Minimal free resolution of the associated graded ring of monomial curves of generalized arithmetic sequences

Let A(C) be the coordinate ring of a monomial curve C⊆Aⁿ corresponding to the numerical semigroup S minimally generated by a sequence a₀,…,a_n. In the literature, little is known about the Betti numbers of the corresponding associated graded ring gr_m(A) with respect to the maximal ideal m of A=A(C). In this paper we characterize the numerical invariants of a minimal free resolution of gr_m(A) in the case a₀,…,a_n is a generalized arithmetic sequence.     

A note on matrix subalgebras containing a full Jordan block

Let A be a subalgebra of the full matrix algebra M_n(F), and suppose J∈A, where J is the Jordan block corresponding to xⁿ. Let $\text{S}$ be a set of generators of A. It is shown that the graph of $\text{S}$ determines whether A is the full matrix algebra M_n(F).     

Set theoretic complete intersection for curves in a smooth affine algebra

Let A be a regular ring of dimension d (d≥3) containing an infinite field k. Let n be an integer such that 2n≥d+3. Let I be an ideal in A of height n and P be a projective A-module of rank n. Suppose P⊕A≈Aⁿ⁺¹ and there is a surjection α: P→I. It is proved in this note that I is a set theoretic complete intersection ideal. As a consequence, a smooth curve in a smooth affine C-algebra with trivial conormal bundle is a set theoretic complete intersection if its corresponding class in the Grothendieck group is torsion.     

The minimum distance of sets of points and the minimum socle degree

Let K be a field of characteristic 0. Let be a reduced finite set of points, not all contained in a hyperplane. Let be the maximum number of points of Γ contained in any hyperplane, and let . If I⊂R=K[x₀,…,x_n] is the ideal of Γ, then in Tohaˇneanu (2009) [12] it is shown that for n=2,3, d(Γ) has a lower bound expressed in terms of some shift in the graded minimal free resolution of R/I. In these notes we show that this behavior holds true in general, for any n≥2: d(Γ)≥A_n, where A_n=min{a_i−n} and ⊕_iR(−a_i) is the last module in the graded minimal free resolution of R/I. In the end we also prove that this bound is sharp for a whole class of examples due to Juan Migliore (2010) [10].     

Semidefinite descriptions of low-dimensional separable matrix cones

Let K⊂E, K^′⊂E^′ be convex cones residing in finite-dimensional real vector spaces. An element y in the tensor product E⊗E^′ is K⊗K^′-separable if it can be represented as finite sum , where x_l∈K and for all l. Let S(n), H(n), Q(n) be the spaces of n×n real symmetric, complex Hermitian and quaternionic Hermitian matrices, respectively. Let further S₊(n), H₊(n), Q₊(n) be the cones of positive semidefinite matrices in these spaces. If a matrix A∈H(mn)=H(m)⊗H(n) is H₊(m)⊗H₊(n)-separable, then it fulfills also the so-called PPT condition, i.e. it is positive semidefinite and has a positive semidefinite partial transpose. The same implication holds for matrices in the spaces S(m)⊗S(n), H(m)⊗S(n), and for m?2 in the space Q(m)⊗S(n). We provide a complete enumeration of all pairs (n,m) when the inverse implication is also true for each of the above spaces, i.e. the PPT condition is sufficient for separability. We also show that a matrix in Q(n)⊗S(2) is Q₊(n)⊗S₊(2)- separable if and only if it is positive semidefinite.     

A finiteness criterion and asymptotics for codimensions of generalized identities

Let A be an associative algebra over a field of characteristic zero. Then either all codimensions gc _n (A) of its generalized polynomial identities are infinite or A is the sum of ideals I and J such that dim _F I < ∞ and J is nilpotent. In the latter case, there exist numbers n ₀ ∈ ?, C ∈ ?₊, and t ∈ ?₊ for which gc _n (A) < +∞ if n ≥ n ₀ and gc _n (A) ～ Cn ^t d ⁿ as n → ∞, where d = PIexp(A) ∈ ?₊. Thus, in the latter case, conjectures of Amitsur and Regev on generalized codimensions hold.     

On the dimension of global sections of adjoint bundles for polarized 3-folds and 4-folds

Let (X,L) be a polarized manifold of dimension n defined over the field of complex numbers. In this paper, we treat the case where n=3 and 4. First we study the case of n=3 and we give an explicit lower bound for h⁰(K_X+L) if κ(X)≥0. Moreover, we show the following: if κ(K_X+L)≥0, then h⁰(K_X+L)>0 unless κ(X)=−∞ and h¹(O_X)=0. This gives us a partial answer of Effective Non-vanishing Conjecture for polarized 3-folds. Next for n=4 we investigate the dimension of H⁰(K_X+mL) for m≥2. If n=4 and κ(X)≥0, then a lower bound for h⁰(K_X+mL) is obtained. We also consider a conjecture of Beltrametti-Sommese for 4-folds and we can prove that this conjecture is true unless κ(X)=−∞ and h¹(O_X)=0. Furthermore we prove the following: if (X,L) is a polarized 4-fold with κ(X)≥0 and h¹(O_X)>0, then h⁰(K_X+L)>0.     

Local cohomology modules with respect to an ideal containing the irrelevant ideal

Let R=?_n≥0R_n be a homogeneous Noetherian ring, let M be a finitely generated graded R-module and let R₊=?_n>0R_n. Let b?b₀+R₊, where b₀ is an ideal of R₀. In this paper, we first study the finiteness and vanishing of the n-th graded component of the i-th local cohomology module of M with respect to b. Then, among other things, we show that the set becomes ultimately constant, as n→−∞, in the following cases:

(i)

and (R₀,m₀) is a local ring;

(ii)

dim(R₀)≤1 and R₀ is either a finite integral extension of a domain or essentially of finite type over a field;

(iii)

i≤g_b(M), where g_b(M) denotes the cohomological finite length dimension of M with respect to b.

Also, we establish some results about the Artinian property of certain submodules and quotient modules of .     

Hamiltonian cycles and dominating cycles passing through a linear forest

Let G be an (m+2)-graph on n vertices, and F be a linear forest in G with |E(F)|=m and ω₁(F)=s, where ω₁(F) is the number of components of order one in F. We denote by σ₃(G) the minimum value of the degree sum of three vertices which are pairwise non-adjacent. In this paper, we give several σ₃ conditions for a dominating cycle or a hamiltonian cycle passing through a linear forest. We first prove that if σ₃(G)≥n+2m+2+max{s−3,0}, then every longest cycle passing through F is dominating. Using this result, we prove that if σ₃(G)≥n+κ(G)+2m−1 then G contains a hamiltonian cycle passing through F. As a corollary, we obtain a result that if G is a 3-connected graph and σ₃(G)≥n+κ(G)+2, then G is hamiltonian-connected.     

Identifications in modular group algebras

Let G be a group, S a subgroup of G, and F a field of characteristic p. We denote the augmentation ideal of the group algebra FG by ω(G). The Zassenhaus-Jennings-Lazard series of G is defined by D_n(G)=G∩(1+ωⁿ(G)). We give a constructive proof of a theorem of Quillen stating that the graded algebra associated with FG is isomorphic as an algebra to the enveloping algebra of the restricted Lie algebra associated with the D_n(G). We then extend a theorem of Jennings that provides a basis for the quotient ωⁿ(G)/ωⁿ⁺¹(G) in terms of a basis of the restricted Lie algebra associated with the D_n(G). We shall use these theorems to prove the main results of this paper. For G a finite p-group and n a positive integer, we prove that G∩(1+ω(G)ωⁿ(S))=D_n+1(S) and G∩(1+ω²(G)ωⁿ(S))=D_n+2(S)D_n+1(S∩D₂(G)). The analogous results for integral group rings of free groups have been previously obtained by Gruenberg, Hurley, and Sehgal.     

On classical adjoint-commuting mappings between matrix algebras

Let F be a field and let m and n be integers with m,n?3. Let M_n denote the algebra of n×n matrices over F. In this note, we characterize mappings ψ:M_n→M_m that satisfy one of the following conditions:

1.

|F|=2 or |F|>n+1, and ψ(adj(A+αB))=adj(ψ(A)+αψ(B)) for all A,B∈M_n and α∈F with ψ(I_n)≠0.

2.

ψ is surjective and ψ(adj(A-B))=adj(ψ(A)-ψ(B)) for every A,B∈M_n.

Here, adjA denotes the classical adjoint of the matrix A, and I_n is the identity matrix of order n. We give examples showing the indispensability of the assumption ψ(I_n)≠0 in our results.     

A numerical characterization of the S2-ification of a Rees algebra

Let A be a local ring with maximal ideal . For an arbitrary ideal I of A, we define the generalized Hilbert coefficients . When the ideal I is -primary, j_k(I)=(0,…,0,(−1)^ke_k(I)), where e_k(I) is the classical kth Hilbert coefficient of I. Using these coefficients we give a numerical characterization of the homogeneous components of the S₂-ification of S=A[It,t⁻¹], extending previous results obtained by the author to not necessarily -primary ideals.     

Representations of positive polynomials on noncompact semialgebraic sets via KKT ideals

This paper studies the representation of a positive polynomial f(x) on a noncompact semialgebraic set S={x∈Rⁿ:g₁(x)≥0,…,g_s(x)≥0} modulo its KKT (Karush-Kuhn-Tucker) ideal. Under the assumption that the minimum value of f(x) on S is attained at some KKT point, we show that f(x) can be represented as sum of squares (SOS) of polynomials modulo the KKT ideal if f(x)>0 on S; furthermore, when the KKT ideal is radical, we argue that f(x) can be represented as a sum of squares (SOS) of polynomials modulo the KKT ideal if f(x)≥0 on S. This is a generalization of results in [J. Nie, J. Demmel, B. Sturmfels, Minimizing polynomials via sum of squares over the gradient ideal, Mathematical Programming (in press)], which discusses the SOS representations of nonnegative polynomials over gradient ideals.     

Large spaces of symmetric matrices of bounded rank are decomposable ∗

Let k and n be positive integers such that kn. Let S_n (F) denote the space of all n×n symmetric matrices over the field F with char F≠2. A subspace L of S_n (F) is said to be a k-subspace if rank Ak for every A?L.

Now suppose that k is even, and write k=2r. We say a k∥-subspace of S_n (F) is decomposable if there exists in Fⁿ a subspace W of dimension n?r such that x^tAx=0 for every x?W A?L.

We show here, under some mild assumptions on k n and F, that every k∥-subspace of S_n (F) of sufficiently large dimension must be decomposable. This is an analogue of a result obtained by Atkinson and Lloyd for corresponding subspaces of F^m,n .     

A new topological degree theory for perturbations of the sum of two maximal monotone operators

Let X be an infinite dimensional real reflexive Banach space with dual space X^∗ and G⊂X, open and bounded. Assume that X and X^∗ are locally uniformly convex. Let T:X⊃D(T)→2^X∗ be maximal monotone and strongly quasibounded, S:X⊃D(S)→X^∗ maximal monotone, and C:X⊃D(C)→X^∗ strongly quasibounded w.r.t. S and such that it satisfies a generalized (S₊)-condition w.r.t. S. Assume that D(S)=L⊂D(T)∩D(C), where L is a dense subspace of X, and 0∈T(0),S(0)=0. A new topological degree theory is introduced for the sum T+S+C, with degree mapping d(T+S+C,G,0). The reason for this development is the creation of a useful tool for the study of a class of time-dependent problems involving three operators. This degree theory is based on a degree theory that was recently developed by Kartsatos and Skrypnik just for the single-valued sum S+C, as above.     

Degree functions and projectively full ideals in 2-dimensional rational singularities that can be desingularized by blowing up the unique maximal ideal

Let (R,m) be a 2-dimensional rational singularity with algebraically closed residue field and for which the associated graded ring is an integrally closed domain. According to Göhner, (R,m) satisfies condition (N): given a prime divisor v, there exists a unique complete m-primary ideal A_v in R with T(A_v)={v} and such that any complete m-primary ideal with unique Rees valuation v, is a power of A_v. We use the theory of degree functions developed by Rees and Sharp as well as some results about regular local rings, to investigate the degree coefficients d(A_v,v). As an immediate corollary, we find that for a simple complete m₁-primary ideal I₁ in an immediate quadratic transform (R₁,m₁) of (R,m); the inverse transform of I₁ in R is projectively full.     

Higher syzygies of hyperelliptic curves

Let X be a hyperelliptic curve of arithmetic genus g and let f:X→P¹ be the hyperelliptic involution map of X. In this paper we study higher syzygies of linearly normal embeddings of X of degree d≤2g. Note that the minimal free resolution of X of degree ≥2g+1 is already completely known. Let A=f^∗O_P¹(1), and let L be a very ample line bundle on X of degree d≤2g. For , we call the pair (m,d−2m)the factorization type ofL. Our main result is that the Hartshorne-Rao module and the graded Betti numbers of the linearly normal curve embedded by |L| are precisely determined by the factorization type of L.     

Henriksen?s contributions to residue class rings of analytic and entire functions

The author surveys, summarizes and generalizes results of Golasiński and Henriksen, and of others, concerning certain residue class rings.Let A(R) denote the ring of analytic functions over reals R and E(K) the ring of entire functions over R or complex numbers C. It is shown that if m is a maximal ideal of A(R), then A(R)/m is isomorphic either to the reals or a real-closed field that is η₁-set, while if m is a maximal ideal of E(K), then E(K)/m is isomorphic to one of these latter two fields or to complex numbers.     

Hankel planes

Starting from an (m+1)×(n+1) matrix A one can construct (m+p+1)×(n+1)(p+1) block Toeplitz matrices , p≥0, based on the rows of A. The connections between the ranks of the two matrices is studied by comparing the corresponding vector spaces of row relations R and R(p). A main tool are the Hankel matrices with rows in R. The dimension of R(p) is determined in terms of geometric invariants attached to the Hankel matrices with rows in R. The study of Hankel r-planes of P^m, for r≥1, turns out to be very useful and interesting in itself since they constitute a subvariety of the Grassmannian G(r,m).