Search

Consider the projective coordinate ring of the Geometric Invariant Theory (GIT) quotient (ℙ1) n //SL(2), with the usual linearization, where n is even. In 1894, Kempe proved that this ring is generated in degree one. In [2] we showed that, over ℚ, the relations between degree one invariants are generated by a class of quadratic relations—the simplest binomial relations—with the exception of n = 6, where there is a single cubic relation. The purpose of this article is to show that these results hold over , and to suggest why they may be true over . [ABSTRACT FROM PUBLISHER]

Let R be a commutative Noetherian ring. In this article, we provide some new criteria for a semidualizing module to be dualizing in terms of special homological properties of module categories. The purpose of this article is twofold: first, it aims at improving Christensen's and Takahashi et al.'s characterizations of dualizing modules; secondly, while applying these criteria to the ring itself, we not only recover some results of Jenda and Xu, respectively, but also obtain a new characterization of Gorenstein rings. [ABSTRACT FROM PUBLISHER]

We investigate the least studied class of differential rings—the class of differential rings of nonzero characteristic. We present the notion of differentially closed quasifield and develop geometrical theory of differential equations in nonzero characteristic. The notions of quasivariety and its morphisms are scrutinized. Presented machinery is a basis for reduction modulo p for differential equations. [ABSTRACT FROM PUBLISHER]

We call an ideal I of a ring R radically perfect if among all ideals whose radical is equal to the radical of I, the one with the least number of generators has this number of generators equal to the height of I. Let R be a ring and R[X] be the polynomial ring over R. We prove that if R is a strong S-domain of finite Krull dimension and if each nonzero element of R is contained in finitely many maximal ideals of R, then each maximal ideal of R[X] of maximal height is the Jmax-radical of an ideal generated by two elements. We also show that if R is a Prufer domain of finite Krull dimension with coprimely packed set of maximal ideals, then for each maximal ideal M of R, the prime ideal MR[X] of R[X] is radically perfect if and only if R is of dimension one and each maximal ideal of R is the radical of a principal ideal. We then prove that the above conditions on the Prufer domain R also imply that a power of each finitely generated maximal ideal of R is principal. This result naturally raises the question whether the same conditions on R imply that the Picard group of R is torsion, and we prove this to be so when either R is an almost Dedekind domain or a Prufer domain with an extra condition imposed on it. [ABSTRACT FROM AUTHOR]

The real closed valuation rings, i.e., convex subrings of real closed fields, form a proper subclass of the class of real closed domains. It is shown how one can recognize whether a real closed domain is a valuation ring. This leads to a characterization of the totally ordered domains whose real closure is a valuation ring. Real closures of totally ordered factor rings of coordinate rings of real algebraic varieties are very frequently valuation rings. In particular, the real closure of the coordinate ring of a curve is an SV-ring (i.e., the factor rings modulo prime ideals are valuation rings). Real closed valuation rings play a role in the definition of real closed rings, as well as in the construction of real closures of rings and porings. They can also be used for the study of univariate differentiable semi-algebraic functions. This leads to the notion of differentiablility of semi-algebraic functions along half branches of curves. [ABSTRACT FROM AUTHOR]

Let A be a noetherian local k-algebra with residue field k and associated graded ring G(A). Let  be the conductor of A in its normalization [image omitted], which we assume to be regular and finite over A. Assume also that Proj(G(A)) is reduced and [image omitted]. Let r be the embedding dimension of A and [image omitted] be the Hilbert function of [image omitted]. We show that if the Hilbert function of Proj(G(A)) is equal to [image omitted], then G(A) is reduced. This allows us to compute, in the general case, the conductor of the local ring of singular isolated points on surfaces whose tangent cone is multiplanar; that is, it consists—as a set—of planes. We explicitly construct a class of such surfaces in the rational case. [ABSTRACT FROM AUTHOR]

Let k be an algebraically closed field of characteristic 0, and let A = k[x, y]/(f) be a quasi-homogeneous plane curve. We show that for any graded torsion free A-module M, there exists a natural graded integrable connection, i.e., a graded A-linear homomorphism ∇: Derk(A) → Endk(M), that satisfies the derivation property and preserves the Lie product. In particular, a torsion free module N over the complete local ring [image omitted] admits a natural integrable connection if A is a simple curve singularity, or if A is irreducible and N is a gradable module. [ABSTRACT FROM AUTHOR]

We give a presentation for the stack [image omitted] of rational curves with at most 1 node as the quotient by GL3 of an open set in a 6-dimensional irreducible representation. We then use equivariant intersection theory to calculate the integral Chow ring of this stack. [ABSTRACT FROM AUTHOR]

We introduce and study the Picard group of a coring. We give an exact sequence relating the Picard group of a coring and its automorphisms generalizing the known exact sequences associated to an algebra and a coalgebra over a field. We also extend to corings the Aut-Pic property and we give some new corings satisfying this property. [ABSTRACT FROM AUTHOR]

We present a class of homogeneous ideals which are generated by monomials and binomials of degree 2 and are set-theoretic complete intersections. This class includes certain reducible varieties of minimal degree and, in particular, the presentation ideals of the fiber cone algebras of monomial varieties of codimension 2. [ABSTRACT FROM AUTHOR]

The purpose of this article is to give criteria for the Gorenstein property of the blowup of the spectrum of a Gorenstein local ring A along an ideal I⊂⃒A by means of the coefficient ideal of I. Moreover, we investigate the conductor ideal and the Gorenstein property of the fiber cone. [ABSTRACT FROM AUTHOR]

Recently Jain, López-Permouth and Syed classified indecomposable non-local right continuous right π-rings. In the present article we describe the structure of right continuous right π-rings. [ABSTRACT FROM AUTHOR]

This paper deals with the classification of plane algebroid branches, modulo contact equivalence, within the equisingularity class determined by the semigroup of values generated by the integers 6, 9 and 19. This illustrates how one can use the techniques developed in Hefez and Hernandes (Hefez, A., Hernandes, M. E. (2001). Computational Methods in the Local Theory of Curves. 23° Colóquio Brasileiro de Matemática. Rio de Janeiro: Publicações Matemáticas, IMPA), to make computations in the module of Kähler differentials of the local ring of an algebroid branch. [ABSTRACT FROM AUTHOR]