Preprints

Preprints

Ordinal length


Binary modules and their endomorphisms.

 ArXiv

Based upon properties of ordinal length, we introduce a new class of modules, the binary modules, and study their endomorphism ring. The nilpotent endomorphisms form a two-sided ideal, and after factoring this out, we get a commutative ring. In particular, any binary module without embedded primes is isomorphic to an ideal in a reduced ring.


Semi-additivity and acyclicity.

 ArXiv

We generalize the notion of length to an ordinal-valued invariant defined on the class of finitely generated modules over a Noetherian ring. A key property of this invariant is its semi-additivity on short exact sequences. As an application, we prove some general acyclicity theorems.

 

 

Schemic Grothendieck rings

Schemic Grothendieck rings I: motivic sites.
      
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.

Schemic Grothendieck rings II: jet schemes and motivic integration.
      
We generalize the notion of a jet scheme (truncated arc space) to arbitrary fat points via adjunction, and show that this yields for each fat point, an endomorphism on each schemic Grothendieck ring as defined in part I. We prove that some of the analogues for linear jets still hold true, like locally trivial fibration over the smooth locus. In this formalism, we can define several generating zeta series, motivic series, the rationality of which can now be investigated. We use the theory of jet schemes to define a local motivic integration with values in the formal Grothendieck ring.

 

The following two papers on ArXiv contain older treatment of the subject. The first of these papers is essentially the same as the two newer prepints above:


The yoga of schemic Grothendieck rings, a topos-theoretical approach, preprint, 2010.

 ArXiv

We propose a suitable substitute for the classical Grothendieck ring of an algebraically closed field, in which any quasi-projective scheme  is represented while maintaining its  non-reduced structure. This yields a more subtle invariant, called the schemic Grothendieck ring, in which we can formulate a form of integration resembling Kontsevich's motivic integration via arc schemes.  Although originally cast in terms of definability, everything  in this paper has been phrased in  a topos-theoretic framework.


The dogma of schemic Grothendieck rings, a model-theoretic approach (formerly: Schemic Grothendieck rings and motivic rationality), preprint, 2009.
 ArXiv

We propose a suitable substitute for the classical Grothendieck ring of an algebraically closed field, in which any quasi-projective scheme  is represented, while maintaining its  non-reduced structure. This yields a more subtle invariant, called the schemic Grothendieck ring, in which we can formulate a form of integration resembling Kontsevich's motivic integration via arc schemes. In view of its  more   functorial properties, we can present a characteristic-free proof of the rationality of the geometric Igusa zeta series  for certain hypersurfaces, thus generalizing the ground-breaking work on motivic integration by Denef and Loeser. The construction  uses first-order formulae, and some infinitary versions, called formularies.


O-minimalism

Prolegomena to o-minimalism: definable completeness, type completeness, and tameness.
      
An ordered structure is called o-minimalistic if it has all the first-order features of an o-minimal structure. In this paper, we concentrate on a well-known fragment $Ded$ (Definable Completeness/Type Completeness) and generalize o-minimal properties to this more general situation (dimension theory, monotonicity, Hardy structures, quasi-cell decomposition) upon replacing finiteness by discreteness in all of these. Failure of cell decomposition leads to the related notion of a tame structure.

O-minimalism.
      
An ordered structure is called o-minimalistic if it has all the first-order features of an o-minimal structure. 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 an o-minimalistic property, the Discrete Pigeonhole Principle, which in turn allows us to define discretely valued Euler characteristics. As an application, we study certain analytic subsets, called Taylor sets.

The two papers above were orginally combined in the following prepint on ArXiv:


O-minimalism, preprint, 2011.

 ArXiv

An ordered structure is called o-minimalistic if it has all the first-order features of an o-minimal structure. We propose a theory, $Ded$ (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.

 

Other preprints




Drafts



A generalization of the  Auslander-Buchsbaum formula, preprint 2005.
      

Let $R$ be a Noetherian local ring and $M$ an arbitrary $R$-module of finite depth and finite projective dimension. The flat dimension of $M$ is at least $depth(R)- depth(M)$ with equality in the following cases: (i) $M$ is finitely generated over some Noetherian local $R$-algebra $S$; (ii) $dim(R)=1$; (iii) $dim(R)=2$ and $M$ is separated;  (iv) $R$ is Cohen-Macaulay, $dim(R)=3$ and $M$ is complete.


t-minimality, preprint 2001.
      

A model theoretic minimality notion for structures with a definable topology, called t-minimality, is introduced. Cells are defined in analogy with the o-minimal or the $p$-adic case. It is shown that any definable set can be written as a finite union of cells, provided definable Skolem functions exist. This allows for the definition of the dimension of a definable set, and some basic properties of dimension are derived. In particular, dimension is preserved under definable bijections. Under some mild topological conditions on the definable topology, every definable function is continuous outside a set without interior. As a consequence, one can write the domain of the function as a union of finitely many cells, such that the restriction of the function to each such cell is continuous. Examples of t-minimal structures are o-minimal structures and $p$-adic fields, so that we recover the Cell Decomposition theorems in each of these setups.

Muchnik's proof of Tarski-Seidenberg, notes (2001).
      

These notes arose in an attempt to understand a preprint by Semenov entitled 'Decidability of the Field of Reals' regarding a proof due to A. Muchnik of the Tarski-Seidenberg algebraic quantifier elimination over the reals. The method of proof is extremely simple: it consists of determining from the coefficients of a polynomial a finite list of polynomial expressions in these coefficients, such that the knowledge of the signs of these expressions yields (in an effective way) the knowledge of the sign table of the original function. These expressions in the coefficients are obtained from the original polynomial by the Khovanskii paradigm "divide, differentiate and use Rolle's Theorem". As such this proof is truly an 'undergraduate' proof for a Theorem that without doubt belongs to the Pantheon of Mathematics. Moreover, the method extends to include an effective quantifier elimination procedure for any algebraically closed field of characteristic zero.

Ideals in local rings of finite embedding dimension, (in progress).

Topics include: primary decomposition for closed ideals, Noetherian ideals, rings of finite Krull dimension and coherence criteria.

The order dimension of a Noetherian scheme, (in progress).

Two new ordinal invariants are defined on the category of Noetherian schemes, measuring the complexity of the 'subscheme relation'.