Publications

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Abstracts

The more recent articles have an abstract written in a TeX-patois (with some AMS-LaTeX words thrown in). Some papers are listed on ArXiv and a link is provided to that site, from which the paper can be downloaded.

Books

  1. Rigid subanalytic sets, (under construction--the plan is to eventually publish this book in the Asterisque series).

Table of contents:

Journals and Proceedings

In print


  1. On the vanishing of Tor for the absolute integral closure, (ArXiv e-print math.AC/0304051), J. Algebra 275 (2004), 567-574.
    Let $R$ be an excellent local domain of positive characteristic with residue field $k$. This paper investigates properties of $R$ in case $Tor^R_1(R^+,k)$ vanishes, where $R^+$ denotes the absolute integral closure of $R$. Such a ring is F-rational and F-pure. If $R$ has at most an isolated singularity or has dimension at most two, then $R$ is regular.

  2. Canonical big Cohen-Macaulay algebras and rational singularities, Illinois J. Math. 41 (2004), 131-150. (ArXiv e-print math.RA/0206250)
    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.

  3. Lefschetz Principle applied to symbolic powers, J. of Algebra and its Appl. 2 (2003), 177-187.
    In this paper, an alternative proof is presented of the following result on symbolic powers due to Ein-Lazarsfeld-Smith (for the affine case over $\mathbb C$) and to Hochster-Huneke (for the general case). Let $A$ be a regular ring containing a field $K$. Let $I$ be a radical ideal of $A$ and let $h$ be the maximum of the heights of its minimal primes. Then for all $n$, we have an inclusion $I^{(hn)}\subset I^n$, where the first ideal denotes the $hn$-th symbolic power of $I$. In prime characteristic, this result admits an easy tight closure proof due to Hochster-Huneke. In this paper, the characteristic zero version is obtained from this by an application of the Lefschetz Principle. The paper is entirely self-contained.

  4. Non-standard tight closure for affine $\mathbb C$-algebras,  Manu. Math., 111 (2003), 379-412.
    In this paper, non-standard tight closure is proposed as an alternative for classical tight closure on finitely generated algebras over $\mathbb C$.  It has the advantage that it admits a functional definition, similar to the characteristic $p$ definition of tight closure, where instead of the characteristic $p$ Frobenius, we use now its ultraproduct, the non-standard Frobenius.  This new closure operation $cl(A)$ on ideals $I$ of $A$, has the same properties as classical tight closure, to wit, (1) if $A$ is regular, then $I=cl(I)$; (2) if $A\subset B$ is an integral extension of domains, then $cl(IB)\cap A\subset cl(I)$; (3) if $A$ is local and $(x_1,\dots,x_n)$ is a system of parameters, then $((x_1,\dots,x_i)A:x_{i+1})$ is contained in $cl((x_1,\dots,x_i)A)$ (Colon-Capturing); (4) if $I$ is generated by $m$ elements, then $cl(I)$ is contained in the integral closure of $I$ and contains the integral closure of $I^m$ (Briancon-Skoda).

  5. Projective dimension and the singular locus, Comm. Algebra 31 (2003), 217-239.
    For a Noetherian local ring, the prime ideals in the singular locus completely determine the category of finitely generated modules up to direct summand, extensions and syzygies. From this some simple homological criteria are derived for testing whether an arbitrary module has finite projective dimension.

  6. Mixed characteristic homological theorems in low degrees,  Comp. Rend. Ac. Sci. 336 (2003), 463-466.
    Let $R$ be a locally finitely generated algebra over a discrete valuation ring $V$ of mixed characteristic. For any of the homological properties, the Direct Summand Theorem, the Monomial Theorem, the Improved New Intersection Theorem, the Vanishing of Maps of Tors and the Hochster-Roberts Theorem, we show that it holds for $R$ and possibly some other data defined over $R$, provided the residual characteristic of $V$ is sufficiently large in terms of the complexity of the data, where the complexity is primarily given in terms of the degrees of the polynomials over $V$ that define the data, but possibly also by some additional invariants.

  7. Number of generators of a Cohen-Macaulay ideal, J. Algebra 259 (2003), 235-242.
    For a Noetherian local ring $R$, if $R/I$ is Cohen-Macaulay, then the ideal $I$ can be generated by at most $(e-2)(v-d-1)+2$ elements, where $v$ is the embedding dimension of $R$ and where $d$ and $e\geq 3$ are the dimension and the multiplicity of $R/I$ respectively. This bound is in general much sharper than the bounds given by Sally or Boratynski-Eisenbud-Rees in case $I$ has height bigger than $2$. Moreover, no Cohen-Macaulay assumption on $R$ is required.

  8. Computing the minimal number of equations defining an affine curve ideal-theoretically, J. Pure Applied Alg. 177 (2003), 95-101.
    There is an algorithm which computes the minimal number of generators of the ideal of a reduced curve $C$ in affine $n$-space over an algebraically closed field $K$, provided $C$ is not a local complete intersection. The existence of such an algorithm follows from the fact that given $d$, there exists $d'\in\nat$, such that if $I$ is a height $n-1$ radical ideal in $K[X]$ with an $n$-tuple of variables, generated by polynomials of degree at most $d$, then $I$ admits a set of generators of minimal cardinality, with each generator having degree at most $d'$, except possibly when $K[X]/I$ is an (unmixed) local complete intersection.

  9. A non-standard proof of the Briancon-Skoda Theorem, Proc. Amer. Math. Soc. 131 (2003), 103-112.
    Using a tight closure argument in characteristic $p$ and then lifting the argument to characteristic zero with aid of ultraproducts, I present an elementary proof of the Brian\c{c}on-Skoda Theorem: for an $m$-generated ideal $I$ of $\mathbb C[[X_1,\dots,X_n]]$, the $m$-th power of its integral closure is contained in $I$. It is well-known that as a corollary, one gets a solution to the following classical problem. Let $f$ be a convergent power series in $n$ variables over $\mathbb C$ which vanishes at the origin. Then $f^n$ lies in the ideal generated by the partial derivatives of $f$.

  10. J. Denef and H. Schoutens, On the decidability of the existential theory of $F_p[[t]]$, Valuation Theory and its Applications, Vol. 2 (Saskatoon, 1999), 43-60, Fields Inst. Commun., 33, AMS, 2003.
    We show how Resolution of Singularities in characteristic $p$ implies the decidability of the existential theory of $\mathbb F_p[[t]]$ in the language of discrete valuation rings, where $t$ is a single variable and $\mathbb F_p$ the $p$-element field.

  11. T.S. Gardener and H. Schoutens, Rigid subanalytic sets, Valuation Theory and its Applications, Vol. 1 (Saskatoon, 1999), 141-150, Fields Inst. Commun., 32, AMS, 2002.
    Let $K$ be an algebraically closed field endowed with a complete non-archimedean norm. Let $f:Y\to X$ be a map of $K$-affinoid varieties. In this paper we study the analytic structure of the image $f(Y)\subset X$; such an image is a typical example of a subanalytic set. We show that the subanalytic sets are precisely the $\mathbf D$-semianalytic sets, where $\mathbf D$ is the truncated division function first introduced by Denef and van den Dries.
    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. More precisely, 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 Gruson showing that the image of a flat map of affinoid varieties is open in the Grothendieck topology.
    Using Embedded Resolution of Singularities, we derive in the zero characteristic case a Uniformization Theorem for subanalytic sets: a subanalytic set can be rendered semianalytic using only finitely many local blowing ups with smooth centers. As a corollary we obtain that any subanalytic set in the plane is semianalytic.

  12. Recursive sequences and flatness, Rocky Mountain J. of Math. 31, (2001), pp. 1423-1427.
    If $A\to B$ is a faithfully flat ring homomorphism of Noetherian rings and $(x_n|n)$ is a sequence of elements in $A$ satisfying a linear recursion relation with coefficients in $B$, then this sequence already satisfies such a recursion relation (of the same length) with coefficients in $A$. As a corollary, we obtain that, if $A$ and $B$ are moreover normal domains, then any power series over $A$ which is rational over $B$, is already rational over $A$.
  13. Uniform bounds in Algebraic Geometry and Commutative Algebra, in "Connections between Model Theory and Algebraic and Analytic Geometry" (ed. A. Macintyre), Quaderni di Mathematica 6, (2000), pp. 43-93.
    In this survey article, we introduce various measures of complexity for algebraic constructions in polynomial rings over fields and show how they are often uniformly bounded by the complexity of the starting data. In problems which have a linear nature, the degree of the polynomials provide a sufficient notion of complexity. However, in the non-linear case, the more sophisticated measure of etale complexity is needed.
    These bounds lead often to the constructible nature of geometric problems, where in the non-linear case, one should work in the etale site rather than in the Zariski site. As another application of the existence of these bounds we mention the possibility of transferring results from one characteristic to another by means of the Lefschetz Principle. We will give some examples of new results as well as some new proofs to old results.
  14. T.S. Gardener and H. Schoutens, Flattening and subanalytic sets in rigid analytic geometry, Proc. of Lond. Math. Soc. 83 (2001), 681-707 (ArXiv e-print link.)
    Let $K$ be an algebraically closed field endowed with a complete non-archimedean norm with valuation ring $R$. Let $f: Y\to X$ be a map of $K$-affinoid varieties. In this paper we study the analytic structure of the image $f(Y)\subset X$; such an image is a typical example of a subanalytic set. Using Embedded Resolution of Singularities, we derive in the zero characteristic case a Uniformization Theorem for subanalytic sets: after finitely many local blowing ups with smooth centers, a subanalytic set becomes semi-analytic. 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. Specifically we show that a map of affinoid varieties can be rendered flat by using only finitely many local blowing ups. The case of an image under a flat map is then dealt with by a small extension of a result of Raynaud. Our result can be conveniently stated as a Quantifier Elimination theorem for the valuation ring $R$ in an analytic expansion of the language of valued fields. This formulation is in the style of Denef and van den Dries.
  15. Artin approximation via the model theory of Cohen-Macaulay rings, Logic Colloquium '98: proceedings of the 1998 ASL Meeting held in Prague, Czech Republic (S. Buss, P. Hajek and P. Pudlak), Lect. Notes in Logic, 13 (2000), pp. 409-425.
    We show the existence of a first order theory $\text{CMM}_{d,e}$ whose Noetherian models are precisely the local Cohen-Macaulay rings of dimension $d$ and multiplicity $e$. The completion of a model of $\text{CMM}_{d,e}$ is again a model and is moreover Noetherian. If $R$ is an equicharacteristic local Gorenstein ring of dimension $d$ and multiplicity $e$ with algebraically closed residue field and if the Artin Approximation Property holds for $R$, then $R$ is an existentially closed model in the subclass of all Noetherian models of $\text{CMM}_{d,e}$. In case $R$ is moreover excellent, Spivakovski proved that the weaker Henselian assumption implies the Artin Approximation. This suggests an alternative, model theoretic strategy for proving Artin Approximation under the additional assumptions that $R$ is Gorenstein, equicharacteristic and has algebraically closed residue field.

  16. Bounds in cohomology, Israel J. of Math. 116 (2000), 125-169.
    We introduce a measure of complexity for affine algebras and their finitely generated modules, in terms of the degrees of the polynomials used in their description. We then study how various cohomological operations and numerical invariants are uniformly bounded with respect to these complexities. We apply this to give first order characterisations of certain algebraic-geometric properties. This enables us to apply the Lefschetz Principle to transfer properties between various characteristics. As an application, we obtain the following version of the Zariski-Lipman Conjecture in positive \ch: let $R$ be the local ring of a point $P$ on a hypersurface over an algebraically closed field $K$ such that the module of $K$-invariant derivatives on $R$ is free, then $P$ is a non-singular point, provided the characteristic is larger than some bound only depending on the degree of the hypersurface.

  17. Embedded resolution of singularities in rigid analytic geometry, Ann. Fac. Sci. Toulouse VIII (1999), 297-330.
    We give a rigid analytic version of Hironaka's Embedded Resolution of Singularities over an algebraically closed field of characteristic zero, complete with respect to a non-archimedean norm. This resolution is local with respect to the Grothendieck topology. The proof uses Hironaka's original result, together with an application of our analytization functor.

  18. Existentially closed models of the theory of Artinian local rings, J. Symb. Logic, 64 (1999), 825-845. (ArXiv e-print link.)
    The class of all Artinian local rings of length at most $l$ is $\forall_2$-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 $GOR_l$.

  19. Rigid analytic flatificators, Quart. J. Math. Oxford Ser. (2) 50 (1999), 321-353. (ArXiv e-print link).
    Let $K$ be an algebraically closed field endowed with a complete non-archimedean norm. Let $f: Y\to X$ be a map of $K$-affinoid varieties. We prove that for each point $x\in X$, either $f$ is flat at $x$, or there exists, at least locally around $x$, a maximal locally closed analytic subvariety $Z\subset X$ containing $x$, such that the base change $f^{-1}(Z)\to Z$ is flat at $x$, and, moreover, $g^{-1}(Z)$ has again this property in any point of the fiber of $x$ after base change over an arbitrary map $g: X'\to X$ of affinoid varieties. If we take the local blowing up $\pi:\tilde X\to X$ with this centre $Z$, then the fiber with respect to the strict transform $\tilde f$ of $f$ under $\pi$, of any point of $\tilde X$ lying above $x$, has grown strictly smaller. Among the corollaries to these results we quote, that flatness in rigid analytic geometry is local in the source; that flatness over a reduced quasi-compact rigid analytic variety can be tested by surjective families; that an inclusion of affinoid domains is flat in a point, if it is unramified in that point.

  20. The closure of rigid semianalytic sets, J. Algebra. 198 (1997), 120-134. (ArXiv e-print link.)
  21. Blowing up in rigid analytic geometry, Bull. Soc. Math. Belg. 2 (1994), 401-419.
  22. Rigid subanalytic sets in the plane, J. Algebra 170 (1994), no. 1, 266-276.
  23. Rigid subanalytic sets, Compositio Math. 94 (1994), 269-295.
  24. Uniformization of rigid subanalytic sets, Compositio Math. 94 (1994), 227-245.
  25. Approximation properties for some non-noetherian local rings, Pacific J. Math. 131 (1988), 331-359.

To appear


Preprints

Submitted

  1. Closure operations and pure subrings of regular rings, 2002.

The tight closure proof of the fact that a pure subring of an affine regular ring of characteristic $p$ is Cohen-Macaulay, is put in an axiomatic framework of ideal closure operations, which allows for a more direct argument of the same result in characteristic $0$ via non-standard tight closure. As a corollary, a self-contained and more direct proof of the Hochster-Roberts Theorem is obtained.

  1. Constructible invariants, submitted to New York J. of Math., 2002.

In this paper, a local invariant is a map $\omega$ which assigns to a local ring $R$ and a finitely generated $R$-module $M$, a value $\omega(R,M)$ in some set $\mathbb S$. For $X$ a scheme and $\mathcal F$ an $\mathcal O_X$-module, the invariant $\omega$ induces a partition of $X$ by the sets consisting of all points $x$ of $X$ for which $\omega(\mathcal O_{X,x},\mathcal F_x)$ is constant. Criteria are given for this partition to be constructible, in case $X$ is a scheme of finite type over an algebraically closed field. It follows that if the partition is constructible, then it is finite, so that the invariant takes only finitely many different values on $X$. Examples of local invariants to which these results apply, are the regularity defect, the Cohen-Macaulay defect, the Gorenstein defect, the complete intersection defect, the Betti numbers and the (twisted) Bass numbers.

As an application, we obtain that for an affine scheme $X$ of finite type over a perfect field $K$, there is a number $\delta(X)$, such that for any $n$ and any closed immersion $X\subset\mathbb A_K^n$, we can realize $X$ as the scheme-theoretic intersection of $\delta(X)+n$ hypersurfaces. Moreover, this bound $\delta(X)$ is uniform in families.

  1. Log-terminal Singularities and Vanishing Theorems via non-standard tight closure, submitted to J. of Alg. Geom. (ArXiv e-print math.AG/0303189).

Generalizing work of Smith and Hara, we give a new characterization of log-terminal singularities for finitely generated algebras over C, in terms of purity properties of ultraproducts of characteristic p Frobenii. As a first application we obtain a Boutot-type theorem for log-terminal singularities: given a pure morphism  $Y\to X$ between affine Q-Gorenstein varieties of finite type over 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 $A\subseteq R$ is a Noether Normalization of a finitely generated C-algebra $R$ and $S$ is an $R$-algebra of finite type with log-terminal singularities, then the natural morphism  $\text{Tor}^A_i(M,R) \to \text{Tor}^A_i(M,S)$ is zero, for every $A$-module $M$ and every $i\geq 1$. The final application is Kawamata-Viehweg Vanishing for  a connected projective variety $X$ of finite type over C whose affine cone has a log-terminal vertex (for some choice of polarization). As a corollary, we obtain a proof of the following conjecture of Smith: if $G$ is the complexification of a real Lie group acting algebraically on a projective smooth Fano variety $X$, then for any numerically effective line bundle $L$ on any GIT quotient $Y:=X//G$, each cohomology module $H^i(Y,L)$ vanishes for $i>0$, and, if $L$ is moreover big, then $H^i(Y,L^{-1})$ vanishes for $i<dim Y$.

  1. Asymptotic homological conjectures in mixed characteristic, (ArXiv e-print math.AC/0303383).

In this paper, various Homological Conjectures are studied for local rings which are locally finitely generated over a discrete valuation ring $V$ of mixed characteristic. Typically, we can only conclude that a particular Conjecture holds for such a ring provided the residual characteristic of $V$ is sufficiently large in terms of the complexity of the data, where the complexity is primarily given in terms of the degrees of the polynomials over $V$ that define the data, but possibly also by some additional invariants such as (homological) multiplicity. Thus asymptotic versions of the Improved New Intersection Theorem, the Monomial Conjecture, the Direct Summand Conjecture, the Hochster-Roberts Theorem and the Vanishing of Maps of Tors Conjecture are given.

That the results only hold asymptotically, is due to the fact that non-standard arguments are used, relying on the Ax-Kochen-Ershov Principle, to infer their validity from their positive characteristic counterparts. A key role in this transfer is played by the Hochster-Huneke canonical construction of big Cohen-Macaulay algebras in positive characteristic via absolute integral closures.

Drafts

  1. Absolute bounds on the number of generators of Cohen-Macaulay ideals of height at most 2, (ArXiv e-print math.RA/0304050).

For a non-degenerate Noetherian local ring $R$, there exists an upperbound $N_t=N_t(R)$ on the number of generators of a height two ideal $I$ for which $R/I$ is Cohen-Macaulay of type $t$. More precisely, one can take $N_t=(t+1)e$, where $e$ is the homological multiplicity of $R$, and if $R$ is moreover Cohen-Macaulay and equicharacteristic, we can take $e$ equal to the multiplicity of $R$.

  1. An algorithm for computing algebraic solutions, preprint, 2001.

Consider an algebraic system of equations with coefficients in a polynomial ring $K[X_1,\dots,X_n]$ over a field $K$. In general, there is no algorithm for finding (polynomial) solutions in $K[X_1,\dots,X_n]$, but there is one for finding solutions in an etale covering of $\mathbb A_K^n$. This algorithm is obtained by a uniform version of Artin Approximation; the proof uses non-standard methods. As an application, we describe an algorithm which verifies whether two schemes of finite type are isomorphic in the etale topology.

  1. (Under construction) Reduction modulo p of power series with integer coefficients, preprint, 2001.

A Uniform Strong Artin Approximation with Parameters for excellent Henselian local rings in mixed characteristic is shown. As an application of this Theorem and the Ax-Kochen-Ershov Principle, one obtains that given power series $f,g_i\in\mathbb Z[[X]]$ such that, for every prime $p$, we have that the reduction of $f$ modulo $p$ lies in the ideal of $\mathbb Z/p\mathbb Z[[X]]$ generated by the reductions of the $g_i$ modulo $p$, then there is a non-zero integer $d$ and power series $h_i$ in $\mathbb Z[1/d][[X]]$, such that $f=h_1g_1+\dots+h_sg_s$.

 

  1. 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.

  1. 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.

  1. The weak Auslander-Buchsbaum formula, preprint, 2001.

Let $R$ be a $d$-dimensional local Cohen-Macaulay ring and let $M$ be an arbitrary $R$-module. The Auslander-Buchsbaum defect of $M$ is defined to be the number $fl.dim (M)+depth(M)-d$. It is shown that this defect when finite, is always non-negative. The following conditions guarantee that the defect is zero: (i) $M$ is finitely generated over some local $R$-algebra $S$; (ii) $d=1$; (iii) $d=2$ and $M$ is separated; (iv) $d=3$ and $M$ is complete.

  1. Rational singularities and non-standard tight closure, preprint, 2002.

(Under construction)

  1. A local flatness criterion for complete modules, preprint 2001.

This paper treats various extensions of the Local Flatness Criterion over a Noetherian local ring $R$ with residue field $k$. For instance, if $M$ is a complete $R$-module of finite projective dimension, then $M$ is flat if, and only if, $Tor^R_n(M,k)=0$, for all $n=1,...,dim(R)$. As a corollary, one obtains that the completion of a flat module is again flat. If $R$ has dimension at most two, then in order for $M$ to be flat, it suffices that it is separated, that its projective dimension is finite and that $Tor^R_1(M,k)=0$.

  1. M. Aschenbrenner and H. Schoutens, Artin Approximation and Lefschetz Extensions, preprint, 2003.

(In preparation)

  1. Uniform bounds and gauges, preprint, 2002.

(In preparation)

  1. Local rings of finite embeding dimension, preprint 2004.



  2. Bounds in polynomial rings over Artinian local rings, preprint 2002.

(In preparation)

Other

* Definability, Constructibility and Transfer, Research Proposal of my current NSF Grant.

* An introduction to rigid analytic geometry, lecture notes of a graduate course on rigid analytic geometry taught at Paris VII (2002).

* Bounds in Cohomology, video and lecture notes of a talk given at MSRI in June '98.

* Approximation and subanalytic sets over a complete valuation ring, Phd. thesis, Cath. Univ. Leuven (1991).

* Flatificators, secondary Phd. thesis, Cath. Univ. Leuven (1991).

* Artin Approximation, Master thesis, Cath. Univ. Leuven (1981).