On Schneider’s Continued Fraction Map on a Complete Non-Archimedean Field

Let $${\mathcal {M}}$$ M denote the maximal ideal of the ring of integers of a non-Archimedean field K with residue class field k whose invertible elements, we denote $$k^{\times }$$ k × , and a uniformizer we denote $$\pi $$ π . In this paper, we consider the map $$T_{v}: {\mathcal {M}} \rightarrow {\mathcal {M}}$$ T v : M → M defined by $$\begin{aligned} T_v(x) = \frac{\pi ^{v(x)}}{x} - b(x), \end{aligned}$$ T v ( x ) = π v ( x ) x - b ( x ) , where b(x) denotes the equivalence class to which $$\frac{\pi ^{v(x)}}{x}$$ π v ( x ) x belongs in $$k^{\times }$$ k × . We show that $$T_v$$ T v preserves Haar measure $$\mu $$ μ on the compact abelian topological group $${\mathcal {M}}$$ M . Let $${\mathcal {B}}$$ B denote the Haar $$\sigma $$ σ -algebra on $${\mathcal {M}}$$ M . We show the natural extension of the dynamical system $$({\mathcal {M}}, {\mathcal {B}}, \mu , T_v)$$ ( M , B , μ , T v ) is Bernoulli and has entropy $$\frac{\#( k)}{\#( k^{\times })}\log (\#( k))$$ # ( k ) # ( k × ) log ( # ( k ) ) . The first of these two properties is used to study the average behaviour of the convergents arising from $$T_v$$ T v . Here for a finite set A its cardinality has been denoted by $$\# (A)$$ # ( A ) . In the case $$K = {\mathbb {Q}}_p$$ K = Q p , i.e. the field of p-adic numbers, the map $$T_v$$ T v reduces to the well-studied continued fraction map due to Schneider.

Homology of Powers of Ideals: Artin-Rees Numbers of Syzygies and the Golod Property

Let R0 be a Noetherian local ring and R a standard graded R0-algebra with maximal ideal 𝔪 and residue class field 𝕂 = R/𝔪. For a graded ideal I in R we show that for k ≫ 0: (1) the Artin-Rees number of the syzygy modules of Ik as submodules of the free modules from a free resolution is constant, and thereby the Artin-Rees number can be presented as a proper replacement of regularity in the local situation; and (2) R is a polynomial ring over the regular R0, the ring R/Ik is Golod, its Poincaré-Betti series is rational and the Betti numbers of the free resolution of 𝕂 over R/Ik are polynomials in k of a specific degree. The first result is an extension of the work by Swanson on the regularity of Ik for k ≫ 0 from the graded situation to the local situation. The polynomiality consequence of the second result is an analog of the work by Kodiyalam on the behaviour of Betti numbers of the minimal free resolution of R/Ik over R.

DEFINABLE HENSELIAN VALUATION RINGS

We give model theoretic criteria for the existence of ∃∀ and ∀∃- formulas in the ring language to define uniformly the valuation rings ${\cal O}$ of models $\left( {K,\,{\cal O}} \right)$ of an elementary theory Σ of henselian valued fields. As one of the applications we obtain the existence of an ∃∀-formula defining uniformly the valuation rings ${\cal O}$ of valued henselian fields $\left( {K,\,{\cal O}} \right)$ whose residue class field k is finite, pseudofinite, or hilbertian. We also obtain ∀∃-formulas φ2 and φ4 such that φ2 defines uniformly k[[t]] in k(t) whenever k is finite or the function field of a real or complex curve, and φ4 replaces φ2 if k is any number field.

Relative Brauer group and pro-p Galois group of pre-p-henselian fields

Let p be a prime number and (F, v) a valued field. In this paper, we find a presentation for the p-torsion part of the Brauer group Br (F), by means of the valuation v. We only assume that F has a primitive pth root of the unity and the residue class field has characteristic not equal to p. This result naturally leads to consider valued fields that we call pre-p-henselian fields. It concerns valuations compatible with Rp, the p-radical of the field. To be precise, Rp is the radical of the skew-symmetric pairing which associates to a pair (a, b) the class of the symbol algebra (F; a, b) in Br F. In our main result, we state that pre-p-henselian fields are precisely the fields for which the Galois group of the maximal Galois p-extension admits a particular decomposition as a free pro-p product.

Knowability

This chapter introduces the Knowability Lemma, which explains when products of Gauss sums associated to elements of a preaccordion are explicitly evaluable as polynomials in q, the order of the residue class field. It considers an episode in the cartoon associated to the short Gelfand-Tsetlin pattern and the three cases that apply according to the Knowability Lemma, two of which are maximality and knowability. Knowability is not important for the proof that Statement C implies Statement B. The chapter discusses the cases where ε‎ is Class II or Class I, leaving the remaining two cases to the reader. It also describes the variant of the argument for the case that ε‎ is of Class I, again leaving the two other cases to the reader.

Zeros of systems of 𝔭-adic quadratic forms

We show that a system of r quadratic forms over a 𝔭-adic field in at least 4r+1 variables will have a non-trivial zero as soon as the cardinality of the residue field is large enough. In contrast, the Ax–Kochen theorem [J. Ax and S. Kochen, Diophantine problems over local fields. I, Amer. J. Math. 87 (1965), 605–630] requires the characteristic to be large in terms of the degree of the field over ℚp. The proofs use a 𝔭-adic minimization technique, together with counting arguments over the residue class field, based on considerations from algebraic geometry.

Direct Summands of Syzygy Modules of the Residue Class Field

Let R be a commutative Noetherian local ring. This paper deals with the problem asking whether R is Gorenstein if the nth syzygy module of the residue class field of R has a non-trivial direct summand of finite G-dimension for some n. It is proved that if n is at most two then it is true, and moreover, the structure of the ring R is determined essentially uniquely.

The equation y′ = fy in zero residue characteristic

Let K be an algebraically closed field complete with respect to an ultrametric absolute value |.| and let k be its residue class field. We assume k to have characteristic zero (hence K has characteristic zero too).Let D be a clopen bounded infraconnected set [3] in K, let R(D) be the algebra of the rational functions with no pole in D, let ‖.‖D be the norm of uniform convergence on D defined on R(D), and let H(D) be the algebra of the analytic elements on D i.e. the completion of R(D) for the norm ‖.‖D.

Minimal linkage and the Gorenstein locus of an ideal

Let I be a Cohen-Macaulay ideal of grade g > 0 in a local Gorenstein ring (R, m) with residue class field k. An R-ideal J is said to be linked to I with respect to the regular sequence α = α1 …, αg ⊂ I ∩ J if J = (α): I and I = (α): J ([6]). In this paper we are concerned with the following question: how big is dimk ((α, mJ)/mJ)? Obviously this dimension is at most g, but it could be as small as 0. If it is g then the link from J to I is called a minimal link, which is in most respects the desired type of link. The only general result known in this direction is that if I is Gorenstein, then dimk {(α, mJ)/mJ) = g unless both I and J are complete intersections (see [1], Proposition 5.2). We are able to generalize this fact to the case where (R/I)p is Gorenstein for all prime ideals p in R/I with dim (R/J)p ≤ 4; however we have to assume that I is generically a complete intersection ideal, and that R is a complete intersection (Theorem 2.3). Without the assumption on R we prove that if I is generically a complete intersection, and if for a fixed integer r the type of (R/I)p is at most r for all prime ideals p in R/I with dim (R/I)P ≤ (r + 1)2, then dimk ((α, mJ/mJ)) ≥ g — r (Proposition 2.1). If r = 1, i.e. if R/I is Gorenstein in codimension 4, then this estimate shows the dimension is at least g — 1. Theorem 2.3 can also be interpreted to yield a strong upper bound for the codimension of the non-Gorenstein-locus of certain perfect ideals: Let R be a regular local ring. Let I be an R-ideal which is generically a complete intersection, and assume that I is in the even linkage class of a Gorenstein ideal (i.e., there exists a sequence of links I ~ I1 ~ I2 ~ … ~ I2n with I2n a Gorenstein ideal); then I is a Gorenstein ideal provided that {R/I)p is Gorenstein for all prime ideals p of R/I with dim (R/I)p ≤ 4 (Corollary 3.1).