Prime T-Ideals in Polynomial and Power Series Rings Over a Pseudo-Valuation Domain

Let R be an m-dimensional pseudo-valuation domain with residue field k, let V be the associated valuation domain with residue field K, and let k ₀ be the maximal separable extension of k in K. We compute the t-dimension of polynomial and power series rings over R. It is easy to see that t-dim R[x ₁,…, x _n ] = 2 if m = 1 and K is transcendental over k, but equals m otherwise, and that t-dim R[[x ₁,…, x _n ]] = ∞ if R is a nonSFT-ring. When R is an SFT-ring, we also show that: (1) t-dim R[[x]] = m; (2) t-dim R[[x ₁,…, x _n ]] = 2m ? 1, if n ≥ 2, K has finite exponent over k ₀, and [k ₀: k] < ∞; (3) t-dim R[[x ₁,…, x _n ]] = 2m, otherwise.     

Minimal graded free resolutions for monomial curves in 𝔸4 defined by almost arithmetic sequences

Let m = (m₀, m₁, m₂, n) be an almost arithmetic sequence, i.e., a sequence of positive integers with gcd(m₀, m₁, m₂, n) = 1, such that m₀ < m₁ < m₂ form an arithmetic progression, n is arbitrary and they minimally generate the numerical semigroup Γ =m₀? +m₁? +m₂? +n?. Let k be a field. The homogeneous coordinate ring k[Γ] of the affine monomial curve parametrically defined by X₀ = t^m₀, X₁ = t^m₁, X₂ = t^m₂, Y = tⁿ is a graded R-module, where R is the polynomial ring k[X₀, X₁, X₂, Y] with the grading degX_i: = m_i, degY: = n. In this paper, we construct a minimal graded free resolution for k[Γ].     

Stability of Numerical Invariants in Free Groups

Let d be an odd integer, and let k be a field which contains a primitive dth root of unity. Let l ₁ and l ₂ be cyclic field extensions of k of degree d with norms n _{l 1/k} and n _{l 2/k} . Minà?'s approach which showed that quadratic Pfister forms are strongly multiplicative is applied to the form n _{l 1/k}  ? n _{l 2/k} of degree d. Let K = k(X ₁,…, X _{d ²} ). We compute polynomials which are similarity factors of a form of the kind N ? (n _{l 2/k}  ? _k K) over K, where N is the norm of a certain field extension of K of degree d. These polynomials arise by specializing certain indeterminates of the homogeneous polynomial representing the form n _{l 1/k}  ? n _{l 2/k} to be zero. Similar results are obtained for the tensor product of the norm of a cubic division algebra and a cubic norm n _{l 1/k} .     

THE STRUCTURE OF THE TANGENT CONE: AN INTERESTING BIJECTION#

ABSTRACT

Let X = Spec(R) be a reduced equidimensional algebraic variety over an algebraically closed field k. Let Y = Spec(R/𝔮) be a codimension one ordinary multiple subvariety, where 𝔮 is a prime ideal of height 1 of R. If U is a nonempty open subset of Y and 𝔪 a closed point of U, we denote by A ? R _𝔪 its local ring in X, by 𝔭 the extension of 𝔮 in A, and by K the algebraic closure of the residue field k(𝔭).

Then there exists a bijection γ^𝔪:Proj(G _𝔭(A) ?  _A/𝔭 k) → Proj(G(A _𝔭) ?  _k(𝔭)K) such that for every subset Σ of Proj(G _𝔭(A) ?  _A/𝔭 k), the Hilbert function of Σ coincides with the Hilbert function of γ^𝔪(Σ). We examine some applications. We study the structure of the tangent cone at a closed point of a codimension one ordinary multiple subvariety.     

Catenarian Property of Mixed Polynomial and Power Series Rings Over a Pullback

Let T be an integral domain with a maximal ideal M, ?: T → K: = T/M the natural surjection, and R the pullback ?^?1(D), where D is a proper subring of K. We give necessary and sufficient conditions for the mixed extensions R[x ₁]]…[x _n ]] to be catenarian, where each [x _i ]] is fixed as either [x _i ] or [[x _i ]]. We also give a complete answer to the question of determining the field extensions k ? K such that the contraction map Spec(K[x ₁]]…[x _n ]]) → Spec(k[x ₁]]…[x _n ]]) is a homeomorphism. As an application, we characterize the globalized pseudo-valuation domains R such that R[x ₁]]…[x _n ]] is catenarian.     

Rank One Discrete Valuations of Power Series Fields

In this article we study rank one discrete valuations of the field k((X ₁,…, X _n )) whose center in k[[X ₁,…, X _n ]] is the maximal ideal. In Sections 2 to 6 we give a construction of a system of parametric equations describing such valuations. This amounts to finding a parameter and a field of coefficients. We devote Section 2 to finding an element of value 1, that is, a parameter. The field of coefficients is the residue field of the valuation, and it is given in Section 5.

The constructions given in these sections are not effective in the general case, because we need either to use Zorn's lemma or to know explicitly a section σ of the natural homomorphism R _v  → Δ _v between the ring and the residue field of the valuation v.

However, as a consequence of this construction, in Section 7, we prove that k((X ₁,…, X _n )) can be embedded into a field L((Y ₁,…, Y _n )), where L is an algebraic extension of k and the “extended valuation” is as close as possible to the usual order function.     

On the Splitting Ring of a Polynomial

Let f(Z) = Zⁿ ? a₁Z^n?1 + … + (?1)^n?1a_n?1Z + (?1)ⁿa_n be a monic polynomial with coefficients in a ring R with identity, not necessarily commutative. We study the ideal I_f of R[X₁,…, X_n] generated by σ_i(X₁,…, X_n) ? a_i, where σ₁,…, σ_n are the elementary symmetric polynomials, as well as the quotient ring R[X₁,…, X_n]/I_f.     

Length Four Polynomial Automorphisms

We study the structure of length four polynomial automorphisms of R[X, Y] when R is a unique factorization domain. The results from this study are used to prove that, if SL _m (R[X ₁, X ₂,…, X _n ]) = E _m (R[X ₁, X ₂,…, X _n ]) for all n, m ≥ 0, then all length four polynomial automorphisms of R[X, Y] that are commutators are stably tame.     

Linear Groups Over Integral Extensions of Semilocal Commutative Rings

Let K be an associative and commutative ring with 1, k a subring of K such that 1 ∈ k, K is an integral finitely generated extension of k, the element 2 invertible in k, and k is semilocal. The paper studies subgroups of the general linear group GL _n (K) with n ≥ 2 containing the special linear group SL _n (k).     

Resolutions of subsets of finite sets of points in projective space

Abstract

Eisenbud et al. proved a number of results regarding Gröbner bases and initial ideals of those ideals J in the free associative algebra K ?X ₁,…, X _n ? which contain the commutator ideal. We prove similar results for ideals which contains the anti-commutator ideal (the defining ideal of the exterior algebra). We define one weak notion of generic initial ideals in K ?X ₁,…, X _n ?, and show that generic initial ideals of ideals containing the anti-commutator ideal, or the commutator ideal, are finitely generated.     

Localization and Catenarity in Iterated Differential Operator Rings

ABSTRACT

Let k be a field, R an associative k-algebra with identity, Δ a finite set of derivations of R, and R[Θ₁, δ₁] ··· [Θ _n , δ _n ] an iterated differential operator k-algebra over R such that δ _j (Θ _i ) ∈ R[Θ₁, δ₁] ··· [Θ _i?1, δ _i?1]; 1 ≤ i < j ≤ n. If R is Noetherian Δ-hypercentral, then every prime ideal P of A is classically localizable. The aim of this article is to show that under some additional hypotheses on the Δ-prime ideals of R, the local ring A _P is regular in the sense of Robert Walker. We use this result to study the catenarity of A and to compute the numbers μ _i of Bass. Let g be a nilpotent Lie algebra of finite dimension n acting on R by derivations and U(g) the enveloping algebra of g. Then the crossed product of R by U(g) is an iterated differential operator k-algebra as above. In this particular case, our results are known if k has characteristic zero.     

Generic Initial Ideals,Graded Betti Numbers,and k-Lefschetz Properties

We introduce the k-strong Lefschetz property and the k-weak Lefschetz property for graded Artinian K-algebras, which are generalizations of the Lefschetz properties. The main results are:

1. Let I be an ideal of R = K[x ₁, x ₂,…, x _n ] whose quotient ring R/I has the n-SLP. Suppose that all kth differences of the Hilbert function of R/I are quasi-symmetric. Then the generic initial ideal of I is the unique almost revlex ideal with the same Hilbert function as R/I.

2. We give a sharp upper bound on the graded Betti numbers of Artinian K-algebras with the k-WLP and a fixed Hilbert function.     

Generalized Derivations with Engel Conditions on Polynomials

Let R be a noncommutative prime ring with extended centroid C, and let D: R → R be a nonzero generalized derivation, f(X ₁,…, X _t ) a nonzero polynomial in noncommutative indeterminates X ₁,…, X _t over C with zero constant term, and k ≥ 1 a fixed integer. In this article, D and f(X ₁,…, X _t ) are characterized if the Engel identity is satisfied: [D(f(x ₁,…, x _t )), f(x ₁,…, x _t )] _k  = 0 for all x ₁,…, x _t  ∈ R.     

Metrizability of Spaces of ℝ-places of Function Fields of Transcendence Degree 1 Over Real Closed Fields

In this article we discuss the following question “When do different orderings of the rational function field R(X) (where R is a real closed field) induce the same ?-place?” We use this to show that if R contains a dense real closed subfield R′, then the spaces of ?-places of R(X) and R′(X) are homeomorphic. For the function field K = R(X), we prove that its space M(K) of ?-places is metrizible if and only if R contains a countable dense subfield. Moreover, we show that this condition is neccessary for the metrizability of M(F) for any function field F of transcendence degree 1 over R.     

Relatively Free Nilpotent Torsion-Free Groups and Their Lie Algebras

Let K be a field of characteristic zero. For a torsion-free finitely generated nilpotent group G, we naturally associate four finite dimensional nilpotent Lie algebras over K, ? _K (G), grad^(?)(? _K (G)), grad^(g)(exp ? _K (G)), and L _K (G). Let 𝔗 _c be a torsion-free variety of nilpotent groups of class at most c. For a positive integer n, with n ≥ 2, let F _n (𝔗 _c ) be the relatively free group of rank n in 𝔗 _c . We prove that ? _K (F _n (𝔗 _c )) is relatively free in some variety of nilpotent Lie algebras, and ? _K (F _n (𝔗 _c )) ? L _K (F _n (𝔗 _c )) ? grad^(?)(? _K (F _n (𝔗 _c ))) ? grad^(g)(exp ? _K (F _n (𝔗 _c ))) as Lie algebras in a natural way. Furthermore, F _n (𝔗 _c ) is a Magnus nilpotent group. Let G ₁ and G ₂ be torsion-free finitely generated nilpotent groups which are quasi-isometric. We prove that if G ₁ and G ₂ are relatively free of finite rank, then they are isomorphic. Let L be a relatively free nilpotent Lie algebra over ? of finite rank freely generated by a set X. Give on L the structure of a group R, say, by means of the Baker–Campbell–Hausdorff formula, and let H be the subgroup of R generated by the set X. We show that H is relatively free in some variety of nilpotent groups; freely generated by the set X, H is Magnus and L ? ?_?(H) ? L _?(H) as Lie algebras. For relatively free residually torsion-free nilpotent groups, we prove that ? _K and L _K are isomorphic as Lie algebras. We also give an example of a finitely generated Magnus nilpotent group G, not relatively free, such that ?_?(G) is not isomorphic to L _?(G) as Lie algebras.     

Interval partitions and Stanley depth

In this paper, we answer a question posed by Herzog, Vladoiu, and Zheng. Their motivation involves a 1982 conjecture of Richard Stanley concerning what is now called the Stanley depth of a module. The question of Herzog et al., concerns partitions of the non-empty subsets of {1,2,…,n} into intervals. Specifically, given a positive integer n, they asked whether there exists a partition P(n) of the non-empty subsets of {1,2,…,n} into intervals, so that |B|?n/2 for each interval [A,B] in P(n). We answer this question in the affirmative by first embedding it in a stronger result. We then provide two alternative proofs of this second result. The two proofs use entirely different methods and yield non-isomorphic partitions. As a consequence, we establish that the Stanley depth of the ideal (x₁,…,xn)⊆K[x₁,…,xn] (K a field) is ⌈n/2⌉.     

The multiplier ideal is a universal test ideal

Abstract

For a canonical threefold X, we know that h ⁰(X, 𝒪 _X (nK _X )) ≥ 1 for a sufficiently large n. When χ(𝒪 _X ) > 0, it is not easy to get such an integer n. Fletcher showed that h ⁰(X, 𝒪 _X (12K _X )) ≥ 1 and h ⁰(X, 𝒪 _X (24K _X )) ≥ 2 when χ(𝒪 _X ) = 1. He inquired about existence of a canonical threefold with given conditions which shows the result sharp. We show that such an example does not exist. Using a different technique, we prove h ⁰(X, 𝒪 _X (12K _X )) ≥ 2.