Mathematics

Let A be a finite subset of an abelian group (G, +). Let h ≥ 2 be an integer. If |A| ≥ 2 and the cardinality |hA| of the h-fold iterated sumset hA = A + × × × + A is known, what can one say about |(h -- 1)A| and |(h + 1)A|? It is known that |(h -- 1)A| ≥ |hA| (h--1)/h , a consequence of Pl{ü}nnecke's inequality. Here we improve this bound with a new approach. Namely, we model the sequence |hA| h ≥ 0 with the Hilbert function of a standard graded algebra. We then apply Macaulay's 1927 theorem on the growth of Hilbert functions, and more specifically a recent condensed version of it. Our bound implies |(h -- 1)A| ≥ θ (x, h) |hA| (h--1)/h for some factor θ (x, h) > 1, where x is a real number closely linked to |hA|. Moreover, we show that θ (x, h) asymptotically tends to e ≈ 2.718 as |A| grows and h lies in a suitable range varying with |A|.

We let A=R/I be a standard graded Artinian algebra quotient of R=k[x,y] , the polynomial ring in two variables over a field k by an ideal I , and let n be its vector space dimension. The Jordan type P ℓ of a linear form ℓ∈ A 1 is the partition of n determining the Jordan block decomposition of the multiplication on A by ℓ - which is nilpotent. The first three authors previously determined which partitions of n= dim k A may occur as the Jordan type for some linear form ℓ on a graded complete intersection Artinian quotient A=R/(f,g) of R , and they counted the number of such partitions for each complete intersection Hilbert function T [arXiv:1810.00716]. We here consider the family G T of graded Artinian quotients A=R/I of R=k[x,y] , having arbitrary Hilbert function H(A)=T . The cell V( E P ) corresponding to a partition P having diagonal lengths T is comprised of all ideals I in R whose initial ideal is the monomial ideal E P determined by P . These cells give a decomposition of the variety G T into affine spaces. We determine the generic number κ(P) of generators for the ideals in each cell V( E P ) , generalizing a result of [arXiv:1810.00716]. In particular, we determine those partitions for which κ(P)=κ(T) , the generic number of generators for an ideal defining an algebra A in G T . We also count the number of partitions P of diagonal lengths T having a given κ(P) . A main tool is a combinatorial and geometric result allowing us to split T and any partition P of diagonal lengths T into simpler T i and partitions P i , such that V( E P ) is the product of the cells V( E P i ) , and T i is single-block: G T i is a Grassmannian.

We study Jordan types of linear forms for graded Artinian Gorenstein algebras having arbitrary codimension. We introduce rank matrices of linear forms for such algebras that represent the ranks of multiplication maps in various degrees. We show that there is a 1-1 correspondence between rank matrices and Jordan degree types. For Artinian Gorenstein algebras with codimension three we classify all rank matrices that occur for linear forms with vanishing third power. As a consequence, we show for such algebras that the possible Jordan types with parts of length at most four are uniquely determined by at most three parameters.

In this paper I consider the construction of K + M, where K is the domain, M is the maximal ideal of a some ring of polynomials with coefficients from the field L, where K is its subring. In addition to the usual domains, we also consider the Noetherian, Prufer and GCD-domains. In particular, polynomial composites are a case of K + M construction. In this paper we will find numerous construction conclusions related to polynomial composites.

Motivated by a work of Knutson, in a recent paper Conca and Varbaro have defined a new class of ideals, namely "Knutson ideals", starting from a polynomial f with squarefree leading term. We will show that the main properties that this class has in polynomial rings over fields of characteristic p are preserved when one introduces the definition of Knutson ideal also in polynomial rings over fields of characteristic zero. Then we will show that determinantal ideals of Hankel matrices are Knutson ideals for a suitable choice of the polynomial f .

In this paper we show that determinantal ideals of generic matrices are Knutson ideals. This fact leads to a useful result about Gröbner bases of certain sums of determinantal ideals. More specifically, given I= I 1 +?? I k a sum of ideals of minors on adjacent columns or rows, we prove that the union of the Gröbner bases of the I j 's is a Gröbner basis of I .

For a local ring (A,m) of dimension n , we study the natural map from the Koszul cohomology module H n (m;A) to the local cohomology module H n m (A) . We prove that the injectivity of this map characterizes the Cohen-Macaulay property of the ring A . We also answer a question of Dutta by constructing normal rings A for which this map is zero.

We prove a Cohen-Macaulay version of a result by Avramov-Golod and Frankild-Jørgensen about Gorenstein rings, showing that if a noetherian ring A is Cohen-Macaulay, and a 1 ,…, a n is any sequence of elements in A , then the Koszul complex K(A; a 1 ,…, a n ) is a Cohen-Macaulay DG-ring. We further generalize this result, showing that it also holds for commutative DG-rings. In the process of proving this, we develop a new technique to study the dimension theory of a noetherian ring A , by finding a Cohen-Macaulay DG-ring B such that H 0 (B)=A , and using the Cohen-Macaulay structure of B to deduce results about A . As application, we prove that if f:X→Y is a morphism of schemes, where X is Cohen-Macaulay and Y is nonsingular, then the homotopy fiber of f at every point is Cohen-Macaulay. As another application, we generalize the miracle flatness theorem. Generalizations of these applications to derived algebraic geometry are also given.

Given a monomial m in a polynomial ring and a subset L of the variables of the polynomial ring, the principal L -Borel ideal generated by m is the ideal generated by all monomials which can be obtained from m by successively replacing variables of m by those which are in L and have smaller index. Given a collection I={ I 1 ,…, I r } where I i is L i -Borel for i=1,…,r (where the subsets L 1 ,…, L r may be different for each ideal), we prove in essence that if the bipartite incidence graph among the subsets L 1 ,…, L r is chordal bipartite, then the defining equations of the multi-Rees algebra of I has a Gröbner basis of quadrics with squarefree lead terms under lexicographic order. Thus the multi-Rees algebra of such a collection of ideals is Koszul, Cohen-Macaulay, and normal. This significantly generalizes a theorem of Ohsugi and Hibi on Koszul bipartite graphs. As a corollary we obtain that the multi-Rees algebra of a collection of principal Borel ideals is Koszul. To prove our main result we use a fiber-wise Gröbner basis criterion for the kernel of a toric map and we introduce a modification of Sturmfels' sorting algorithm.

We explicitly describe the divisor class groups and semidualizing modules for ladder determinantal rings with coefficients in an arbitrary normal domain for arbitrary ladders, not necessarily connected, and all sizes of minors.