic complete discrete valuation ring with perfect residue field.

The strengthened Broué abelian defect group conjecture for SL(2,pn) and GL(2,pn)

We show that each p-block of SL(2,pn) and GL(2,pn) over an arbitrary complete discrete valuation ring is splendidly Rickard equivalent to its Brauer correspondent, hence give new evidence for a refined version of Broué's abelian defect group conjecture proposed by Kessar and Linckelmann.

Milnor K-theory of p-adic rings

Abstract We study the mod p r {p^{r}} Milnor K-groups of p-adically complete and p-henselian rings, establishing in particular a Nesterenko–Suslin-style description in terms of the Milnor range of syntomic cohomology. In the case of smooth schemes over complete discrete valuation rings we prove the mod p r {p^{r}} Gersten conjecture for Milnor K-theory locally in the Nisnevich topology. In characteristic p we show that the Bloch–Kato–Gabber theorem remains true for valuation rings, and for regular formal schemes in a pro sense.

Ramification of torsion points on a curve with superspecial reduction over an absolutely unramified base

Let $p$ be a prime number, $W$ an absolutely unramified $p$-adically complete discrete valuation ring with perfect residue field, and $X$ a curve over the field of fractions of $W$ of genus greater than one. In the present paper, we study the ramification of torsion points on the curve $X$. A consequence of the main result of the present paper is nonexistence of ramified torsion point on $X$ in the case where $p$ is greater than three, the Jacobian variety $J$ of $X$ has good reduction over $W$, and the special fiber of the good model of $J$ is superspecial. This consequence generalizes a theorem proved by Coleman.

On periodic stable Auslander–Reiten components containing Heller lattices over the symmetric Kronecker algebra

Let O be a complete discrete valuation ring, K its quotient field, and A the symmetric Kronecker algebra over O. We consider the full subcategory of the category of A-lattices whose objects are A-lattices M such that M⊗OK is projective A⊗OK-modules. In this paper, we study Heller lattices of indecomposable periodic modules over A. As a main result, we determine the shapes of stable Auslander–Reiten components containing Heller lattices of indecomposable periodic modules over A.

Bökstedt periodicity and quotients of DVRs

In this paper we compute the topological Hochschild homology of quotients of discrete valuation rings (DVRs). Along the way we give a short argument for Bökstedt periodicity and generalizations over various other bases. Our strategy also gives a very efficient way to redo the computations of $\operatorname {THH}$ (respectively, logarithmic $\operatorname {THH}$) of complete DVRs originally due to Lindenstrauss and Madsen (respectively, Hesselholt and Madsen).

On tensor products and almost split sequences for Scott lattices over group rings

Let G be a finite group and Q a p-subgroup of G. Let O be a complete discrete valuation ring of characteristic zero with algebraically closed residue class field of characteristic p. We show that for an indecomposable OG-lattice L with a Q-source T and the almost split sequence A(S(Q)) terminating in the Scott lattice S(Q) with vertex Q, the tensor sequence L⊗OA(S(Q)) is a direct sum of the almost split sequence A(L) terminating in L and a split sequence if and only if L is virtually irreducible and T is a splitting trace lattice.

Purity for Barsotti–Tate groups in some mixed characteristic situations

Let $p$ be a prime. Let $R$ be a regular local ring of dimension $d\ge 2$ whose completion is isomorphic to $C(k)[[x_1,\ldots,x_d]]/(h)$, with $C(k)$ a Cohen ring with the same residue field $k$ as $R$ and with $h\in C(k)[[x_1,\ldots,x_d]]$ such that its reduction modulo $p$ does not belong to the ideal $(x_1^p,\ldots,x_d^p)+(x_1,\ldots,x_d)^{2p-2}$ of $k[[x_1,\ldots,x_d]]$. We extend a result of Vasiu-Zink (for $d=2$) to show that each Barsotti-Tate group over $\text{Frac}(R)$ which extends to every local ring of $\text{Spec}(R)$ of dimension $1$, extends uniquely to a Barsotti-Tate group over $R$. This result corrects in many cases several errors in the literature. As an application, we get that if $Y$ is a regular integral scheme such that the completion of each local ring of $Y$ of residue characteristic $p$ is a formal power series ring over some complete discrete valuation ring of absolute ramification index $e\le p-1$, then each Barsotti-Tate group over the generic point of $Y$ which extends to every local ring of $Y$ of dimension $1$, extends uniquely to a Barsotti-Tate group over $Y$.

Comparison between rigid and crystalline syntomic cohomology for strictly semistable log schemes with boundary

We introduce rigid syntomic cohomology for strictly semistable log schemes over a complete discrete valuation ring of mixed characteristic $(0,p)$ In case a good compactification exists, we compare this cohomology theory to Nekovář–Nizioł’s crystalline syntomic cohomology of the generic fibre. The main ingredients are a modification of Große-Klönne’s rigid Hyodo–Kato theory and a generalization of it for strictly semistable log schemes with boundary.

Cohen–Lenstra distributions via random matrices over complete discrete valuation rings with finite residue fields

Let (R,m) be a complete discrete valuation ring with the finite residue field R∕m= Fq. Given a monic polynomial P(t)∈R[t] whose reduction modulo m gives an irreducible polynomial P‾(t)∈Fq[t], we initiate an investigation of the distribution of coker(P(A)), where A∈Matn(R) is randomly chosen with respect to the Haar probability measure on the additive group Matn(R) of n×n R-matrices. In particular, we provide a generalization of two results of Friedman and Washington about these random matrices. We use some concrete combinatorial connections between Matn(R) and Matn(Fq) to translate our problems about a Haar-random matrix in Matn(R) into problems about a random matrix in Matn(Fq) with respect to the uniform distribution. Our results over Fq are about the distribution of the P‾-part of a random matrix A‾∈Matn(Fq) with respect to the uniform distribution, and one of them generalizes a result of Fulman. We heuristically relate our results to a celebrated conjecture of Cohen and Lenstra, which predicts that given an odd prime p, any finite abelian p-group (i.e., Zp-module) H occurs as the p-part of the class group of a random imaginary quadratic field extension of Q with a probability inversely proportional to |AutZ(H)|. We review three different heuristics for the conjecture of Cohen and Lenstra, and they are all related to special cases of our main conjecture, which we prove as our main theorems.

Non-acyclic SL2-representations of Twist Knots, -3-Dehn Surgeries, and L-functions

Abstract We study irreducible $\mathop{\textrm{SL}}\nolimits _2$-representations of twist knots. We first determine all non-acyclic $\mathop{\textrm{SL}}\nolimits _2({\mathbb{C}})$-representations, which turn out to lie on a line denoted as $x=y$ in ${\mathbb{R}}^2$. Our main tools are character variety, Reidemeister torsion, and Chebyshev polynomials. We also verify a certain common tangent property, which yields a result on $L$-functions, that is, the orders of the knot modules associated to the universal deformations. Secondly, we prove that a representation is on the line $x=y$ if and only if it factors through the $-3$-Dehn surgery, and is non-acyclic if and only if the image of a certain element is of order 3. Finally, we study absolutely irreducible non-acyclic representations $\overline{\rho }$ over a finite field with characteristic $p>2$, to concretely determine all non-trivial $L$-functions $L_{{\boldsymbol{\rho }}}$ of the universal deformations over complete discrete valuation rings. We show among other things that $L_{{\boldsymbol{\rho }}}$ $\dot{=}$ $k_n(x)^2$ holds for a certain series $k_n(x)$ of polynomials.

Morita equivalence classes of principal blocks with elementary abelian defect groups of order 64

We classify the Morita equivalence classes of principal blocks with elementary abelian defect groups of order 64 with respect to a complete discrete valuation ring with an algebraically closed residue field of characteristic two.

Morita equivalence classes of blocks with elementary abelian defect groups of order 32

We describe a general technique to classify blocks of finite groups, and we apply it to determine Morita equivalence classes of blocks with elementary abelian defect groups of order 32 with respect to a complete discrete valuation ring with an algebraically closed residue field of characteristic two. As a consequence we verify that a conjecture of Harada holds on these blocks.

On the structure of certain valued fields

In this article, we study the structure of finitely ramified mixed characteristic valued fields. For any two complete discrete valued fields K1 and K2 of mixed characteristic with perfect residue fields, we show that if the n-th residue rings are isomorphic for each n≥1, then K1 and K2 are isometric and isomorphic. More generally, for n1≥1, there is n2 depending only on the ramification indices of K1 and K2 such that any homomorphism from the n1-th residue ring of K1 to the n2-th residue ring of K2 can be lifted to a homomorphism between the valuation rings. Moreover, we get a functor from the category of certain principal Artinian local rings of length n to the category of certain complete discrete valuation rings of mixed characteristic with perfect residue fields, which naturally generalizes the functorial property of unramified complete discrete valuation rings. Our lifting result improves Basarab's relative completeness theorem for finitely ramified henselian valued fields, which solves a question posed by Basarab, in the case of perfect residue fields.

Lifting Brauer-friendly modules

Let O be a complete discrete valuation ring with a residue field k=O/J(O) of characteristic p, G a finite group, and b a block of kG with lift b^. In this paper, we show that any indecomposable Brauer-friendly kGb-module satisfying certain condition is liftable to an indecomposable Brauer-friendly OGb^-module.

THE GROTHENDIECK–SERRE CONJECTURE OVER SEMILOCAL DEDEKIND RINGS

For a reductive group scheme G over a semilocal Dedekind ring R with total ring of fractions K, we prove that no nontrivial G-torsor trivializes over K. This generalizes a result of Nisnevich–Tits, who settled the case when R is local. Their result, in turn, is a special case of a conjecture of Grothendieck–Serre that predicts the same over any regular local ring. With a patching technique and weak approximation in the style of Harder, we reduce to the case when R is a complete discrete valuation ring. Afterwards, we consider Levi subgroups to reduce to the case when G is semisimple and anisotropic, in which case we take advantage of Bruhat–Tits theory to conclude. Finally, we show that the Grothendieck–Serre conjecture implies that any reductive group over the total ring of fractions of a regular semilocal ring S has at most one reductive S-model.

On the Structure of Affine Flat Group Schemes Over Discrete Valuation Rings, II

Abstract 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.

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