Witt equivalence of algebraic function fields over real closed fields

We examine the conditions for two algebraic function fields over real closed fields to be Witt equivalent. We show that there are only two Witt classes of algebraic function fields with a fixed real closed field of constants: real and non-real ones. The first of them splits further into subclasses corresponding to the tame equivalence. This condition has a natural interpretation in terms of both: orderings (the associated Harrison isomorphism maps 1-pt fans onto 1-pt fans), and geometry and topology of associated real curves (the bijection of points is a homeomorphism and these two curves have the same number of semi-algebraically connected components). Finally, we derive some immediate consequences of those theorems. In particular we describe all the Witt classes of algebraic function fields of genus 0 and 1 over the fixed real closed field. Received: 16 February 2000; in final form: 7 December 2000 / Published online: 18 January 2002     

Rank-sets of indecomposable modules over one-dimensional rings

Let R be a one-dimensional, reduced Noetherian ring with finite normalization, and suppose there exists a positive integer N_R such that, for every indecomposable finitely generated torsion-free R-module M and every minimal prime ideal P of R, the dimension of M_P, as a vector space over the localization R_P (a field), is less than or equal to N_R. For a finitely generated torsion-free R-module M, we call the set of all such vector-space dimensions the rank-set of M. What subsets of the integers arise as rank-sets of indecomposable finitely generated torsion-free R-modules? In this article, we give more information on rank-sets of indecomposable modules, to supplement previous work concerning this question. In particular we provide examples having as rank-sets those intervals of consecutive integers that are not ruled out by an earlier article of Arnavut, Luckas and Wiegand. We also show that certain non-consecutive rank-sets never arise.     

Projective modules over overrings of polynomial rings and a question of Quillen

Let (R,m,K)$(R,\mathfrak{m},K)$ be a regular local ring containing a field k   such that either char k=0$k=0$ or char k=p$k=p$ and tr-deg K/_Fp?1$K/{\mathbb{F}}_p?1$. Let _g1,…,_gt$g_1,\dots ,g_t$ be regular parameters of R   which are linearly independent modulo m²${\mathfrak{m}}^2$. Let A=_Rg1?gt[_Y1,…,_Ym,_f1(_l1)⁻¹,…,_fn(_ln)⁻¹]$A=R_{g_1?g_t}[Y_1,\dots ,Y_m,f_1{(l_1)}^{-1},\dots ,f_n{(l_n)}^{-1}]$, where _fi(T)∈k[T]$f_i(T)\in k[T]$ and _li=_ai1Y1+?+_aimYm$l_i=a_{i1}{Y}_1+?+a_{im}{Y}_m$ with (_ai1,…,_aim)∈k^m−(0)$(a_{i1},\dots ,a_{im})\in k^m-(0)$. Then every projective A-module of rank ?t   is free. Laurent polynomial case _fi(_li)=_Yi$f_i(l_i)=Y_i$ of this result is due to Popescu.     

The vanishing ideal of a finite set of closed points in affine space

Given a finite set of closed rational points of affine space over a field, we give a Gröbner basis for the lexicographic ordering of the ideal of polynomials which vanish at all given points. Our method is an alternative to the Buchberger-Möller algorithm, but in contrast to that, we determine the set of leading terms of the ideal without solving any linear equation but by induction over the dimension of affine space. The elements of the Gröbner basis are also computed by induction over the dimension, using one-dimensional interpolation of coefficients of certain polynomials.     

Self-dual modules over local rings of curve singularities

Let C be a reduced curve singularity. C is called of finite self-dual type if there exist only finitely many isomorphism classes of indecomposable, self-dual, torsion-free modules over the local ring of C. In this paper it is shown that the singularities of finite self-dual type are those which dominate a simple plane singularity.     

Effective computation of singularities of parametric affine curves

Let k be a field of characteristic zero and f(t),g(t) be polynomials in k[t]. For a plane curve parameterized by x=f(t),y=g(t), Abhyankar developed the notion of Taylor resultant (Mathematical Surveys and Monographs, Vol. 35, American Mathematical Society, Providence, RI, 1990) which enables one to find its singularities without knowing its defining polynomial. This concept was generalized as D-resultant by Yu and Van den Essen (Proc. Amer. Math. Soc. 125(3) (1997) 689), which works over an arbitrary field. In this paper, we extend this to a curve in affine n-space parameterized by x₁=f₁(t),…,x_n=f_n(t) over an arbitrary ground field k, where f₁,…,f_n∈k[t]. This approach compares to the usual approach of computing the ideal of the curve first. It provides an efficient algorithm of computing the singularities of such parametric curves using Gröbner bases. Computational examples worked out by symbolic computation packages are included.     

Totally reflexive modules constructed from smooth projective curves of genus <Emphasis Type="Italic">g</Emphasis> ≥ 2

In this paper, from an arbitrary smooth projective curve of genus at least two, we construct a non-Gorenstein Cohen-Macaulay normal domain and a nonfree totally reflexive module over it. Received: 2 May 2006     

Set theoretic complete intersection for curves in a smooth affine algebra

Let A be a regular ring of dimension d (d≥3) containing an infinite field k. Let n be an integer such that 2n≥d+3. Let I be an ideal in A of height n and P be a projective A-module of rank n. Suppose P⊕A≈Aⁿ⁺¹ and there is a surjection α: P→I. It is proved in this note that I is a set theoretic complete intersection ideal. As a consequence, a smooth curve in a smooth affine C-algebra with trivial conormal bundle is a set theoretic complete intersection if its corresponding class in the Grothendieck group is torsion.     

A generalized Sard theorem on real closed fields

We work with semi‐algebraic functions on arbitrary real closed fields. We generalize the notion of critical values and prove a Sard type theorem in our framework.     

Connections on modules over singularities of finite CM representation type

Let A be a commutative k-algebra, where k is an algebraically closed field of characteristic 0, and let M be an A-module. We consider the following question: Under what conditions is it possible to find a connection on M?We consider the maximal Cohen-Macaulay (MCM) modules over complete CM algebras that are isolated singularities, and usually assume that the singularities have finite CM representation type. It is known that any MCM module over a simple singularity of dimension d≤2 admits an integrable connection. We prove that an MCM module over a simple singularity of dimension d≥3 admits a connection if and only if it is free. Among singularities of finite CM representation type, we find examples of curves with MCM modules that do not admit connections, and threefolds with non-free MCM modules that admit connections.Let A be a singularity not necessarily of finite CM representation type, and consider the condition that A is a Gorenstein curve or a -Gorenstein singularity of dimension d≥2. We show that this condition is sufficient for the canonical module ω_A to admit an integrable connection, and conjecture that it is also necessary. In support of the conjecture, we show that if A is a monomial curve singularity, then the canonical module ω_A admits an integrable connection if and only if A is Gorenstein.     

Cancellation of projective modules over polynomial extensions over a two-dimensional ring

Let R be a commutative Noetherian ring of dimension two with $1/2\in R$ and let $A=R[X_1,?,X_n]$. Let P be a projective A-module of rank 2. In this article, we prove that P is cancellative if ${\wedge }^2(P)\oplus A$ is cancellative.     

Strongly Gorenstein projective, injective, and flat modules

In this paper, we study a particular case of Gorenstein projective, injective, and flat modules, which we call, respectively, strongly Gorenstein projective, injective, and flat modules. These last three classes of modules give us a new characterization of the first modules, and confirm that there is an analogy between the notion of “Gorenstein projective, injective, and flat modules” and the notion of the usual “projective, injective, and flat modules”.     

Determinantal varieties over truncated polynomial rings

We study components and dimensions of higher-order determinantal varieties obtained by considering generic m×n (m?n) matrices over rings of the form F[t]/(t^k), and for some fixed r, setting the coefficients of powers of t of all r×r minors to zero. These varieties can be interpreted as spaces of (k−1)th order jets over the classical determinantal varieties; a special case of these varieties first appeared in a problem in commuting matrices. We show that when r=m, the varieties are irreducible, but when r<m, these varieties are reducible. We show that when r=2<m (any k), there are exactly ⌊k/2⌋+1 components, which we determine explicitly, and for general r<m, we show there are at least ⌊k/2⌋+1 components. We also determine the components explicitly for k=2 and 3 for all values of r (for k=3 for all but finitely many pairs of (m,n)).     

Ideals of varieties parameterized by certain symmetric tensors

The ideal of a Segre variety P^n₁×?×P^n_t?P^{(n₁+1)?(n_t+1)−1} is generated by the 2-minors of a generic hypermatrix of indeterminates (see [H.T. Hà, Box-shaped matrices and the defining ideal of certain blowup surface, J. Pure Appl. Algebra 167 (2-3) (2002) 203-224. MR1874542 (2002h:13020)] and [R. Grone, Decomposable tensors as a quadratic variety, Proc. Amer. Math. 43 (2) (1977) 227-230. MR0472853 (57 #12542)]). We extend this result to the case of Segre-Veronese varieties. The main tool is the concept of “weak generic hypermatrix” which allows us to treat also the case of projection of Veronese surfaces from a set of general points and of Veronese varieties from a Cohen-Macaulay subvariety of codimension 2.     

Correction to “Tensor products of modules and the rigidity of Tor”, Math. Annalen, 299 (1994), 449–476

This note makes a correction to the paper “Tensor products of modules and the ridigity of Tor”, a correction which is needed due to an incorrect convention for the depth of the zero module.