Preprints

Submitted

Classifying singularities up to analytic extensions of scalars, submitted to LMS, 2005.
      
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 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.


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.

Uniform bounds and gauges, preprint, 2002.

(In preparation)
 Dimension and singularity theory for local rings of finite embedding dimension, preprint 2004.
           

In this paper, an algebraic theory  for local rings of finite embedding dimension is developed.  Several extensions of (Krull) dimension are proposed, which are then used to generalize singularity notions from commutative algebra.  Finally, variants of the homological theorems are shown to hold in equal characteristic.
This theory is then applied to obtain the following two results on Noetherian local rings: (i) new characterizations in terms of certain uniform behavior for a ring to be respectively analytically unramified, analytically irreducible, unmixed, quasi-unmixed, normal, Cohen-Macaulay or (in prime characteristic) pseudo-rational or weakly F-regular;  (ii) the Improved New Intersection Theorem in mixed characteristic for a ring whose residual characteristic or whose ramification index is large with respect to its dimension (and some other numerical invariants).

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'.