Let A be a discrete valuation domain. It is known that an Eisenstein polynomial f in A[x] is useful in constructing a totally ramified extension B of A, where B≅A[x]/(f) and where the prime of B is the coset x‾ of x in B. We define a counter-Eisenstein formal power series c∈A[[x]], where A is a complete discrete valuation domain. The definition depends on the choice of a prime p∈A and a so-called set of representatives S⊆A. We show that such a counter-Eisenstein formal power series c play a role analogous to that played by an Eisenstein polynomial f in A[x]. Finally, we argue that there is a natural bijective correspondence between the counter-Eisenstein formal power series c’s in A[[x]] and the Eisenstein polynomials f’s in A[x] with the property that A[x]/(f)≅A[[x]]/(c) when f corresponds to c.
Let G be an additive abelian group. A sequence S=g1⋅…⋅gℓ of terms from G is a plus-minus weighted zero-sum sequence if there are ε1,…,εℓ∈{−1,1} such that ε1g1+…+εℓgℓ=0. We study sets of lengths in the monoid ℬ±(G) of plus-minus weighted zero-sum sequences over G. If G is finite, then sets of lengths are highly structured. If G is infinite, then every finite, nonempty subset of ℕ≥2 is the set of lengths of some sequence S∈ℬ±(G).
We begin the study of Lipschitz saturation for germs of toric singularities. By looking at their associated analytic algebras, we prove that if (X, 0) is a germ of toric singularity with smooth normalization then its Lipschitz saturation is again toric. Finally we show how to calculate the Lipschitz saturation for some families of toric singularities starting from the semigroup that defines them.
For 1 <= r <= n-1, let G(r,n) denote the Grassmannian parametrizing r-dimensional subspaces of C-n. Let (r, n) = 1. We show that the GIT quotients of certain Richardson varieties in G(r,n) for the action of a maximal torus in SL(n, C) are the product of projective spaces with respect to the descent of a suitable line bundle.
We provide a description of initial ideals for almost complete intersections generated by powers of general linear forms and prove that WLP in a fixed degree d holds when the number of variables n is sufficiently large compared to d. In particular, we show that if n≥ 3d-2 then WLP holds for the ideal generated by squares at the degree d spot and for n≥3d-3/2 WLP holds for ideal generated by cubes at the degree d spot. Finally, we prove that WLP fails for the ideal generated by squares when n< 3d -2 at the dth spot by finding an explicit element in the kernel of the multiplication by a general linear form. This shows that our bound on n is sharp in the case of the squares.
An integral domain D is a valuation ideal factorization domain (VIFD) if each nonzero principal ideal of D can be written as a finite product of valuation ideals. Clearly, pi-domains are VIFDs. We study the ring-theoretic properties of VIFDs and the *-operation analogs of VIFDs. Among them, we show that if D is treed (resp., *-treed), then D is a VIFD (resp., *-VIFD) if and only if D is an h-local Pr & uuml;fer domain (resp., a *-h-local P*MD) if and only if every nonzero prime ideal of D contains an invertible (resp., a *-invertible) valuation ideal. We also study integral domains D such that for each nonzero nonunit a E D, there is a positive integer n such that an can be written as a finite product of valuation elements.
Let (A, m) be an excellent normal local domain of dimension d > 2 with infinite residue field. Let I be an m-primary ideal. We prove the following assertions are equivalent: (i) The extended Rees algebra A[It, t(-1)] is R-1. (ii) The Rees algebra A[It] is R-1. (iii) Proj(A[It]) is R-1. (iv) (I-n)* = (I-n)(1) for all n >= 1. Here (I-n)* is the integral closure of In and (I-n)1 is the first coefficient ideal of I-n.
Let R be a standard graded, finitely generated algebra over a field, and let M be a graded module over R with all Bass numbers finite. Set (-)^(n) to be the n-th Veronese functor. We compute the Bass numbers of M^(n) over the ring R^(n) for all prime ideals of R^(n) that are not the homogeneous maximal ideal in terms of the Bass numbers of M over R. As an application to local cohomology modules, we determine the Bass numbers of H_I∩ R^(n)^i(R^(n)) over the ring R^(n) in the case where H_I^i(R) has finite Bass numbers over R and I is a graded ideal.
Let I_n,m = (x_1⋯ x_m,x_2 ⋯ x_m+1,…,x_n+1x_n+2⋯ x_n+m) be the m-path ideal of a path of length n + m-1 over a polynomial ring S = k[x_1,…,x_n+m]. We compute all the graded Betti numbers of all powers of I_n,m.
An almost complete intersection ideal can be seen as a d-sequence ideal with the minimal number of generators being one more than its height. In this paper, we give exact formulas for the regularity of powers of graded almost complete intersection ideals. We also present the minimal bigraded free resolutions of Rees algebras associated with linear type ideals having one generator more than their grade and study the properties of Cohen-Macaualyness, Koszulness associated to their diagonals.
Beilinson's resolution of the diagonal for complex projective space was generalized by Bayer-Popescu-Sturmfels for any unimodular toric variety. Here, we give a resolution of the diagonal for any smooth toric variety (viewed as a toric Deligne-Mumford stack) in families by deformation of the cellular complex of Bayer-Popescu-Sturmfels and show that the cokernel of this resolution gives the diagonal, modulo torsion from the irrelevant ideal. Furthermore, we give a resolution of the diagonal for a toric Deligne-Mumford stack associated to the global quotient of a smooth toric variety by a finite abelian group.
Let $M$ be a finitely generated module over a free twisted commutative algebra $A$ that is finitely generated in degree one. We show that the projective dimension of $M({\bf C}^n)$ as an $A({\bf C}^n)$-module is eventually linear as a function of $n$. This confirms a conjecture of Le, Nagel, Nguyen, and R\"omer for a special class of modules.
Richard Stanley introduced the order polytope $\mathcal{O}(P)$ and the chain polytope $\mathcal{C}(P)$ arising from a finite partially ordered set $P$, and showed that the Ehrhart polynomial of $\mathcal{O}(P)$ is equal to that of $\mathcal{C}(P)$. In addition, the unimodular equivalence problem of $\mathcal{O}(P)$ and $\mathcal{C}(P)$ was studied by the first author and Nan Li. In the present paper, three integral convex polytopes $\Gamma(\mathcal{O}(P), -\mathcal{O}(Q))$, $\Gamma(\mathcal{O}(P), -\mathcal{C}(Q))$ and $\Gamma(\mathcal{C}(P), -\mathcal{C}(Q))$, where $P$ and $Q$ are partially ordered sets with $| P | = | Q |$, will be studied. First, it will be shown that the Ehrhart polynomial of $\Gamma(\mathcal{O}(P), -\mathcal{C}(Q))$ coincides with that of $\Gamma(\mathcal{C}(P), -\mathcal{C}(Q))$. Furthermore, when $P$ and $Q$ possess a common linear extension, it will be proved that these three convex polytopes have the same Ehrhart polynomial. Second, the problem of characterizing partially ordered sets $P$ and $Q$ for which $\Gamma(\mathcal{O}(P), -\mathcal{O}(Q))$ or $\Gamma(\mathcal{O}(P), -\mathcal{C}(Q))$ or $\Gamma(\mathcal{C}(P), -\mathcal{C}(Q))$ is a smooth Fano polytope will be solved. Finally, when these three polytopes are smooth Fano polytopes, the unimodular equivalence problem of these three polytopes will be discussed.
Our objects, called polyclones, are generalizations of finitely generated modules: they are modules M Mi/Mi-1 of adjacent submodules are certain uniserial (rather than cyclic) modules. The focus is on modules for which these factors are divisible and the annihilators of elements are principal ideals. The results are of special interest over valuation domains admitting so-called nonstandard uniserial modules. We prove a theorem of Jordan-H & ouml;lder type and characterize the submodules that occur in composition series. Basic submodules are also discussed. It is shown that the isomorphy classes of polyclones over a valuation domain are elements in a semiring which is contained in a commutative ring where addition is direct sum and multiplication is torsion product.
Let R be a commutative local Noetherian ring. We use the theory of semigroups to analyze decompo-sitions of R-modules. By a semigroup (or monoid) of R-modules we mean a nonempty collection of R-modules closed under direct summands and finite direct sums. We give several examples of nonunique decompositions of finitely generated modules as direct sums of indecomposable submodules.
"Group" is understood to mean "abelian group". Let O be the class of all torsion groups. Given any group A the family of subgroups {U <= A A/U E O} serves as a neighborhood base at 0 E A defining a group topology on A in such a way that every group homomorphism is continuous. Thus the "O-topology" does not add new structure to a group, i.e., A similar to= B as groups if and only if A and B are isomorphic as topological groups, but the topology raises new questions, and introduces related concepts and objects such as completions. The 0-topology is thoroughly investigated in this paper.
Trying to finalize in some way the present subject, this paper targets to generalize substantially the notions of Bassian and co-Bassian groups by introducing the so-called finitely (co-)Bassian groups, semi and establishing their crucial properties and characterizations. In fact, some of the concepts give nothing new by coinciding in the reduced case with the well-known (co-)Bassian property. However, in some of the definitions, the situation is slightly more complicated and we obtain a few new and interesting things by showing that these extensions of the Bassian and co-Bassian properties are totally distinct from each other.
The one-dimensional compact connected abelian groups, called solenoids, are classified and constructed as topological subgroups of the torus T aleph 0. For an arbitrary solenoid E not equal T, we exhibit a nonsplitting extension of E by a profinite group, dual to a nonsplitting extension 0 -> tor(A) -> A -> F -> 0 of abelian groups where F is a rank-1 torsion-free group not equal 7L. The constructed groups A are generalizations of examples of Fuchs.
