The singularity space consists of all germs (X,x), with X a Noetherian scheme and x a point, wher... more The singularity space consists of all germs (X,x), with X a Noetherian scheme and x a point, where we identify two such germs if they become the same after an analytic extension of scalars. This is a Polish space for the metric given by the order to which infinitesimal neighborhoods, or jets, agree after base change. In other words, the classification of singularities up to analytic extensions of scalars is a smooth problem in the sense of descriptive set-theory. Over C, the following two classification problems up to isomorphism are now smooth: (i) analytic germs; and (ii) polarized schemes.
The class of all Artinian local rings of length at most l is A_2-elementary, axiomatised by a fin... more The class of all Artinian local rings of length at most l is A_2-elementary, axiomatised by a finite set of axioms Art_l. We show that its existentially closed models are Gorenstein, of length exactly l and their residue fields are algebraically closed, and, conversely, every existentially closed model is of this form. The theory Gor_l of all Artinian local Gorenstein rings of length l with algebraically closed residue field is model complete and the theory Art_l is companionable, with model-companion Gor_l.
An ordered structure is called o-minimalistic if it has all the first-order features of an o-mini... more An ordered structure is called o-minimalistic if it has all the first-order features of an o-minimal structure. We propose a theory, DCTC (Definable Completeness/Type Completeness), that describes many properties of o-minimalistic structures (dimension theory, monotonicity, Hardy structures, quasi-cell decomposition). Failure of cell decomposition leads to the related notion of a tame structure, and we give a criterium for an o-minimalistic structure to be tame. To any o-minimalistic structure, we can associate its Grothendieck ring, which in the non-o-minimal case is a non-trivial invariant. To study this invariant, we identify a third o-minimalistic property, the Discrete Pigeonhole Principle, which in turn allows us to define discretely valued Euler characteristics.
We show that for both the unary relation of transcendence and the finitary relation of algebraic ... more We show that for both the unary relation of transcendence and the finitary relation of algebraic independence on a field, the degree spectra of these relations may consist of any single computably enumerable Turing degree, or of those c.e. degrees above an arbitrary fixed Δ^0_2 degree. In other cases, these spectra may be characterized by the ability to enumerate an arbitrary Σ^0_2 set. This is the first proof that a computable field can fail to have a computable copy with a computable transcendence basis.
We give a canonical construction of a balanced big Cohen-Macaulay algebra for a domain of finite ... more We give a canonical construction of a balanced big Cohen-Macaulay algebra for a domain of finite type over C by taking ultraproducts of absolute integral closures in positive characteristic. This yields a new tight closure characterization of rational singularities in characteristic zero.
Abstract. Let K be an algebraically closed field endowed with a complete non-archimedean norm wit... more Abstract. Let K be an algebraically closed field endowed with a complete non-archimedean norm with valuation ring R. Let f: Y → X be a map of K-affinoid varieties. In this paper we study the analytic structure of the image f(Y) ⊂ X; such an image is a typical example of a subanalytic set. We show that the subanalytic sets are precisely the D-semianalytic sets, where D is the truncated division function first introduced by Denef and van den Dries. This result is most conveniently stated as a Quantifier Elimination result for the valuation ring R in an analytic expansion of the language of valued fields. To prove this we establish a Flattening Theorem for affinoid varieties in the style of Hironaka, which allows a reduction to the study of subanalytic sets arising from flat maps, i.e., we show that a map of affinoid varieties can be rendered flat by using only finitely many local blowing ups. The case of a flat map is then dealt with by a small extension of a result of Raynaud and Gru...
For a Noetherian local ring R, if R/a is Cohen-Macaulay, then the ideal a can be generated by at ... more For a Noetherian local ring R, if R/a is Cohen-Macaulay, then the ideal a can be generated by at most (e−2)(ν−d−1)+2 elements, where ν is the embedding dimension of R and where d and e ≥ 3 are the dimension and the multiplicity of R/a respectively. This bound is in general much sharper than the bounds given by Sally or Boratyński-Eisenbud-Rees in case a has height bigger than 2. Moreover, no Cohen-Macaulay assumption on R is required.
We show how Resolution of Singularities in characteristic p implies the decidability of the exist... more We show how Resolution of Singularities in characteristic p implies the decidability of the existential theory of Fp [[t]] in the language of discrete valuation rings, where t is a single variable and Fp the p- element eld.
We propose a suitable substitute for the classical Grothendieck ring of an algebraically closed f... more We propose a suitable substitute for the classical Grothendieck ring of an algebraically closed field, in which any quasi-projective scheme is represented with its non-reduced structure. This yields a more subtle invariant, called the schemic Grothendieck ring. In order to include open subschemes and their complements, we introduce formal motives. Although originally cast in terms of definability, everything in this paper has been phrased in a topos-theoretic framework.
For a Noetherian local domain $A$, there exists an upper bound $N_\tau(A)$ on the minimal number ... more For a Noetherian local domain $A$, there exists an upper bound $N_\tau(A)$ on the minimal number of generators of any height two ideal $I$ for which $A/I$ is Cohen-Macaulay of type $\tau$. More precisely, we may take $N_\tau(A):=(\tau+1)e_{\text{h}}(A)$, where $e_{\text{h}}(A)$ is the homological multiplicity of $A$.
Generalizing work of Smith and Hara, we give a new characterization of log-terminal singularities... more Generalizing work of Smith and Hara, we give a new characterization of log-terminal singularities for finitely generated algebras over $\mathbb C$, in terms of purity properties of ultraproducts of characteristic $p$ Frobenii. The first application is a Bout\^ot-type theorem for log-terminal singularities: given a pure morphism $Y\to X$ between affine $\mathbb Q$-Gorenstein varieties of finite type over $\mathbb C$, if $Y$ has at most a log-terminal singularities, then so does $X$. The second application is the Vanishing for Maps of Tor for log-terminal singularities: if $A\subset R$ is a Noether Normalization of a finitely generated $\mathbb C$-algebra $R$ and $S$ is a finitely generated $R$-algebra with log-terminal singularities, then the natural morphism $\operatorname{Tor}^A_i(M,R) \to \operatorname{Tor}^A_i(M,S)$ is zero, for every $A$-module $M$ and every $i\geq 1$. The final application is the Kawamata-Viehweg Vanishing Theorem for a connected projective variety $X$ of finit...
Throughout, we will fix an algebraically closed field K. Classical algebraic geometry studies var... more Throughout, we will fix an algebraically closed field K. Classical algebraic geometry studies varieties defined over K, where in this proposal, a variety means a solution set of some polynomial equation system over K. Therefore, the study of a variety V is equivalent to the study of the maximal spectrum of its coordinate ring A(V ). Grothendieck realized that a relative version of the concept of a variety, termed scheme, would allow for the infinitesimal study of varieties, and at the same time, greatly facilitate the study of algebraic families of varieties (if f : X → S is morphism of schemes, then the collection of all its fibers forms an algebraic family of varieties). To fully exploit this new point of view, the classical notion of point as a K-rational solution has to be replaced by allowing arbitrary L-rational solutions (up to some obvious congruence relation), where L is any extension field of K. Put differently, this leads to the study of all prime ideals of the coordinate...
The singularity space consists of all germs (X,x), with X a Noetherian scheme and x a point, wher... more The singularity space consists of all germs (X,x), with X a Noetherian scheme and x a point, where we identify two such germs if they become the same after an analytic extension of scalars. This is a Polish space for the metric given by the order to which infinitesimal neighborhoods, or jets, agree after base change. In other words, the classification of singularities up to analytic extensions of scalars is a smooth problem in the sense of descriptive set-theory. Over C, the following two classification problems up to isomorphism are now smooth: (i) analytic germs; and (ii) polarized schemes.
The class of all Artinian local rings of length at most l is A_2-elementary, axiomatised by a fin... more The class of all Artinian local rings of length at most l is A_2-elementary, axiomatised by a finite set of axioms Art_l. We show that its existentially closed models are Gorenstein, of length exactly l and their residue fields are algebraically closed, and, conversely, every existentially closed model is of this form. The theory Gor_l of all Artinian local Gorenstein rings of length l with algebraically closed residue field is model complete and the theory Art_l is companionable, with model-companion Gor_l.
An ordered structure is called o-minimalistic if it has all the first-order features of an o-mini... more An ordered structure is called o-minimalistic if it has all the first-order features of an o-minimal structure. We propose a theory, DCTC (Definable Completeness/Type Completeness), that describes many properties of o-minimalistic structures (dimension theory, monotonicity, Hardy structures, quasi-cell decomposition). Failure of cell decomposition leads to the related notion of a tame structure, and we give a criterium for an o-minimalistic structure to be tame. To any o-minimalistic structure, we can associate its Grothendieck ring, which in the non-o-minimal case is a non-trivial invariant. To study this invariant, we identify a third o-minimalistic property, the Discrete Pigeonhole Principle, which in turn allows us to define discretely valued Euler characteristics.
We show that for both the unary relation of transcendence and the finitary relation of algebraic ... more We show that for both the unary relation of transcendence and the finitary relation of algebraic independence on a field, the degree spectra of these relations may consist of any single computably enumerable Turing degree, or of those c.e. degrees above an arbitrary fixed Δ^0_2 degree. In other cases, these spectra may be characterized by the ability to enumerate an arbitrary Σ^0_2 set. This is the first proof that a computable field can fail to have a computable copy with a computable transcendence basis.
We give a canonical construction of a balanced big Cohen-Macaulay algebra for a domain of finite ... more We give a canonical construction of a balanced big Cohen-Macaulay algebra for a domain of finite type over C by taking ultraproducts of absolute integral closures in positive characteristic. This yields a new tight closure characterization of rational singularities in characteristic zero.
Abstract. Let K be an algebraically closed field endowed with a complete non-archimedean norm wit... more Abstract. Let K be an algebraically closed field endowed with a complete non-archimedean norm with valuation ring R. Let f: Y → X be a map of K-affinoid varieties. In this paper we study the analytic structure of the image f(Y) ⊂ X; such an image is a typical example of a subanalytic set. We show that the subanalytic sets are precisely the D-semianalytic sets, where D is the truncated division function first introduced by Denef and van den Dries. This result is most conveniently stated as a Quantifier Elimination result for the valuation ring R in an analytic expansion of the language of valued fields. To prove this we establish a Flattening Theorem for affinoid varieties in the style of Hironaka, which allows a reduction to the study of subanalytic sets arising from flat maps, i.e., we show that a map of affinoid varieties can be rendered flat by using only finitely many local blowing ups. The case of a flat map is then dealt with by a small extension of a result of Raynaud and Gru...
For a Noetherian local ring R, if R/a is Cohen-Macaulay, then the ideal a can be generated by at ... more For a Noetherian local ring R, if R/a is Cohen-Macaulay, then the ideal a can be generated by at most (e−2)(ν−d−1)+2 elements, where ν is the embedding dimension of R and where d and e ≥ 3 are the dimension and the multiplicity of R/a respectively. This bound is in general much sharper than the bounds given by Sally or Boratyński-Eisenbud-Rees in case a has height bigger than 2. Moreover, no Cohen-Macaulay assumption on R is required.
We show how Resolution of Singularities in characteristic p implies the decidability of the exist... more We show how Resolution of Singularities in characteristic p implies the decidability of the existential theory of Fp [[t]] in the language of discrete valuation rings, where t is a single variable and Fp the p- element eld.
We propose a suitable substitute for the classical Grothendieck ring of an algebraically closed f... more We propose a suitable substitute for the classical Grothendieck ring of an algebraically closed field, in which any quasi-projective scheme is represented with its non-reduced structure. This yields a more subtle invariant, called the schemic Grothendieck ring. In order to include open subschemes and their complements, we introduce formal motives. Although originally cast in terms of definability, everything in this paper has been phrased in a topos-theoretic framework.
For a Noetherian local domain $A$, there exists an upper bound $N_\tau(A)$ on the minimal number ... more For a Noetherian local domain $A$, there exists an upper bound $N_\tau(A)$ on the minimal number of generators of any height two ideal $I$ for which $A/I$ is Cohen-Macaulay of type $\tau$. More precisely, we may take $N_\tau(A):=(\tau+1)e_{\text{h}}(A)$, where $e_{\text{h}}(A)$ is the homological multiplicity of $A$.
Generalizing work of Smith and Hara, we give a new characterization of log-terminal singularities... more Generalizing work of Smith and Hara, we give a new characterization of log-terminal singularities for finitely generated algebras over $\mathbb C$, in terms of purity properties of ultraproducts of characteristic $p$ Frobenii. The first application is a Bout\^ot-type theorem for log-terminal singularities: given a pure morphism $Y\to X$ between affine $\mathbb Q$-Gorenstein varieties of finite type over $\mathbb C$, if $Y$ has at most a log-terminal singularities, then so does $X$. The second application is the Vanishing for Maps of Tor for log-terminal singularities: if $A\subset R$ is a Noether Normalization of a finitely generated $\mathbb C$-algebra $R$ and $S$ is a finitely generated $R$-algebra with log-terminal singularities, then the natural morphism $\operatorname{Tor}^A_i(M,R) \to \operatorname{Tor}^A_i(M,S)$ is zero, for every $A$-module $M$ and every $i\geq 1$. The final application is the Kawamata-Viehweg Vanishing Theorem for a connected projective variety $X$ of finit...
Throughout, we will fix an algebraically closed field K. Classical algebraic geometry studies var... more Throughout, we will fix an algebraically closed field K. Classical algebraic geometry studies varieties defined over K, where in this proposal, a variety means a solution set of some polynomial equation system over K. Therefore, the study of a variety V is equivalent to the study of the maximal spectrum of its coordinate ring A(V ). Grothendieck realized that a relative version of the concept of a variety, termed scheme, would allow for the infinitesimal study of varieties, and at the same time, greatly facilitate the study of algebraic families of varieties (if f : X → S is morphism of schemes, then the collection of all its fibers forms an algebraic family of varieties). To fully exploit this new point of view, the classical notion of point as a K-rational solution has to be replaced by allowing arbitrary L-rational solutions (up to some obvious congruence relation), where L is any extension field of K. Put differently, this leads to the study of all prime ideals of the coordinate...
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