Noether Normalization Research Articles

Lim Ulrich sequences and Boij-Söderberg cones

Abstract This paper extends the results of Boij, Eisenbud, Erman, Schreyer and Söderberg on the structure of Betti cones of finitely generated graded modules and finite free complexes over polynomial rings, to all finitely generated graded rings admitting linear Noether normalizations. The key new input is the existence of lim Ulrich sequences of graded modules over such rings.

The super Mumford form and Sato Grassmannian

We describe a supersymmetric generalization of the construction of Kontsevich and Arbarello, De Concini, Kac, and Procesi, which utilizes a relation between the moduli space of curves with the infinite-dimensional Sato Grassmannian. Our main result is the existence of a flat holomorphic connection on the line bundle λ3/2⊗λ1/2−5 on the moduli space of triples: a super Riemann surface, a Neveu-Schwarz puncture, and a formal coordinate system. We also prove a superconformal Noether normalization lemma for families of super Riemann surfaces.

Algorithms for specifying initial data for discrete autonomous nD systems

For nD systems, initial data is obtained by specifying trajectories on special subsets of the domain, known as characteristic sets. In this paper, we consider a special class of systems that admits a union of a coordinate sublattice and finitely many parallel translates of it as a characteristic set; we call such systems as strongly relevant systems. Using the discrete Noether's normalization lemma, every discrete autonomous nD system can be transformed to a strongly relevant system. For a strongly relevant system, the set of allowable initial conditions, obtained by restricting trajectories on the characteristic set, is characterized. We then provide an implementable algorithm, based on Gröbner basis, for obtaining a representation of the set of allowable initial conditions. Once such a representation is obtained, important deductions, such as, arbitrary assignability of initial data can be easily made.

Grothendieck–Serre in the quasi-split unramified case

AbstractThe Grothendieck–Serre conjecture predicts that every generically trivial torsor under a reductive group schemeGover a regular local ringRis trivial. We settle it in the case whenGis quasi-split andRis unramified. Some of the techniques that allow us to overcome obstacles that have so far kept the mixed characteristic case out of reach include a version of Noether normalization over discrete valuation rings, as well as a suitable presentation lemma for smooth relative curves in mixed characteristic that facilitates passage to the relative affine line via excision and patching.

A Simple Form of Noether Normalization

In this note we give a simple version (with a simple proof) of Noether’s Normalization Lemma which implies Zariski’s Lemma and Hilbert’s Nullstellensatz (weak form) more naturally.

Morita theory and singularity categories

We propose an analogue of the bounded derived category for an augmented ring spectrum, defined in terms of a notion of Noether normalization. In many cases we show this category is independent of the chosen normalization. Based on this, we define the singularity and cosingularity categories measuring the failure of regularity and coregularity and prove they are Koszul dual in the style of the BGG correspondence. Examples of interest include Koszul algebras and Ginzburg DG-algebras, C⁎(BG) for finite groups (or for compact Lie groups with orientable adjoint representation), cochains in rational homotopy theory and various examples from chromatic homotopy theory.

A probabilistic approach to systems of parameters and Noether normalization

We study systems of parameters over finite fields from a probabilistic perspective, and use this to give the first effective Noether normalization result over a finite field. Our central technique is an adaptation of Poonen's closed point sieve, where we sieve over higher dimensional subvarieties, and we express the desired probabilities via a zeta function-like power series that enumerates higher dimensional varieties instead of closed points. This also yields a new proof of a recent result of Gabber-Liu-Lorenzini and Chinburg-Moret-Bailly-Pappas-Taylor on Noether normalizations of projective families over the integers.

A new subadditivity formula for test ideals

We exhibit a new subadditivity formula for test ideals on singular varieties using an argument similar to [9] and [16]. Any subadditivity formula for singular varieties must have a correction term that measures the singularities of that variety. Whereas earlier subadditivity formulas accomplished this by multiplying by the Jacobian ideal, our approach is to use the formalism of Cartier algebras [1]. We also show that our subadditivity containment is sharper than ones shown previously in [32] and [11]. The first of these results follows from a Noether normalization technique due to Hochster and Huneke. The second of these results is obtained using ideas from [33] and [11] to show that the adjoint ideal JX(A,Z) reduces mod p to Takagi's adjoint test ideal, even when the ambient space is singular, provided that A is regular at the generic point of X. One difficulty of using this new subadditivity formula in practice is the computational complexity of computing its correction term. Thus, we discuss a combinatorial construction of the relevant Cartier algebra in the toric setting.

Noether resolutions in dimension 2

Let R := K [ x 1 ,..., x n ] be a polynomial ring over an infinite field K , and let I ⊂ R be a homogeneous ideal with respect to a weight vector ω = (ω 1 , ..., ω n ) ∈ (Z + ) n such that dim( R/I ) = d. We consider the minimal graded free resolution of R/I as A -module, that we call the Noether resolution of R/I , whenever A := K [ x n-d +1 ,..., x n ] is a Noether normalization of R/I. When d = 2 and I is saturated, we give an algorithm for obtaining this resolution that involves the computation of a minimal Gröbner basis of I with respect to the weighted degree reverse lexicographic order. In the particular case when R/I is a 2-dimensional semigroup ring, we also describe the multigraded version of this resolution in terms of the underlying semigroup. Whenever we have the Noether resolution of R/I or its multigraded version, we obtain formulas for the corresponding Hilbert series of R/I , and when I is homogeneous, we obtain a formula for the Castelnuovo-Mumford regularity of R/I. As an application of the results for 2-dimensional semigroup rings, we provide a new upper bound for the Castelnuovo-Mumford regularity of the coordinate ring of a projective monomial curve. Finally, we describe the multigraded Noether resolution of either the coordinate ring of a projective monomial curve [EQUATION] associated to an arithmetic sequence or of any of its canonical projections [EQUATION].

Noether Normalization theorem and dynamical Gröbner bases over Bezout domains of Krull dimension 1

We propose a new version of Noether normalization theorem over Bezout domains of Krull dimension one (the ring Z of integers as main example). We also show that one can avoid branching when computing dynamical Gröbner bases over a Bezout domain of Krull dimension one.

Noether resolutions in dimension 2

Let R:=K[x1,…,xn] be a polynomial ring over an infinite field K, and let I⊂R be a homogeneous ideal with respect to a weight vector ω=(ω1,…,ωn)∈(Z+)n such that dim(R/I)=d. In this paper we study the minimal graded free resolution of R/I as A-module, that we call the Noether resolution of R/I, whenever A:=K[xn−d+1,…,xn] is a Noether normalization of R/I. When d=2 and I is saturated, we give an algorithm for obtaining this resolution that involves the computation of a minimal Gröbner basis of I with respect to the weighted degree reverse lexicographic order. In the particular case when R/I is a 2-dimensional semigroup ring, we also describe the multigraded version of this resolution in terms of the underlying semigroup. Whenever we have the Noether resolution of R/I or its multigraded version, we obtain formulas for the corresponding Hilbert series of R/I, and when I is homogeneous, we obtain a formula for the Castelnuovo–Mumford regularity of R/I. Moreover, in the more general setting that R/I is a simplicial semigroup ring of any dimension, we provide its Macaulayfication.As an application of the results for 2-dimensional semigroup rings, we provide a new upper bound for the Castelnuovo–Mumford regularity of the coordinate ring of a projective monomial curve. Finally, we describe the multigraded Noether resolution and the Macaulayfication of either the coordinate ring of a projective monomial curve C⊆PKn associated to an arithmetic sequence or the coordinate ring of any canonical projection πr(C) of C to PKn−1.

A Differential Analog of the Noether Normalization Lemma

In this paper, we prove the following differential analog of the Noether normalization lemma: for every $d$-dimensional differential algebraic variety over differentially closed field of zero characteristic there exists a surjective map onto the $d$-dimensional affine space. Equivalently, for every integral differential algebra $A$ over differential field of zero characteristic there exist differentially independent $b_1, \ldots, b_d$ such that $A$ is differentially algebraic over subalgebra $B$ differentially generated by $b_1, \ldots, b_d$, and whenever $\mathfrak{p} \subset B$ is a prime differential ideal, there exists a prime differential ideal $\mathfrak{q} \subset A$ such that $\mathfrak{p} = B \cap \mathfrak{q}$. We also prove the analogous theorem for differential algebraic varieties over the ring of formal power series over an algebraically closed differential field and present some applications to differential equations.

-neighborhoods and comparison theorems

A technical ingredient in Faltings’ original approach to$p$-adic comparison theorems involves the construction of$K({\it\pi},1)$-neighborhoods for a smooth scheme$X$over a mixed characteristic discrete valuation ring with a perfect residue field: every point$x\in X$has an open neighborhood$U$whose generic fiber is a$K({\it\pi},1)$scheme (a notion analogous to having a contractible universal cover). We show how to extend this result to the logarithmically smooth case, which might help to simplify some proofs in$p$-adic Hodge theory. The main ingredient of the proof is a variant of a trick of Nagata used in his proof of the Noether normalization lemma.

Noether normalizations, reductions of ideals, and matroids

We show that given a finitely generated standard graded algebra of dimension d d over an infinite field, its graded Noether normalizations obey a certain kind of ‘generic exchange’, allowing one to pass between any two of them in at most d d steps. We prove analogous generic exchange theorems for minimal reductions of an ideal, minimal complete reductions of a set of ideals, and minimal complete reductions of multigraded k k -algebras. Finally, we unify all these results into a common axiomatic framework by introducing a new topological-combinatorial structure we call a generic matroid, which is a common generalization of a topological space and a matroid.

Noether normalization guided by monomial cone decompositions

This paper explains the relevance of partitioning the set of standard monomials into cones for constructing a Noether normalization for an ideal in a polynomial ring. Such a decomposition of the complement of the corresponding initial ideal in the set of all monomials–also known as a Stanley decomposition–is constructed in the context of Janet bases, in order to come up with sparse coordinate changes which achieve Noether normal position for the given ideal.

The core of ideals in arbitrary characteristic

In this paper we provide explicit formulas for the core of an ideal. Recall that for an ideal I in a Noetherian ring R, the core of I, core(I ), is the intersection of all reductions of I. For a subideal J ⊂ I we say that J is a reduction of I, or that I is integral over J, if I r+1 = JI r for some r ≥ 0; the smallest such r is called the reduction number of I with respect to J and is denoted by rJ(I ). If (R,m) is local with infinite residue field k then every ideal has a minimal reduction, which is a reduction minimal with respect to inclusion. Minimal reductions of a given ideal I are far from unique, but they all share the same minimal number of generators, called the analytic spread of I and written (I ). Minimal reductions arise from Noether normalizations of the special fiber ring F(I ) = grI(R)⊗k of I, and therefore (I ) = dimF(I ). From this one readily sees that ht I ≤ (I ) ≤ dimR; these inequalities are equalities for any m-primary ideal, and if the first inequality is an equality then I is called equimultiple. Obviously, the core can be obtained as an intersection of minimal reductions of a given ideal. Through the study of the core one hopes to better understand properties shared by all reductions. The notion was introduced by Rees and Sally for the purpose of generalizing the Briancon–Skoda Theorem [17]. As an a priori infinite intersection of reductions, the core is difficult to compute, and there have been considerable efforts to find explicit formulas; see [3; 4; 9; 10; 11; 12; 15]. We quote the following result from [15].

On the structure of the fiber cone of ideals with analytic spread one

For a given a local ring ( A , m ) , we study the fiber cone of ideals in A with analytic spread one. In this case, the fiber cone has a structure as a module over its Noether normalization which is a polynomial ring in one variable over the residue field. One may then apply the structure theorem for modules over a principal domain to get a complete description of the fiber cone as a module. We analyze this structure in order to study and characterize in terms of the ideal itself the arithmetical properties and other numerical invariants of the fiber cone as multiplicity, reduction number or Castelnuovo–Mumford regularity.

Standard Noether normalizations of the graph subring

In this paper, we confirm a conjecture of Alcántar by showing that graph subrings always admit standard Noether normalizations.

Lie algebras and algebraic groups

Preface 1. Results on topological spaces 1.1 Irreducible sets and spaces 1.2 Dimension 1.3 Noetherian spaces 1.4 Constructible sets 1.5 Gluing topological spaces 2. Rings and modules 2.1 Ideals 2.2 Prime and maximal ideals 2.3 Rings of fractions and localization 2.4 Localization of modules 2.5 Radical of an ideal 2.6 Local rings 2.7 Noetherian rings and modules 2.8 Derivations 2.9 Module of differentials 3. Integral extensions 3.1 Integral dependence 3.2 Integrally closed rings 3.3 Extensions of prime ideals 4. Factorial rings 4.1 Generalities 4.2 Unique factorization 4.3 Principal ideal domains and Euclidean domains 4.4 Polynomial and factorial rings 4.5 Symmetric polynomials 4.6 Resultant and discriminant 5. Field extensions 5.1 Extensions 5.2 Algebraic and transcendental elements 5.3 Algebraic extensions 5.4 Transcendence basis 5.5 Norm and trace 5.6 Theorem of the primitive element 5.7 Going Down Theorem 5.8 Fields and derivations 5.9 Conductor 6. Finitely generated algebras 6.1 Dimension 6.2 Noether's Normalization Theorem 6.3 Krull's Principal Ideal Theorem 6.4 Maximal ideals 6.5 Zariski topology 7. Gradings and filtrations 7.1 Graded rings and graded modules 7.2 Graded submodules 7.3 Applications 7.4 Filtrations 7.5 Grading associated to a filtration 8. Inductive limits 8.1 Generalities 8.2 Inductive systems of maps 8.3 Inductive systems of magmas, groups and rings 8.4 An example 8.5 Inductive systems of algebras 9. Sheaves of functions 9.1 Sheaves 9.2 Morphisms 9.3 Sheaf associated to a presheaf 9.4 Gluing 9.5 Ringed space 10. Jordan decomposition and some basic results on groups 10.1 Jordan decomposition 10.2 Generalities on groups 10.3 Commutators 10.4 Solvable groups 10.5 Nilpotent groups 10.6 Group actions 10.7 Generalities on representations 10.8 Examples 11. Algebraic sets 11.1 Affine algebraic sets 11.2 Zariski topology 11.3 Regular functions 11.4 Morphisms 11.5 Examples of morphisms 11.6 Abstract algebraic sets 11.7 Principal open subsets 11.8 Products of algebraic sets 12. Prevarieties and varieties 12.1 Structure sheaf 12.2 Algebraic prevarieties 12.3 Morphisms of prevarieties 12.4 Products of prevarieties 12.5 Algebraic varieties 12.6 Gluing 12.7 Rational functions 12.8 Local rings of a variety 13. Projective varieties 13.1 Projective spaces 13.2 Projective spaces and varieties 13.3 Cones and projective varieties 13.4 Complete varieties 13.5 Products 13.6 Grassmannian variety 14. Dimension 14.1 Dimension of varieties 14.2 Dimension and the number of equations 14.3 System of parameters 14.4 Counterexamples 15. Morphisms

Integrally Closed Modules and Their Divisors

ABSTRACT There is a beautiful theory of integral closure of ideals in regular local rings of dimension two, due to Zariski, several aspects of which were later extended to modules. Our goal is to study integral closures of modules over normal domains by attaching divisors/determinantal ideals to them. They will be of two kinds: the ordinary Fitting ideal and its divisor, and another ‘determinantal’ ideal obtained through Noether normalization. They are useful to describe the integral closure of some class of modules and to study the completeness of the modules of Kähler differentials.