F-Crystals on Schemes with Constant Log Structure Compositio Mathematica, Tome 97, No 1-2 (1995), P
Total Page:16
File Type:pdf, Size:1020Kb
Load more
Recommended publications
-
8 Complete Fields and Valuation Rings
18.785 Number theory I Fall 2016 Lecture #8 10/04/2016 8 Complete fields and valuation rings In order to make further progress in our investigation of finite extensions L=K of the fraction field K of a Dedekind domain A, and in particular, to determine the primes p of K that ramify in L, we introduce a new tool that will allows us to \localize" fields. We have already seen how useful it can be to localize the ring A at a prime ideal p. This process transforms A into a discrete valuation ring Ap; the DVR Ap is a principal ideal domain and has exactly one nonzero prime ideal, which makes it much easier to study than A. By Proposition 2.7, the localizations of A at its prime ideals p collectively determine the ring A. Localizing A does not change its fraction field K. However, there is an operation we can perform on K that is analogous to localizing A: we can construct the completion of K with respect to one of its absolute values. When K is a global field, this process yields a local field (a term we will define in the next lecture), and we can recover essentially everything we might want to know about K by studying its completions. At first glance taking completions might seem to make things more complicated, but as with localization, it actually simplifies matters considerably. For those who have not seen this construction before, we briefly review some background material on completions, topological rings, and inverse limits. -
The Structure Theory of Complete Local Rings
The structure theory of complete local rings Introduction In the study of commutative Noetherian rings, localization at a prime followed by com- pletion at the resulting maximal ideal is a way of life. Many problems, even some that seem \global," can be attacked by first reducing to the local case and then to the complete case. Complete local rings turn out to have extremely good behavior in many respects. A key ingredient in this type of reduction is that when R is local, Rb is local and faithfully flat over R. We shall study the structure of complete local rings. A complete local ring that contains a field always contains a field that maps onto its residue class field: thus, if (R; m; K) contains a field, it contains a field K0 such that the composite map K0 ⊆ R R=m = K is an isomorphism. Then R = K0 ⊕K0 m, and we may identify K with K0. Such a field K0 is called a coefficient field for R. The choice of a coefficient field K0 is not unique in general, although in positive prime characteristic p it is unique if K is perfect, which is a bit surprising. The existence of a coefficient field is a rather hard theorem. Once it is known, one can show that every complete local ring that contains a field is a homomorphic image of a formal power series ring over a field. It is also a module-finite extension of a formal power series ring over a field. This situation is analogous to what is true for finitely generated algebras over a field, where one can make the same statements using polynomial rings instead of formal power series rings. -
Formal Power Series - Wikipedia, the Free Encyclopedia
Formal power series - Wikipedia, the free encyclopedia http://en.wikipedia.org/wiki/Formal_power_series Formal power series From Wikipedia, the free encyclopedia In mathematics, formal power series are a generalization of polynomials as formal objects, where the number of terms is allowed to be infinite; this implies giving up the possibility to substitute arbitrary values for indeterminates. This perspective contrasts with that of power series, whose variables designate numerical values, and which series therefore only have a definite value if convergence can be established. Formal power series are often used merely to represent the whole collection of their coefficients. In combinatorics, they provide representations of numerical sequences and of multisets, and for instance allow giving concise expressions for recursively defined sequences regardless of whether the recursion can be explicitly solved; this is known as the method of generating functions. Contents 1 Introduction 2 The ring of formal power series 2.1 Definition of the formal power series ring 2.1.1 Ring structure 2.1.2 Topological structure 2.1.3 Alternative topologies 2.2 Universal property 3 Operations on formal power series 3.1 Multiplying series 3.2 Power series raised to powers 3.3 Inverting series 3.4 Dividing series 3.5 Extracting coefficients 3.6 Composition of series 3.6.1 Example 3.7 Composition inverse 3.8 Formal differentiation of series 4 Properties 4.1 Algebraic properties of the formal power series ring 4.2 Topological properties of the formal power series -
Forschungsseminar P-Divisible Groups and Crystalline Dieudonné
Forschungsseminar p-divisible Groups and crystalline Dieudonn´etheory Organizers: Moritz Kerz and Kay R¨ulling WS 09/10 Introduction Let A be an Abelian scheme over some base scheme S. Then the kernels A[pn] of multipli- cation by pn on A (p a prime) form an inductive system of finite locally free commutative n n n+1 group schemes over S,(A[p ])n, via the inclusions A[p ] ,→ A[p ]. This inductive system is called the p-divisible (or Barsotti-Tate) group of A and is denoted A(p) = lim A[pn] (or −→ A(∞) or A[p∞]). If none of the residue characteristics of S equals p, it is the same to give A(p) or to give its Tate-module Tp(A) viewed as lisse Zp-sheaf on S´et (since in this case all the A[pn] are ´etaleover S). If S has on the other hand some characteristic p points, A(p) carries much more information than just its ´etalepart A(p)´et and in fact a lot of informations about A itself. For example we have the following classical result of Serre and Tate: Let S0 be a scheme on which p is nilpotent, S0 ,→ S a nilpotent immersion (i.e. S0 is given by a nilpotent ideal sheaf in OS) and let A0 be an Abelian scheme over S0. Then the Abelian schemes A over S, which lift A0 correspond uniquely to the liftings of the p-divisible group A0(p) of A0. It follows from this that ordinary Abelian varieties over a perfect field k admit a canonical lifting over the ring of Witt vectors W (k). -
Introduction
Chapter 1 Introduction In the late 1940s Weil posed the challenge to create a cohomology theory for algebraic varieties X over an arbitrary field k with coefficients in Z. Weil had a deep arithmetic application in mind, namely to prove a conjecture of his concerning the -function attached to a variety over a finite field: Its first part asserts the rationality of this - function, the second claims the existence of a functional equation, and the third predicts the locations of its zeros and poles. Assuming a well-developed cohomology theory with coefficients in Z, possessing in particular a Lefschetz trace formula, he indicated a proof of these conjectures, cf. [48]. As we now know, for fields k of positive characteristic one cannot hope to con- struct meaningful cohomology groups H i .X; Z/. Nevertheless Weil’s quest provided an important stimulus. Starting with Serre and then with the main driving force of Grothendieck, several good substitutes for such a theory have been constructed. Many later developments in arithmetic algebraic geometry hinge crucially on these theories. Among the most important cohomology theories for algebraic varieties over a field k are the following: If k is of characteristic zero, one may pass from k to the field of complex numbers C and consider the singular cohomology of the associated complex analytic space X.C/. Other invariants in the same situation are given by the algebraic de Rham cohomology of X. For general k and a prime ` different from the characteristic of k, there is the cohomology theory of étale `-adic sheaves due to Grothendieck et al. -
The P-Adic Numbers
The p-adic numbers Holly Green∗ 24th September 2018 Abstract This paper is the final product from my place on Imperial College's UROP programme. It provides an introduction to the p-adic numbers and their applications, based loosely on Gouv^ea's \p-adic numbers", [1]. We begin by discussing how they arise in mathem- atics, in particular emphasizing that they are both analytic and algebraic in nature. Proceeding, we explore some of the peculiar results accompanying these numbers before stating and proving Hensel's lemma, which concerns the solvability of p-adic polyno- mials. Specifically, we draw upon how Hensel's lemma allows us to determine the existence of rational solutions to a homogenous polynomial of degree 2 in 3 variables. We conclude with a thorough analysis of the p-adic aspects of the Bernoulli numbers. Contents 1 Introduction 2 1.1 An analytic approach . .2 1.2 An algebraic approach . .9 1.3 The correspondence . 10 2 Discrete valuation rings and fields 15 2.1 Localizations . 16 3 Solving polynomials in Zp 22 3.1 Hensel's lemma . 22 3.2 Local and global . 28 4 Analysis in Qp and its applications 35 4.1 Series . 35 4.2 The Bernoulli numbers and p-adic analyticity . 41 ∗Department of Mathematics, Imperial College London, London, SW7 6AZ, United King- dom. E-mail address: [email protected] 1 1 Introduction The p-adic numbers are most simply a field extension of Q, the rational numbers, which can be formulated in two ways, using either analytic or algebraic methods. -
Valuation Rings Definition 1 ([AK2017, 23.1]). a Discrete Valuation on a Field K Is a Surjective Function V : K × → Z Such Th
Valuation rings Definition 1 ([AK2017, 23.1]). A discrete valuation on a field K is a surjective function × v : K ! Z such that for every x; y 2 K×, (1) v(xy) = v(x) + v(y), (2) v(x + y) ≥ minfv(x); v(y)g if x 6= −y. A = fx 2 K j v(x) ≥ 0g is called the discrete valuation ring of v.A discrete valuation ring, or DVR, is a ring which is a valuation ring of a discrete valuation. Example 2. P1 n (1) The field C((t)) = f n=N ant j N 2 N; an 2 Cg of Laurent series without an essential singularity at t = 0 is the protypical example of a discrete va- P1 n lution ring. The valuation v : C((t)) ! Z sends a series f(t) = n=N ant with aN 6= 0 to N. That is, v(f) is the order of the zero of f at 0 2 C (or minus the order of the pole) when f is considered as a meromorphic function on some neighbourhood of 0 2 C. This can be generalised to any field k by defining 1 X n k((t)) = f ant j N 2 N; an 2 kg: n=N (2) Moreover, if A contains a field k in such a way that k ! A=hπi is an isomorphism for π 2 A with v(π) = 1, then one can show that there is an induced inclusion of fields K ⊂ k((t)), and the valuation of K is induced by that of k((t)). -
An Improvement of De Jong-Oort's Purity Theorem
1 An improvement of de Jong–Oort’s purity theorem Yanhong Yang Abstract. Consider an F -crystal over a noetherian scheme S. De Jong–Oort’s purity theorem states that the associated Newton poly- gons over all points of S are constant if this is true outside a subset of codimension bigger than 1. In this paper we show an improvement of the theorem, which says that the Newton polygons over all points of S have a common break point if this is true outside a subset of codimension bigger than 1. 2000 Mathematics Subject Classification: Primary 14F30; Secondary 11F80,14H30. Keywords and Phrases: F-crystal, Newton slope, Galois representa- tion. 1. introduction arXiv:1004.3090v1 [math.AG] 19 Apr 2010 De Jong–Oort’s purity theorem [1, Theorem 4.1] states that for an F -crystal over a noetherian scheme S of characteristic p the associated Newton polygons over all points of S are constant if this is true outside a subset of codimension bigger than 1. This theorem has been strengthened and generalized by Vasiu [6], who has shown that each stratum of the Newton polygon stratification defined by an F -crystal over any reduced, not necessarily noetherian Fp-scheme S is an affine S-scheme. In the case of a family of p-divisible groups, alternative proofs of the purity have been given by Oort [15] and Zink [16]. In this paper we show an improvement, which implies that for an F -crystal over a noetherian scheme S the Newton polygons over all points have a common break point if this is true outside a subset of codimension bigger than 1. -
Twisted Differential Operators and Q-Crystals Michel Gros, Bernard Le Stum, Adolfo Quirós
Twisted differential operators and q-crystals Michel Gros, Bernard Le Stum, Adolfo Quirós To cite this version: Michel Gros, Bernard Le Stum, Adolfo Quirós. Twisted differential operators and q-crystals. 2020. hal-02559467 HAL Id: hal-02559467 https://hal.archives-ouvertes.fr/hal-02559467 Preprint submitted on 30 Apr 2020 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Twisted differential operators and q-crystals Michel Gros, Bernard Le Stum & Adolfo Quirós∗ Version of April 29, 2020 Abstract We describe explicitly the q-PD-envelopes considered by Bhatt and Scholze in their recent theory of q-crystalline cohomology and explain the relation with our notion of a divided polynomial twisted algebra. Together with an interpretation of crystals on the q-crystalline site, that we call q-crystals, as modules endowed with some kind of stratification, it allows us to associate a module on the ring of twisted differential operators to any q-crystal. Contents Introduction 2 1 δ-structures 3 2 δ-rings and twisted divided powers 5 3 q-divided powers and twisted divided powers 7 4 Complete q-PD-envelopes 14 5 Complete q-PD-envelope of a diagonal embedding 17 6 Hyper q-stratifications 19 7 q-crystals 22 8 Appendix: 1-crystals vs usual crystals 24 References 25 ∗Supported by grant PGC2018-095392-B-I00 (MCIU/AEI/FEDER, UE). -
The P-Adic Numbers and Basic Theory of Valuations
THE P-ADIC NUMBERS AND BASIC THEORY OF VALUATIONS ANDREW DING Abstract. In this paper, we aim to study valuations on finite extensions of Q. These extensions fall under a special type of field called a global field. We shall also cover the topics of the Approximation Theorems and the ring of adeles, or valuation vectors. Contents 1. Valuations 1 2. Types of Valuations 2 3. Ostrowski's Theorem 4 4. Weak Approximation Theorem 5 5. Finite Residue Field 5 6. Haar Measure 7 7. Normed Spaces 8 8. Extensions of (Normalized) Valuations 9 9. Global Fields 10 10. Restricted Topological Product 11 11. Ring of Adeles 12 12. Strong Approximation Theorem 14 13. Conclusion 16 Acknowledgments 16 References 16 The paper will assume knowledge over several topics. In particular, we will as- sume basic familiarity with point-set topology, algebraic number theory, topological groups, and the notion of Haar Measure. Most of the material comes from \Algebraic Number Theory" by Cassels and Fr¨ohlich [1]. 1. Valuations Definition 1.1. A valuation on a field k is a function j · j : k ! R≥0 satisfying these axioms: (1) jαj = 0 () α = 0 (2) jαβj = jαjjβj (3) There is a constant C ≥ 1 such that j1 + αj < C whenever jαj ≤ 1. 1 2 ANDREW DING Definition 1.2. The trivial valuation on k is the following: ( 0 if α = 0; jαj = 1 if otherwise. Every field k has the trivial valuation. If k has a nontrivial valuation, we may say k is a valued field. We will usually not consider this valuation in future theorems. -
Dedekind Domains
NOTES ON DEDEKIND RINGS J. P. MAY Contents 1. Fractional ideals 1 2. The definition of Dedekind rings and DVR’s 2 3. Characterizations and properties of DVR’s 3 4. Completions of discrete valuation rings 6 5. Characterizations and properties of Dedekind rings 7 6. Ideals and fractional ideals in Dedekind rings 10 7. The structure of modules over a Dedekind ring 11 These notes record the basic results about DVR’s (discrete valuation rings) and Dedekind rings, with at least sketches of the non-trivial proofs, none of which are hard. This is standard material that any educated mathematician with even a mild interest in number theory should know. It has often slipped through the cracks of Chicago’s first year graduate program, but then we would need at least three years to cover all of the basic algebra that every educated mathematician should know. Throughout these notes, R is an integral domain with field of fractions K. 1. Fractional ideals Definition 1.1. A fractional ideal A of R is a sub R-module of K for which there is a non-zero element d of R such that dA ⊂ R. Define A−1 to be the set of all k ∈ K, including zero, such that kA ⊂ R. For fractional ideals A and B define AB to be the set of finite linear combinations of elements ab with a ∈ A and b ∈ B. Observe that AB and A−1 are fractional ideals. The set of isomorphism classes of non-zero fractional ideals is a commutative monoid with unit R under this product. -
Henselian Rings and Weierstrass Polynomials
proceedings of the american mathematical society Volume 112, Number 3, July 1991 HENSELIAN RINGS AND WEIERSTRASS POLYNOMIALS BUDH NASHIER (Communicated by Louis J. Ratliff, Jr.) Dedicated to Professor Shreeram S. Abhyankar on his sixtieth birthday Abstract. We give two characterizations of a one-dimensional Henselian do- main. If (A, Âf) is a local domain of Krull dimension at least two, or if (A, JK) is a one-dimensional Henselian local domain, then a polynomial / in A[T] is Weierstrass if and only if (Jf, T) is the only maximal ideal of A[T] that contains /. 1. Introduction Let (A, J?) be a local ring. A polynomial / in the polynomial ring A[T] is called a Weierstrass polynomial if / is monic and all the other coefficients of / are in Jf. A Weierstrass polynomial in A[T] is contained in only one maximal ideal of A[T] ; namely, (Jf, T). Does this property characterize Weierstrass polynomials? The answer to this question is affirmative in the case when A is a domain of Krull dimension at least two. In the one-dimensional case, we find a connection between Weierstrass polynomials and Henselian rings. As an interesting application of this connection, we prove that if A is a Henselian ring and if Af is a maximal ideal in the Laurent polynomial ring A[T, T~ ], then either Mr\A[T] or M DA[T~l] is a maximal ideal. In the following section, we prove two theorems giving characterizations for one-dimensional Henselian domains. In §3, we prove results about Weierstrass polynomials and establish a connection between them and Henselian domains.