Jet closures and the local isomorphism problem

We make a systematic study of the infinitesimal lifting conditions of a pseudo finite type map of noetherian formal schemes. We recover the usual general properties in this context, and, more importantly, we uncover some new phenomena. We define a completion map of formal schemes as the one that arises canonically by performing the completion of a noetherian formal scheme along a subscheme, following the well-known pattern of ordinary schemes. These maps are etale in the sense of this work (but not adic). They allow us to give a local description of smooth morphisms. This morphisms can be factored locally as a completion map followed from a smooth adic morphism. The latter kind of morphisms can be factored locally as an etale adic morphism followed by a (formal) affine space. We also characterize etale adic morphisms giving an equivalence of categories between the category of etale adic formal schemes over a noetherian formal scheme (X, O_X) and the category of etale schemes over th...

Communications in Algebra, 2003

For an ideal I of a Noetherian local ring (R, m) we consider properties of I and its powers as reflected in the fiber cone F (I) of I. In particular, we examine behavior of the fiber cone under homomorphic image R → R/J = R as related to analytic spread and generators for the kernel of the induced map on fiber cones ψJ : FR(I) → F R (IR ). We consider the structure of fiber cones F (I) for which ker ψJ = 0 for each nonzero ideal J of R. If dimF (I) = d > 0, µ(I) = d + 1 and there exists a minimal reduction J of I generated by a regular sequence, we prove that if grade(G+(I)) ≥ d − 1, then F (I) is Cohen-Macaulay and thus a hypersurface.

Journal of Pure and Applied Algebra, 2009

We continue our study on infinitesimal lifting properties of maps between locally noetherian formal schemes started in [3]. In this paper, we focus on some properties which arise specifically in the formal context. In this vein, we make a detailed study of the relationship between the infinitesimal lifting properties of a morphism of formal schemes and those of the corresponding maps of usual schemes associated to the directed systems that define the corresponding formal schemes. Among our main results, we obtain the characterization of completion morphisms as pseudo-closed immersions that are flat. Also, the local structure of smooth andétale morphisms between locally noetherian formal schemes is described: the former factors locally as a completion morphism followed by a smooth adic morphism and the latter as a completion morphism followed by anétale adic morphism.

2001

We analyze the local structure of A-jet spaces, where A is a Weil algebra;by the way, we introduce the bundles of A-jets of sections of a regular projection and describe their vertical tangent spaces.

Illinois Journal of Mathematics, 1997

Computer Aided Geometric Design, 2006

We introduce a new family of subdivision schemes called jet subdivision schemes. Jet subdivision schemes are a natural generalization of the commonly used subdivision schemes for free-form surface modeling. In an order r jet subdivision scheme, rth order Taylor expansions, or r-jets, of functions are the essential objects being generated in a coarse-to-fine fashion. Standard subdivision surface methods correspond to the case r = 0. Just as the standard free-form subdivision surface schemes, jet subdivision schemes are based on combining (i) a symmetric subdivision scheme in the shift-invariant setting with (ii) an extraordinary vertex rule. We formulate the notions of stationarity, symmetry, and smoothness for jet subdivision schemes. We then extend some well-known results in the theory of subdivision surfaces to the setting of jet subdivision schemes. By incorporating high order data into the subdivision rules, jet subdivision schemes offer more degrees of freedom for the design of extraordinary vertex rules than do standard subdivision schemes. Using a simple 2-point stencil, we construct an order 1 jet subdivision scheme, which is interpolatory, C 1 everywhere, and free from polar artifacts at extraordinary vertices of any valence. It is similar to the popular butterfly scheme, but unlike the butterfly scheme, it produces surfaces that can be explicitly parameterized by spline functions. In addition, our jet subdivision scheme allows for explicit control of gradient information on the resulting surface. We address the problem of constructing curvature continuous subdivision schemes at extraordinary vertices by giving an example of a 'flexible C 2 scheme' for extraordinary vertex of valence k = 3. The stencils for this subdivision scheme coincide with those of the popular Loop scheme. We discuss the mathematical structure of this example of a C 2 scheme.

Taiwanese Journal of Mathematics, 2020

Given a 0-dimensional scheme X in a n-dimensional projective space P n K over an arbitrary field K, we use Liaison theory to characterize the Cayley-Bacharach property of X. Our result extends the result for sets of Krational points given in [7]. In addition, we examine and bound the Hilbert function and regularity index of the Dedekind different of X when X has the Cayley-Bacharach property. Theorem 1.1. Let W be a set of points in P n K which is a complete intersection, let X ⊆ W, let Y = W \ X, and let I W , I X and I Y denote the homogeneous vanishing ideals of W, X and Y in P = K[X 0 , ..., X n ], respectively. Set α Y/W = min{i ∈ N | (I Y /I W) i = 0 }. Then the following conditions are equivalent. (a) X is a Cayley-Bachrach scheme. (b) A generic element of (I Y) α Y/W does not vanish at any point of X. (c) We have I W : (I Y) α Y/W = I X .

Mathematical Research Letters, 1996

arXiv (Cornell University), 2022

Let A and B be abelian varieties defined over the function field k(S) of a smooth algebraic variety S/k. We establish criteria, in terms of restriction maps to subvarieties of S, for existence of various important classes of k(S)-homomorphisms from A to B, e.g., for existence of k(S)-isogenies. Our main tools consist of Hilbertianity methods, Tate conjecture as proven by Tate, Zarhin and Faltings, and of the minuscule weights conjecture of Zarhin in the case, when the base field is finite.

Transactions of the American Mathematical Society, 2001

Let R be a reduced ring that is essentially of finite type over an excellent regular local ring of prime characteristic. Then it is shown that the test ideal of R commutes with localization and, if R is local, with completion, under the additional hypothesis that the tight closure of zero in the injective hull E of the residue field of every local ring of R is equal to the finitistic tight closure of zero in E. It is conjectured that this latter condition holds for all local rings of prime characteristic; it is proved here for all Cohen-Macaulay singularities with at most isolated non-Gorenstein singularities, and in general for all isolated singularities. In order to prove the result on the commutation of the test ideal with localization and completion, a ring of Frobenius operators associated to each R-module is introduced and studied. This theory gives rise to an ideal of R which defines the non-strongly F-regular locus, and which commutes with localization and completion. This ideal is conjectured to be the test ideal of R in general, and shown to equal the test ideal under the hypothesis that 0 * E = 0 f g * E