1.1. Reduction of coverings of degree p. Let R be a discrete valuation ring of mixed characteristics, S = SpecR, which has generic point η with fraction field K and special point s with residue field k of characteristic p > 0. Consider a generically smooth, stable marked curve Y → S with an action of a finite group G of order divisible by p. Denote X = Y/G. We assume that G acts freely on the c...

The Witt group of smooth real projective curves was first computed by Knebusch in [Kn]. If the curve is not complete but still smooth, the Witt group is also studied in [Kn] but not explicitely calculated. However, for some precise examples of smooth affine curves, we may find explicit calculations ([Knu] and [Ay-Oj]). In this paper, the Witt group of a general smooth curve is explicitely calcu...

Our goal is to establish an efficient decomposition of an ideal A of a commutative ring R as an intersection of primal ideals. We prove the existence of a canonical primal decomposition: A = ⋂ P∈XA A(P ), where the A(P ) are isolated components of A that are primal ideals having distinct and incomparable adjoint primes P . For this purpose we define the set Ass(A) of associated primes of the id...

A conjecture on vanishing and non-negativity of intersection multiplicity on the blow-up of a regular local ring at its closed point has been proposed. Proofs of vanishing, several special cases of non-negativity and a sufficient condition for non-negativity of this conjecture are described. The topic on intersection multiplicity on vector bundles is also addressed. Introduction. Let X be a reg...

We show that cores of ideals do not preserve the inclusion. Huneke and Swanson[2] raised the question of whether, given integrally closed ideals I ⊂ I ′ in a ring R, it is necessarily true that core(I) ⊂ core(I ). Hyry and Smith [4, Corollary 5.5.1] gave a partial answer. The purpose of this note is to show that the answer is no in general. Although the simplest counterexample is a principal id...

Over an algebraic field K with discrete valuation, we study the line bundles on a smooth curve CK whose rth power is isomorphic to a given line bundle FK . For FK = O, these objects form a finite group K-scheme O/r; furthermore, as soon as the degree of FK is a multiple of r, we get a finite K-torsor FK/r under the group scheme O/r. On the discrete valuation ring R ⊂ K, there exist Néron models...

One of the central problems of algebraic K-theory is to compute the K-groups KX of a scheme X. Since these groups are, by definition, the homotopy groups of a spectrum KX, it makes sense to analyze the homotopy-type of the spectrum, rather than just the disembodied homotopy groups. In addition to facilitating the computation of the K-groups themselves, knowledge of the spectrum KX can be applie...

We give a formalism of arithmetic mixed sheaves, including the case of arithmetic mixed Hodge structures which are recently studied by P. Griffiths, M. Green [22], [23] and M. Asakura [1] using other methods. This notion became necessary for us to describe the image of Griffiths’ Abel-Jacobi map for a generic hypersurface (inspired by previous work of M. Green [21] and C. Voisin [45], [46]), an...

We study tight closure and test ideals in rings of characteristic p 0 using resolution of singularities. The notions of F -rational and F regular rings are defined via tight closure, and they are known to correspond with rational and log terminal singularities, respectively. In this paper, we reformulate this correspondence by means of the notion of the test ideal, and generalize it to wider cl...