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For an arbitrary field K and a K-variety V, we introduce the étale-open topology on the set V(K) of K-points of V. This topology agrees with the Zariski topology, Euclidean topology, or valuation topology when K is separably closed, real closed, or p-adically closed, respectively. Topological properties of the étale-open topology correspond to algebraic properties of K. For example, the étale-open topology on 픸¹ (K) is not discrete if and only if K is large. As an application, we show that a large stable field is separably closed. [ABSTRACT FROM AUTHOR]

Suppose that R is a local domain with fraction field K. If R is Henselian, then the R‐adic topology over K refines the étale open topology. If R is regular, then the étale open topology over K refines the R‐adic topology. In particular, the étale open topology over L((t1,...,tn))$L((t_1,\ldots ,t_n))$ agrees with the L[[t1,...,tn]]$L[[t_1,\ldots ,t_n]]$‐adic topology for any field L and n≥1$n \ge 1$. [ABSTRACT FROM AUTHOR]

Continuing the work of Johnson (2018), we classify the pure fields and pure valued fields of dp-rank 1, up to elementary equivalence. In the process, we give a few new examples of dp-minimal fields. Specializing to the case of dp-small fields, we prove that dp-small fields and VC-minimal fields are all algebraically closed or real closed, as had been conjectured by Guingona (2014). This generalizes earlier work of Haskell and Macpherson (1994) and Macpherson et al. (2000), showing that C -minimal fields are algebraically closed, and weakly o-minimal fields are real closed. In fact, we obtain slight strengthenings of these earlier results, since we assume no compatibility between the field structure on the one hand and the C -relation or order on the other hand. We also give a new proof of Jahnke, Simon, and Walsberg's (2017) theorem that dp-minimal valued fields are henselian. Lastly, we apply the Kaplan-Scanlon-Wagner (2011) theorem (NIP fields are Artin-Schreier closed) to prove some general properties of strongly dependent valued fields. For example, strongly dependent henselian valued fields are defectless. [ABSTRACT FROM AUTHOR]

We put forward a uniform narrative that weaves together several variants of Hrushovski-Kazhdan style integral, and describe how it can facilitate the understanding of the Denef-Loeser motivic Milnor fiber and closely related objects. Our study focuses on the so-called "nonarchimedean Milnor fiber" that was introduced by Hrushovski and Loeser, and our thesis is that it is a richer embodiment of the underlying philosophy of the Milnor construction. The said narrative is first developed in the more natural complex environment, and is then extended to the real one via descent. In the process of doing so, we are able to provide more illuminating new proofs, free of resolution of singularities, of a few pivotal results in the literature, both complex and real. To begin with, the real motivic zeta function is shown to be rational, which yields the real motivic Milnor fiber; this is an analogue of the Hrushovski-Loeser construction. Then, applying T -convex integration after descent, matching the Euler characteristics of the topological Milnor fiber and the motivic Milnor fiber becomes a matter of simple computation, which is not only free of resolution of singularities as in the Hrushovski-Loeser proof, but is also free of other sophisticated algebro-geometric machineries. Finally, we also establish, in a much more intuitive manner, a new Thom-Sebastiani formula, which can be specialized to the one given by Guibert-Loeser-Merle. [ABSTRACT FROM AUTHOR]

Let (A,m) be a Cohen-Macaulay local ring and let CM(A) be the category of maximal Cohen-Macaulay A-modules. We construct T \colon CM(A) × CM(A) \× mod(A), a subfunctor of Ex1A(−,−) and use it to study properties of associated graded modules over G(A) = ⊕n≥0mn/n+1, the associated graded ring of A. As an application we give several examples of complete Cohen-Macaulay local rings A with G(A) Cohen-Macaulay and having distinct indecomposable maximal Cohen-Macaulay modules Mn with G(Mn) Cohen-Macaulay and the set {e(Mn)} bounded (here e(M) denotes multiplicity of M). [ABSTRACT FROM AUTHOR]

Abstract Let R be an excellent Henselian discrete valuation ring with algebraically closed residue field k of any characteristic. Fix integers r , d with r ≥ 2. Let X R be a regular fibred surface over Spec (R) with special fibre denoted X k , a generalised tree-like curve of genus g ≥ 2. Let L R be a line bundle on X R of degree d such that the degree of the restriction of L R on the rational components of X k is a multiple of r. In this article we prove the existence of a rank r locally free sheaf on X R of determinant L R such that it is semistable on the fibres. [ABSTRACT FROM AUTHOR]

Abstract We study subfields of large fields which are generated by infinite existentially definable subsets. We say that such subfields are existentially generated. Let L be a large field of characteristic exponent p , and let E ⊆ L be an infinite existentially generated subfield. We show that E contains L (p n) , the p n -th powers in L , for some n < ω. This generalises a result of Fehm from [4] , which shows E = L , under the assumption that L is perfect. Our method is to first study existentially generated subfields of henselian fields. Since L is existentially closed in the henselian field L ((t)) , our result follows. [ABSTRACT FROM AUTHOR]

This paper determines the Brauer p -dimension Brd p ( K ) and the absolute Brauer p -dimension abrd p ( K ) of a Henselian valued field ( K , v ) , for a prime p ≠ char ( K ˆ ) , under restrictions on the residue field K ˆ , such as the condition abrd p ( K ˆ ) = 0 . It describes the set Σ 0 of sequences abrd p ( E ) , Brd p ( E ) , p ∈ P , where P is the set of prime numbers and E runs across the class of Henselian fields with char ( E ˆ ) = 0 and a projective absolute Galois group G E ˆ . Specifically, Σ 0 contains a sequence a p , b p ∈ N ∪ { 0 , ∞ } , p ∈ P , whenever a 2 ≤ 2 b 2 and a p ≥ b p , for each p . Similar results are obtained in characteristic q > 0 . [ABSTRACT FROM AUTHOR]

Elkik established a remarkable theorem that can be applied for any noetherian henselian ring. For algebraic equations with a formal solution (restricted by some smoothness assumption), this theorem provides a solution adically close to the formal one in the base ring. In this paper, we show that the theorem would fail for some non-noetherian henselian rings. These rings do not satisfy several conditions weaker than noetherianness, such as weak proregularity (due to Grothendieck et al.) of the defining ideal. We describe the resulting pathologies. [ABSTRACT FROM AUTHOR]

A ring R is called n -coherent if every (n − 1) -presented ideal of R is n -presented. In this paper, the n -coherence of the perfect closure R ∞ is investigated. We show that if R ∞ is n -coherent, then it is normal. Also, it is shown that the perfect closure of a Noetherian reduced ring is n -coherent if and only if the perfect closure of its Henselization is n -coherent. Among other results, we study the n -coherence of R ∞ when R is a retract algebra. [ABSTRACT FROM AUTHOR]

A rigidity property for the homotopy invariant stable linear framed presheaves is established. As a consequence a variant of Gabber rigidity theorem is obtained for a cohomology theory representable in the motivic stable homotopy category by a ϕ -torsion spectrum with ϕ ∈ GW ( k ) of rank coprime to the (exponential) characteristic of the base field k . It is shown that the values of such cohomology theories at an essentially smooth Henselian ring and its residue field coincide. The result is applicable to cohomology theories representable by n -torsion spectra as well as to the ones representable by η -periodic spectra and spectra related to Witt groups. [ABSTRACT FROM AUTHOR]

Let K be a discretly henselian field whose residue field is separably closed. Answering a question raised by G. Prasad, we show that a semisimple K-group G is quasi-split if and only if it quasi-splits after a finite tamely ramified extension of K. [ABSTRACT FROM AUTHOR]

The article discusses an ax-kochen-ershov theorem for monotone differential-henselian fields. It discusses Ax-Kochen-Ershov type results for differential-henselian monotone valued differential fields with many constants. It shows how to get rid of the condition with many constants. It keeps k as a differential field characteristic 0 with a single distinguished derivation along with an ordered abelian group Γ which gives rise to the Hahn field K = k((tΓ)).

Let K be the field of fractions of a local Henselian discrete valuation ring OK of characteristic zero with perfect residue field k. Assuming potential semi-stable reduction, we show that an unramified Galois action on the second l-adic cohomology group of a K3 surface over K implies that the surface has good reduction after a finite and unramified extension. We give examples where this unramified extension is really needed. Moreover, we give applications to good reduction after tame extensions and Kuga–Satake Abelian varieties. On our way, we settle existence and termination of certain flops in mixed characteristic, and study group actions and their quotients on models of varieties. [ABSTRACT FROM AUTHOR]

We show the surjectivity of a restriction map for higher (0; 1)-cycles for a smooth projective scheme over an excellent henselian discrete valuation ring. This gives evidence for a conjecture by Kerz, Esnault and Wittenberg saying that base change holds for such schemes in general for motivic cohomology in degrees (i; d) for fixed d being the relative dimension over the base. Furthermore, the restriction map we study is related to a finiteness conjecture for the n-torsion of CH0(X), where X is a variety over a p-adic field. [ABSTRACT FROM AUTHOR]

We present two algorithms for computing hypergeometric solutions of second order linear differential operators with rational function coefficients. Our first algorithm searches for solutions of the form (1) exp ⁡ ( ∫ r d x ) ⋅ 2 F 1 ( a 1 , a 2 ; b 1 ; f ) where r , f ∈ Q ( x ) ‾ , and a 1 , a 2 , b 1 ∈ Q . It uses modular reduction and Hensel lifting. Our second algorithm tries to find solutions in the form (2) exp ⁡ ( ∫ r d x ) ⋅ ( r 0 ⋅ 2 F 1 ( a 1 , a 2 ; b 1 ; f ) + r 1 ⋅ 2 F 1 ′ ( a 1 , a 2 ; b 1 ; f ) ) where r 0 , r 1 ∈ Q ( x ) ‾ , as follows: It tries to transform the input equation to another equation with solutions of type (1) , and then uses the first algorithm. [ABSTRACT FROM AUTHOR]

LetFbe a Henselian field such thatis a global field. It is shown that ifDis anF-central division algebra such thatthenDcontains a cyclic division algebra of prime degree even thoughDcan be a noncrossed product. Here, our approach is based on the Albert’s cyclicity criterion and centralizer theorem. [ABSTRACT FROM AUTHOR]

In this paper we prove what we call Local Bézout Theorem (Theorem 3.7). It is a formal abstract algebraic version, in the frame of Henselian rings and $$\mathfrak {m}$$ -adic topology, of a well known theorem in the analytic complex case. This classical theorem says that, given an isolated point of multiplicity r as a zero of a local complete intersection, after deforming the coefficients of these equations we find in a sufficiently small neighborhood of this point exactly r isolated zeroes counted with multiplicities. Our main tools are, first the border bases [11], which turned out to be an efficient computational tool to deal with deformations of algebras. Second we use an important result of de Smit and Lenstra [7], for which there exists a constructive proof in [13]. Using these tools we find a very simple proof of our theorem, which seems new in the classical literature. [ABSTRACT FROM AUTHOR]

We give a characterization, in terms of the residue field, of those henselian valuation rings and those henselian valuation ideals that are diophantine. This characterization gives a common generalization of all the positive and negative results on diophantine henselian valuation rings and diophantine valuation ideals in the literature. We also treat questions of uniformity and we apply the results to show that a given field can carry at most one diophantine nontrivial equicharacteristic henselian valuation ring or valuation ideal. [ABSTRACT FROM AUTHOR]

Let K be the fraction field of a two-dimensional henselian, excellent, and equicharacteristic local domain. We prove a local-global principle for Galois cohomology with certain finite coefficients over K. We use classical machinery from étale cohomology theory, drawing upon an idea in Saito's work on two-dimensional local class field theory. This approach works equally well over the function field of a curve over an equi-characteristic henselian discrete valuation field, thereby giving a different proof of (a slightly generalized version of) a recent result of Harbater, Hartmann, and Krashen. We also present two applications. One is the Hasse principle for torsors under quasisplit semisimple simply connected groups without E8 factor. The other gives an explicit upper bound for the Pythagoras number of a Laurent series field in three variables. This bound is sharper than earlier estimates. [ABSTRACT FROM AUTHOR]

We develop geometry of affine algebraic varieties in $$K^{n}$$ over Henselian rank one valued fields K of equicharacteristic zero. Several results are provided including: the projection $$K^{n} \times {\mathbb {P}}^{m}(K) \rightarrow K^{n}$$ and blowups of the K-rational points of smooth K-varieties are definably closed maps; a descent property for blowups; curve selection for definable sets; a general version of the Łojasiewicz inequality for continuous definable functions on subsets locally closed in the K-topology; and extending continuous hereditarily rational functions, established for the real and p-adic varieties in our joint paper with J. Kollár. The descent property enables application of resolution of singularities and transformation to a normal crossing by blowing up in much the same way as over the locally compact ground field. Our approach relies on quantifier elimination due to Pas and a concept of fiber shrinking for definable sets, which is a relaxed version of curve selection. The last three sections are devoted to the theory of regulous functions and sets over such valued fields. Regulous geometry over the real ground field $${\mathbb {R}}$$ was developed by Fichou-Huisman-Mangolte-Monnier. The main results here are regulous versions of Nullstellensatz and Cartan's theorems A and B. [ABSTRACT FROM AUTHOR]

A henselian valued field K is called a tame field if its algebraic closure is a tame extension, that is, the ramification field of the normal extension is algebraically closed. Every algebraically maximal Kaplansky field is a tame field, but not conversely. We develop the algebraic theory of tame fields and then prove Ax-Kochen-Ershov Principles for tame fields. This leads to model completeness and completeness results relative to value group and residue field. As the maximal immediate extensions of tame fields will in general not be unique, the proofs have to use much deeper valuation theoretical results than those for other classes of valued fields which have already been shown to satisfy Ax-Kochen-Ershov Principles. The results of this paper have been applied to gain insight in the Zariski space of places of an algebraic function field, and in the model theory of large fields. [ABSTRACT FROM AUTHOR]

A henselian valued field K is called separably tame if its separable-algebraic closure K sep is a tame extension, that is, the ramification field of the normal extension K sep | K is separable-algebraically closed. Every separable-algebraically maximal Kaplansky field is a separably tame field, but not conversely. In this paper, we prove Ax–Kochen–Ershov Principles for separably tame fields. This leads to model completeness and completeness results relative to the value group and residue field. As the maximal immediate extensions of separably tame fields are in general not unique, the proofs have to use much deeper valuation theoretical results than those for other classes of valued fields which have already been shown to satisfy Ax–Kochen–Ershov Principles. Our approach also yields alternate proofs of known results for separably closed valued fields. [ABSTRACT FROM AUTHOR]

This paper studies the structure of special preenvelopes and envelopes with respect to maximal Cohen–Macaulay modules. We investigate the structure of them in terms of their kernels and cokernels. Moreover, using this result, we also study the structure of special proper coresolutions with respect to maximal Cohen–Macaulay modules over a Henselian Cohen–Macaulay local ring. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM PUBLISHER]

Let F be a Henselian field. For a finite extension K of F, the norm factor group is computed. As an application, structure of the tame Brauer group of a generalized local field is determined. In particular, we observe that every torsion divisible abelian group is realizable as the tame Brauer group of a generalized local field. [ABSTRACT FROM AUTHOR]

Let K be a Henselian field with a residue field $$ \overline{K} $$, and let A, A be finite-dimensional division unramified K-algebras with residue algebras Ā and Ā Further, let Hom(A,A) be the set of nonzero K-homomorphisms from A to A. It is proved that there is a natural bijection between the set of nonzero $$ \overline{K} $$ -homomorphisms from Ā to Ā and the factor set of Hom and the factor set of Hom(A,A) under the equivalence relation: ϕ ∼ ϕ ⇔ there exists m ∈ 1 +M such that ϕ = ϕm, where im is the inner automorphism of A induced by m. A similar result is obtained for unramified algebras with involutions. Bibliography: 7 titles. [ABSTRACT FROM AUTHOR]

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 (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 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. [ABSTRACT FROM AUTHOR]

We give an example of a parameter-free definable Henselian valuation ring which is neither definable by a parameter-free [ABSTRACT FROM AUTHOR]

We discuss definability in the language of rings without parameters of the unique canonical Henselian valuation of a field. We show that in most cases where the canonical Henselian valuation is definable, it is already definable by a universal-existential or an existential-universal formula. [ABSTRACT FROM AUTHOR]

We discuss definability of henselian valuation rings in the Macintyre language LMac, the language of rings expanded by nth power predicates. In particular, we show that henselian valuation rings with finite or Hilbertian residue field are uniformly ∃-∅-definable in LMac, and henselian valuation rings with value group Z are uniformly ∃-∅-definable in the ring language, but not uniformly ∃-∅-definable in LMac. We apply these results to local fields Qp and Fp((t)), as well as to higher dimensional local fields. [ABSTRACT FROM AUTHOR]

Admitting a non-trivial p -henselian valuation is a weaker assumption on a field than admitting a non-trivial henselian valuation. Unlike henselianity, p -henselianity is an elementary property in the language of rings. We are interested in the question when a field admits a non-trivial 0-definable p -henselian valuation (in the language of rings). We give a classification of elementary classes of fields in which the canonical p -henselian valuation is uniformly 0-definable. We then apply this to show that there is a definable valuation inducing the ( t -)henselian topology on any ( t -)henselian field which is neither separably closed nor real closed. [ABSTRACT FROM AUTHOR]

An important result of Ito and Izumi about a Diophantine inequality and the linearity of the Artin approximation function has been generalized recently by Hickel, Ito and Izumi for any excellent Henselian local domain. Our objective is to show that their results are also true in the case when R is an analytically irreducible Henselian local domain (without the excellence hypothesis). [ABSTRACT FROM AUTHOR]

Let v be a henselian valuation of arbitrary rank of a field K with value group G(K) and residue field R(K) and be the unique extension of v to a fixed algebraic closure of K with value group . It is known that a complete distinguished chain for an element θ belonging to with respect to (K, v) gives rise to several invariants associated to θ, including a chain of subgroups of , a tower of fields, together with a sequence of elements belonging to which are the same for all K-conjugates of θ. These invariants satisfy some fundamental relations. In this paper, we deal with the converse: Given a chain of subgroups of containing G(K), a tower of extension fields of R(K), and a finite sequence of elements of satisfying certain properties, it is shown that there exists a complete distinguished chain for an element associated to these invariants. We use the notion of lifting of polynomials to construct it. [ABSTRACT FROM AUTHOR]

We extend the formalism and results on motivic integration from [Invent. Math. 173 (2008), 23-121] to mixed characteristic discretely valued Henselian fields with bounded ramification. We also generalize the equicharacteristic zero case of loc. cit. by giving, in all residue characteristics, an axiomatic approach (instead of only using Denef-Pas languages) and by using richer angular component maps. In this setting we prove a general change of variables formula and a general Fubini Theorem. Our set-up can be specialized to previously known versions of motivic integration by e.g. the second author and J. Sebag and to classical p-adic integrals. [ABSTRACT FROM AUTHOR]

In [1], Anscombe and Koenigsmann give an existential ∅-definition of the ring of formal power series F[[t]] in its quotient field in the case where F is finite. We extend their method in several directions to give general definability results for henselian valued fields with finite or pseudo-algebraically closed residue fields. [ABSTRACT FROM PUBLISHER]

In this note we investigate the question when a henselian valued field carries a nontrivial ∅-definable henselian valuation (in the language of rings). This is clearly not possible when the field is either separably or real closed, and, by the work of Prestel and Ziegler, there are further examples of henselian valued fields which do not admit a ∅-definable nontrivial henselian valuation. We give conditions on the residue field which ensure the existence of a parameter-free definition. In particular, we show that a henselian valued field admits a nontrivial henselian ∅-definable valuation when the residue field is separably closed or sufficiently nonhenselian, or when the absolute Galois group of the (residue) field is nonuniversal. [ABSTRACT FROM PUBLISHER]

NTP2 is a large class of first-order theories defined by Shelah generalizing simple and NIP theories. Algebraic examples of NTP2 structures are given by ultra-products of p-adics and certain valued difference fields (such as a non-standard Frobenius automorphism living on an algebraically closed valued field of characteristic 0). In this note we present some results on groups and fields definable in NTP2 structures. Most importantly, we isolate a chain condition for definable normal subgroups and use it to show that any NTP2 field has only finitely many Artin-Schreier extensions. We also discuss a stronger chain condition coming from imposing bounds on burden of the theory (an appropriate analogue of weight) and show that every strongly dependent valued field is Kaplansky. [ABSTRACT FROM AUTHOR]

NTP2 is a large class of first-order theories defined by Shelah generalizing simple and NIP theories. Algebraic examples of NTP2 structures are given by ultra-products of p-adics and certain valued difference fields (such as a non-standard Frobenius automorphism living on an algebraically closed valued field of characteristic 0). In this note we present some results on groups and fields definable in NTP2 structures. Most importantly, we isolate a chain condition for definable normal subgroups and use it to show that any NTP2 field has only finitely many Artin-Schreier extensions. We also discuss a stronger chain condition coming from imposing bounds on burden of the theory (an appropriate analogue of weight) and show that every strongly dependent valued field is Kaplansky. [ABSTRACT FROM AUTHOR]

Henselian elements are roots of polynomials which satisfy the conditions of Hensel's Lemma. In this paper we prove that for a finite field extension ( F | L , v ) , if F is contained in the absolute inertia field of L , then the valuation ring O F of ( F , v ) is generated as an O L -algebra by henselian elements. Moreover, we give a list of equivalent conditions under which O F is generated over O L by finitely many henselian elements. We prove that if the chain of prime ideals of O L is well-ordered, then these conditions are satisfied. We give an example of a finite valued inertial extension ( F | L , v ) for which O F is not a finitely generated O L -algebra. We also present a theorem that relates the problem of local uniformization to the theory of henselian elements. [ABSTRACT FROM AUTHOR]

LetRbe a commutative Noetherian ring andAan ArtinianR-module. We prove that ifAhas finite Gorenstein injective dimension, thenApossesses a Gorenstein injective envelope which is special and Artinian. This, in particular, yields that over a Gorenstein ring any Artinian module possesses a Gorenstein injective envelope which is special and Artinian. [ABSTRACT FROM AUTHOR]

We define 't-stratifications', a strong notion of stratifications for Henselian-valued fields [ABSTRACT FROM AUTHOR]

Abstract: In this article, we define and study the discriminant of symplectic graded involutions on non-inertially split graded simple algebras with simple 0-component. In particular, we show that if F is a graded field of characteristic different from 2, D is a graded central division algebra over F with and (see the preliminaries below), , and σ is a graded involution of symplectic type on A, then there is only a finite number of values for the discriminants , where τ describes all graded involutions of symplectic type on A (see Proposition 2.11). Consequently, for any graded central simple algebra C over F with simple non-split, , and even, we have for any graded involutions of symplectic type σ and τ on C (see Corollary 2.12). We prove also that if E is a Henselian valued field with residue characteristic different from 2, D is a central division algebra of exponent 2 over E with , and with n even, then for any symplectic involutions σ, τ on B, preserving a tame gauge defined on B, we have (see Corollary 3.5). [Copyright &y& Elsevier]

On a Henselian valued field (K, V), where V is the valuation ring, if the value group contains a convex p-regular subgroup that is not p-divisible, then V is definable in the language of rings. A Henselian valuation ring with a regular non-divisible value group is always 0-definable. In particular, some results of Ax and of Konenigmann are generalized. [ABSTRACT FROM AUTHOR]

Abstract: We give a definition, in the ring language, of inside and of inside , which works uniformly for all p and all finite field extensions of these fields, and in many other Henselian valued fields as well. The formula can be taken existential-universal in the ring language, and in fact existential in a modification of the language of Macintyre. Furthermore, we show the negative result that in the language of rings there does not exist a uniform definition by an existential formula and neither by a universal formula for the valuation rings of all the finite extensions of a given Henselian valued field. We also show that there is no existential formula of the ring language defining inside uniformly for all p. For any fixed finite extension of , we give an existential formula and a universal formula in the ring language which define the valuation ring. [Copyright &y& Elsevier]

Let K be the field of fractions of a Henselian discrete valuation ring ${{\mathcal {O}}_{K}}$. Let X/ K be a smooth proper geometrically connected scheme admitting a regular model $X/{{\mathcal {O}}_{K}}$. We show that the index δ( X/ K) of X/ K can be explicitly computed using data pertaining only to the special fiber X/ k of the model X. We give two proofs of this theorem, using two moving lemmas. One moving lemma pertains to horizontal 1-cycles on a regular projective scheme X over the spectrum of a semi-local Dedekind domain, and the second moving lemma can be applied to 0-cycles on an $\operatorname {FA} $-scheme X which need not be regular. The study of the local algebra needed to prove these moving lemmas led us to introduce an invariant γ( A) of a singular local ring $(A, {\mathfrak {m}})$: the greatest common divisor of all the Hilbert-Samuel multiplicities e( Q, A), over all ${\mathfrak {m}}$-primary ideals Q in ${\mathfrak {m}}$. We relate this invariant γ( A) to the index of the exceptional divisor in a resolution of the singularity of $\operatorname {Spec}A$, and we give a new way of computing the index of a smooth subvariety X/ K of ${\mathbb{P}}^{n}_{K}$ over any field K, using the invariant γ of the local ring at the vertex of a cone over X. [ABSTRACT FROM AUTHOR]