Pullback of the Normal Module of Ideals with Low Codimension

2021 ◽  
Author(s):  
Cleto B Miranda-Neto
Keyword(s):  
Algebraic Geometry ◽  
Commutative Algebra ◽  
Difficult Problem ◽  
Natural Projection ◽  
Analytic Spread ◽  
Reduction Number ◽  
Numerical Invariants ◽  
Special Classes

Abstract The normal module (or sheaf) of an ideal is a celebrated object in commutative algebra and algebraic geometry. In this paper, we prove results about its pullback under the natural projection, focusing on subtle numerical invariants such as, for instance, the reduction number. For certain codimension 2 perfect ideals, we show that the pullback has reduction number two. This is of interest since the determination of this invariant in the context of modules (even for special classes) is a mostly open, difficult problem. The analytic spread is also computed. Finally, for codimension 3 Gorenstein ideals, we determine the depth of the pullback, and we also consider a broader class of ideals provided that the Auslander transpose of the conormal module is almost Cohen–Macaulay.

scholarly journals THE STRUCTURE OF THE SALLY MODULE OF INTEGRALLY CLOSED IDEALS

2016 ◽  
Vol 227 ◽  
pp. 49-76 ◽  
Author(s):  
KAZUHO OZEKI ◽  
MARIA EVELINA ROSSI
Keyword(s):  
Algebraic Geometry ◽  
Commutative Algebra ◽  
Associated Graded Ring ◽  
Graded Ring ◽  
Hilbert Coefficients ◽  
Integrally Closed ◽  
Numerical Invariants ◽  
Minimal Reduction ◽  
Integrally Closed Ideals ◽  
Extremal Value

The first two Hilbert coefficients of a primary ideal play an important role in commutative algebra and in algebraic geometry. In this paper we give a complete algebraic structure of the Sally module of integrally closed ideals $I$ in a Cohen–Macaulay local ring $A$ satisfying the equality $\text{e}_{1}(I)=\text{e}_{0}(I)-\ell _{A}(A/I)+\ell _{A}(I^{2}/QI)+1,$ where $Q$ is a minimal reduction of $I$, and $\text{e}_{0}(I)$ and $\text{e}_{1}(I)$ denote the first two Hilbert coefficients of $I,$ respectively, the multiplicity and the Chern number of $I.$ This almost extremal value of $\text{e}_{1}(I)$ with respect to classical inequalities holds a complete description of the homological and the numerical invariants of the associated graded ring. Examples are given.

Tribology of rail bogie

2018 ◽  
Vol 77 (4) ◽  
pp. 230-240
Author(s):  
D. P. Markov
Keyword(s):  
Contact Fatigue ◽  
Basic Element ◽  
Difficult Problem ◽  
Railway Track ◽  
Rolling Stock ◽  
Kinematic Parameters ◽  
Approximate Estimation ◽  
Railway Bogie ◽  
Classical Calculation

Railway bogie is the basic element that determines the force, kinematic, power and other parameters of the rolling stock, and its movement in the railway track has not been studied enough. Classical calculation of the kinematic and dynamic parameters of the bogie's motion with the determination of the position of its center of rotation, the instantaneous axes of rotation of wheelsets, the magnitudes and directions of all forces present a difficult problem even in quasi-static theory. The paper shows a simplified method that allows one to explain, within the limits of one article, the main kinematic and force parameters of the bogie movement (installation angles, clearance between the wheel flanges and side surfaces of the rails), wear and contact damage to the wheels and rails. Tribology of the railway bogie is an important part of transport tribology, the foundation of the theory of wheel-rail tribosystem, without which it is impossible to understand the mechanisms of catastrophic wear, derailments, contact fatigue, cohesion of wheels and rails. In the article basic questions are considered, without which it is impossible to analyze the movement of the bogie: physical foundations of wheel movement along the rail, types of relative motion of contacting bodies, tribological characteristics linking the force and kinematic parameters of the bogie. Kinematics and dynamics of a two-wheeled bogie-rail bicycle are analyzed instead of a single wheel and a wheelset, which makes it clearer and easier to explain how and what forces act on the bogie and how they affect on its position in the rail track. To calculate the motion parameters of a four-wheeled bogie, it is represented as two two-wheeled, moving each on its own rail. Connections between them are replaced by moments with respect to the point of contact between the flange of the guide wheel and the rail. This approach made it possible to give an approximate estimation of the main kinematic and force parameters of the motion of an ideal bogie (without axes skewing) in curves, to understand how the corners of the bogie installation and the gaps between the flanges of the wheels and rails vary when moving with different speeds, how wear and contact injuries arise and to give recommendations for their assessment and elimination.

On Grothendieck's local duality theorem

1979 ◽  
Vol 85 (3) ◽  
pp. 431-437 ◽  
Author(s):  
M. H. Bijan-Zadeh ◽  
R. Y. Sharp
Keyword(s):  
Algebraic Geometry ◽  
Commutative Algebra ◽  
Noetherian Ring ◽  
Duality Theorem ◽  
Great Emphasis ◽  
Triangulated Category ◽  
Elementary Approach ◽  
Commutative Noetherian Ring ◽  
Local Duality ◽  
Dualizing Complexes

In (11) and (12), a comparatively elementary approach to the use of dualizing complexes in commutative algebra has been developed. Dualizing complexes were introduced by Grothendieck and Hartshorne in (2) for use in algebraic geometry; the approach to dualizing complexes in (11) and (12) differs from that of Grothendieck and Hartshorne in that it avoids use of the concepts of triangulated category, derived category, and localization of categories, and instead places great emphasis on the concept of quasi-isomorphism of complexes of modules over a commutative Noetherian ring.

Cellular reductions of the Pommaret-Seiler resolution for Quasi-stable ideals

2021 ◽  
Vol 55 (3) ◽  
pp. 102-106
Author(s):  
Rodrigo Iglesias ◽  
Eduardo Sáenz de Cabezón
Keyword(s):  
Algebraic Geometry ◽  
Commutative Algebra ◽  
Gröbner Bases ◽  
Grobner Bases ◽  
Combinatorial Properties ◽  
Pommaret Bases ◽  
Involutive Bases

Involutive bases were introduced in [6] as a type of Gröbner bases with additional combinatorial properties. Pommaret bases are a particular kind of involutive bases with strong relations to commutative algebra and algebraic geometry[11, 12].

Random packing of an interval. II

1979 ◽  
Vol 11 (03) ◽  
pp. 591-602
Author(s):  
David Mannion
Keyword(s):  
Fine Particles ◽  
Random Variables ◽  
Random Variable ◽  
Difficult Problem ◽  
Random Packing ◽  
Point Of View ◽  
Initial Size ◽  
Moderate Size ◽  
The Individual

We showed in [2] that if an object of initial size x (x large) is subjected to a succession of random partitions, then the object is decomposed into a large number of terminal cells, each of relatively small size, where if Z(x, B) denotes the number of such cells whose sizes are points in the set B, then there exists c, (0 < ≦ 1), such that Z(x, B)x −c converges in probability, as x → ∞, to a random variable W. We show here that if a parent object of size x produces k offspring of sizes y 1, y 2, ···, y k and if for each k x - y 1 - y 2 - ··· - y k (the ‘waste’ or the ‘cover’, depending on the point of view) is relatively small, then for each n the nth cumulant, Ψ n (x, B), of Z(x, B) satisfies Ψ n (x, B)x -c → κ n (B), as x → ∞, for some κ n (B). Thus, writing N = x c , Z(x, B) has approximately the same distribution as the sum of N independent and identically distributed random variables (The determination of the distribution of the individual appears to be a difficult problem.) The theory also applies when an object of moderate size is broken down into very fine particles or granules.

scholarly journals The topological properties of QCD at high temperature: problems and perspectives

2018 ◽  
Vol 175 ◽  
pp. 01011 ◽  
Author(s):  
Claudio Bonati
Keyword(s):  
High Temperature ◽  
Difficult Problem ◽  
High Temperature Phase ◽  
Temperature Phase ◽  
First Principle ◽  
Numerical Determination ◽  
Topological Properties

Lattice computations are the only first principle method capable of quantitatively assessing the topological properties of QCD at high temperature, however the numerical determination of the topological properties of QCD, especially in the high temperature phase, is a notoriously difficult problem. We will discuss the difficulties encountered in such a computation and some strategies that have been proposed to avoid (or at least to alleviate) them.

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