scholarly journals Differential operators on a hypersurface

1986 ◽  
Vol 103 ◽  
pp. 67-84 ◽  
Author(s):  
Balwant Singh
Keyword(s):  
Algebraic Variety ◽  
Characteristic Zero ◽  
Differential Operators ◽  
Coordinate Ring ◽  
Irreducible Curve ◽  
Additional Hypothesis

We study differential operators on an affine algebraic variety, especially a hypersurface, in the context of Nakai’s Conjecture. We work over a field k of characteristic zero. Let X be a reduced affine algebraic variety over k and let A be its coordinate ring. Let be the A-module of differential operators of A over k of order ≤ n. Nakai’s Conjecture asserts that if is generated by for every n ≥ 2 then A is regular. In 1973 Mount and Villamayor [6] proved this in the case when X is an irreducible curve. In the general case no progress seems to have been made on the conjecture, except for a result of Brown [2], where the assertion is proved under an additional hypothesis. An interesting result proved by Becker [1] and Rego [8] says that Nakai’s Conjecture implies the Conjecture of Zariski-Lipman, which is still open in the general case and which asserts that if the module of k-derivations of A is A-projective then A is regular.

scholarly journals Higher Nash blowups

2007 ◽  
Vol 143 (6) ◽  
pp. 1493-1510 ◽  
Author(s):  
Takehiko Yasuda
Keyword(s):  
Algebraic Variety ◽  
Characteristic Zero ◽  
Negative Integer ◽  
Curve Singularity ◽  
Irreducible Curve ◽  
Numerical Monoid

AbstractFor each non-negative integer n we define the nth Nash blowup of an algebraic variety, and call them all higher Nash blowups. When n=1, it coincides with the classical Nash blowup. We study higher Nash blowups of curves in detail and prove that any curve in characteristic zero can be desingularized by its nth Nash blowup with n large enough. Moreover, we completely determine for which n the nth Nash blowup of an analytically irreducible curve singularity in characteristic zero is normal, in terms of the associated numerical monoid.

A Module-theoretic Characterization of Algebraic Hypersurfaces

2018 ◽  
Vol 61 (1) ◽  
pp. 166-173
Author(s):  
Cleto B. Miranda-Neto
Keyword(s):  
Algebraic Variety ◽  
Characteristic Zero ◽  
Vector Fields ◽  
Exterior Power ◽  
Algebraic Hypersurfaces ◽  
Theoretic Characterization

AbstractIn this note we prove the following surprising characterization: if X ⊂ is an (embedded, non-empty, proper) algebraic variety deûned over a field k of characteristic zero, then X is a hypersurface if and only if the module of logarithmic vector fields of X is a reflexive -module. As a consequence of this result, we derive that if is a free -module, which is shown to be equivalent to the freeness of the t-th exterior power of for some (in fact, any) t ≤ n, then necessarily X is a Saito free divisor.

scholarly journals Corrections to “Seminormal rings and weakly normal varieties”

1987 ◽  
Vol 107 ◽  
pp. 147-157 ◽  
Author(s):  
Marie A. Vitulli
Keyword(s):  
Algebraic Variety ◽  
Characteristic Zero ◽  
Regular Function ◽  
Closed Field ◽  
Algebraically Closed Field ◽  
Regular Functions ◽  
Normal Varieties

In “Seminormal rings and weakly normal varieties” we introduced the notion of a c-regular function on an algebraic variety defined over an algebraically closed field of characteristic zero. Our intention was to describe those k-valued functions on a variety X that become regular functions when lifted to the normalization of X, but without any reference to the normalization in the definition. That is, we aspired to identify the c-regular functions on X with the regular functions on the weak normalization of X

Locally free (ℙn)-modules

1992 ◽  
Vol 112 (2) ◽  
pp. 233-245 ◽  
Author(s):  
S. C. Coutinho ◽  
M. P. Holland
Keyword(s):  
Projective Space ◽  
Algebraic Variety ◽  
Differential Operators ◽  
Module Theory ◽  
Rings Of Differential Operators ◽  
Complex Algebraic Variety ◽  
Free Modules

The purpose of this paper is to study the structure of locally free modules over the ring of differential operators on projective space. Let be a non-singular, complex, algebraic variety. Denote by the sheaf of rings of differential operators over and by its ring of global sections. A -module M is called locally free if the associated sheaf ⊗ M is locally free as a sheaf of -modules. Locally free modules arise naturally in -module theory as inverse images of determined modules; see [1] for definitions and examples.

scholarly journals Cohomology of induced modules in rings of differential operators

1988 ◽  
Vol 31 (1) ◽  
pp. 41-47
Author(s):  
Kenneth A. Brown ◽  
Thierry Levasseur
Keyword(s):  
Maximal Ideal ◽  
Characteristic Zero ◽  
Differential Operators ◽  
Finite Dimension ◽  
Rings Of Differential Operators ◽  
Image Position

Let K be a field of characteristic zero and let Δ ={δ1,…,δn} be a set of commuting K-derivations of the commutative Noetherian K-algebra R. Let S = R[X1,…,Xn] be the corresponding ring of differential operators, so [Xi, r] = Xir − rXi=δi(r and [Xi, Xj]=0, for 1≦i, j≦n. Let M be a maximal ideal of R with R/M of finite dimension over K. The purpose of this note is to describe the groups

The global dimension of the algebras of polynomial integro-differential operators 𝕀n and the Jacobian algebras 𝔸n

2019 ◽  
Vol 19 (02) ◽  
pp. 2050030
Author(s):  
V. V. Bavula
Keyword(s):  
Characteristic Zero ◽  
London Math ◽  
Differential Operators ◽  
Polynomial Algebra ◽  
Prime Factor ◽  
Global Dimension ◽  
Direct Sums ◽  
Jacobian Algebra ◽  
Left And Right

The aim of the paper is to prove two conjectures from the paper [V. V. Bavula, The algebra of integro-differential operators on a polynomial algebra, J. London Math. Soc. (2) 83 (2011) 517–543, arXiv:math.RA/0912.0723] that the (left and right) global dimension of the algebra [Formula: see text] of polynomial integro-differential operators and the Jacobian algebra [Formula: see text] is equal to [Formula: see text] (over a field of characteristic zero). The algebras [Formula: see text] and [Formula: see text] are neither left nor right Noetherian and [Formula: see text]. Furthermore, they contain infinite direct sums of nonzero left/right ideals and are not domains. An analogue of Hilbert’s Syzygy Theorem is proven for the algebras [Formula: see text], [Formula: see text] and their factor algebras. It is proven that the global dimension of all prime factor algebras of the algebras [Formula: see text] and [Formula: see text] is [Formula: see text] and the weak global dimension of all the factor algebras of [Formula: see text] and [Formula: see text] is [Formula: see text].

scholarly journals Length of Local Cohomology in Positive Characteristic and Ordinarity

2018 ◽  
Vol 2020 (7) ◽  
pp. 1921-1932 ◽  
Author(s):  
Thomas Bitoun
Keyword(s):  
Characteristic Zero ◽  
Local Cohomology ◽  
Differential Operators ◽  
Positive Characteristic ◽  
Isolated Singularity ◽  
Local Cohomology Module ◽  
Perfect Field ◽  
Tight Closure ◽  
Characteristic P

Abstract Let D be the ring of Grothendieck differential operators of the ring R of polynomials in d ≥ 3 variables with coefficients in a perfect field of characteristic p. We compute the D-module length of the 1st local cohomology module ${H^{1}_{f}}(R)$ with respect to a polynomial f with an isolated singularity, for p large enough. The expression we give is in terms of the Frobenius action on the top coherent cohomology of the exceptional fibre of a resolution of the singularity. Our proof rests on a tight closure computation of Hara. Since the above length is quite different from that of the corresponding local cohomology module in characteristic zero, we also consider a characteristic zero D-module whose length is expected to equal that above, for ordinary primes.

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