arXiv math/05011792005-01-12math.ACmath.CO

Resolution of the residue class field via algebraic discrete Morse theory

Michael Joellenbeck, Volkmar Welker

备注：: 40 pages

英文摘要：

Forman's Discrete Morse theory is studied from an algebraic viewpoint. Analogous to independent work of Emil Skoeldberg we show that this theory can be extended to chain complexes of free modules over a ring. We provide three applications of this theory: We construct new resolutions of the residue class field $k$ over $A$, where $A$ is the quotient of a (i) commutative polynomial ring or (ii) non-commutaitve polynomial ring by a (twosided) ideal and (iii) we construct a new resolution of $A$ as an $A \otimes A^{op}$-module in the situation (ii). In either case we prove minimality of the resolution for certain classes of algebras $A$.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

阅读原文

阅读 PDF ↗

arXiv 页面 ↗HTML 原文 ↗

本文内容

英文摘要相关论文

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

继续探索

相关论文

Algebraic discrete Morse theory for the hull resolution

We study how powerful algebraic discrete Morse theory is when applied to hull resolutions. The main result describes all cases when the hull resolution of the edge ideal of the complement of a triangle-free graph can be made minimal using algebraic discrete Morse theory.

2015-12-09math.ACmath.CO

Some algebraic invariants of the residue class rings of the edge ideals of perfect semiregular trees

Let S be a polynomial algebra over a field. If I is the edge ideal of a perfect semiregular tree, then we give precise formulas for values of depth, Stanley depth, projective dimension, regularity and Krull dimension of S/I.

2022-11-10math.AC

La trace via le calcul residuel: une nouvelle version du theoreme d'Abel-inverse, formes abeliennes

We show here how residue calculus (residue currents, Grothendieck residues, duality theorem) can be used to obtain an algebraic characterization of the Abel-transform of a meromorphic form on germs of analytic sets. We prove by this way a stronger version of Abel-inverse theorem with an "algebraic" approach and we show the link with Wood's theorem. Furthermore, we obtain a new method to bound easily the dimension of the vector space of abelian forms on an algebraic projective hypersurface.

2004-06-02math.ACmath.CV

Value groups, residue fields and bad places of rational function fields

We classify all possible extensions of a valuation from a ground field $K$ to a rational function field in one or several variables over $K$. We determine which value groups and residue fields can appear, and we show how to construct extensions having these value groups and residue fields. In particular, we give several constructions of extensions whose corresponding value group and residue field extensions are not finitely generated. In the case of a rational function field $K(x)$ in one variable, we consider the relative algebraic closure of $K$ in the henselization of $K(x)$ with respect to…

2010-03-29math.AC

代数整数的p-adic逼近与整数值多项式环的剩余类环

P-adic approximation of algebraic integers and residue class rings of rings of integer-valued polynomials

本文通过分析Galois扩张中O_K的Q-adic闭包，刻画了使整数值多项式环Int(O_K)模非零素理想的剩余类环满足GE2性质的整数环O_K。

2024-12-20math.ACmath.NT

Residues and Differential Operators on Schemes

Beilinson Completion Algebras (BCAs) are generalizations of complete local rings, and have a rich algebraic-analytic structure. These algebras were introduced in my paper "Traces and Differential Operators over Beilinson Completion Algebras", Compositio Math. 99 (1995). In the present paper BCAs are used to give an explicit construction of the Grothendieck residue complex on an algebraic scheme. This construction reveals new properties of the residue complex, and in particular its interaction with differential operators. Applications include: (i) results on the algebraic structure of rings of …

1998-02-11alg-geommath.ACmath.AG

更多相关论文收起更多论文

Higher Stanley-Reisner rings and Toric Residues

We give a purely algebraic proof of the hypersurface case of Toric Residue Mirror Conjecture recently proposed by Batyrev and Materov.

2003-06-20math.ACmath.AG

某些图的冠积的边理想之代数性质

Algebraic Properties of Edge Ideals of Corona Product of Certain Graphs

本文研究某些图与任意图冠积所对应边理想的剩余类环的深度、Stanley深度、正则性与投射维数，并给出Krull维数表达式，进而刻画部分Cohen-Macaulay图。

2023-07-04math.AC

Global residues for sparse polynomial systems

We consider families of sparse Laurent polynomials f_1,...,f_n with a finite set of common zeroes Z_f in the complex algebraic n-torus. The global residue assigns to every Laurent polynomial g the sum of its Grothendieck residues over the set Z_f. We present a new symbolic algorithm for computing the global residue as a rational function of the coefficients of the f_i when the Newton polytopes of the f_i are full-dimensional. Our results have consequences in sparse polynomial interpolation and lattice point enumeration in Minkowski sums of polytopes.

2006-05-30math.ACmath.AG

Residue fields for a class of rational $\mathbf{E}_\infty$-rings and applications

Let $A$ be an $\mathbf{E}_{\infty}$-ring spectrum over the rational numbers. If $A$ satisfies a noetherian condition on its homotopy groups $π_*(A)$, we construct a collection of $\mathbf{E}_{\infty}$-$A$-algebras that realize on homotopy the residue fields of $π_*(A)$. We prove an analog of the nilpotence theorem for these residue fields. As a result, we are able to give a complete algebraic description of the Galois theory of $A$ and of the thick subcategories of perfect $A$-modules. We also obtain partial information on the Picard group of $A$.

2016-07-18math.ACmath.ATmath.CT

Existentially closed models of the theory of Artinian local rings

The class of all Artinian local rings of length at most l is A_2-elementary, axiomatised by a finite set of axioms Art_l. We show that its existentially closed models are Gorenstein, of length exactly l and their residue fields are algebraically closed, and, conversely, every existentially closed model is of this form. The theory Gor_l of all Artinian local Gorenstein rings of length l with algebraically closed residue field is model complete and the theory Art_l is companionable, with model-companion Gor_l.

1997-07-11math.ACmath.RA

Local Beilinson-Tate Operators

In 1968 Tate introduced a new approach to residues on algebraic curves, based on a certain ring of operators that acts on the completion at a point of the function field of the curve. This approach was generalized to higher dimensional algebraic varieties by Beilinson in 1980. However Beilinson's paper had very few details, and his operator-theoretic construction remained cryptic for many years. Currently there is a renewed interest in the Beilinson-Tate approach to residues in higher dimensions. Our paper presents a variant of Beilinson's operator-theoretic construction. We consider an n-dime…

2014-12-27math.ACmath.AGmath.NT