Free Z-module Research Articles

The second integral cohomology of moduli spaces of sheaves on K3 and Abelian surfaces

In this paper we study the second integral cohomology of moduli spaces of semistable sheaves on projective K3 surfaces. If S is a projective K3 surface, v a Mukai vector and H a polarization on S that is general with respect to v, we show that H2(Mv,Z) is a free Z-module of rank 23 carrying a pure weight-two Hodge structure and a lattice structure, with respect to which H2(Mv,Z) is Hodge isometric to the Hodge sublattice v⊥ of the Mukai lattice of S. Similar results are proved for Abelian surfaces.

A note on Grothendieck groups of periodic derived categories

The m-periodic derived category of an abelian category is a natural Z/mZ-periodic analogue of the usual derived category. We determine the Grothendieck group of the periodic derived category of a skeletally small abelian category with enough projectives. In particular, we prove that the Grothendieck group of the m-periodic derived category of finitely generated modules over an Artin algebra is a free Z-module if m is even, and is an F2-vector space if m is odd. Moreover, in both cases of parity of m, the rank of the Grothendieck group is equal to the number of isomorphism classes of simple modules in both cases. As an application, we prove that the number of non-isomorphic summands of a strict periodic tilting object T, which was introduced by the author in [6] as a periodic analogue of tilting objects, is independent of the choice of T.

The number of ideals of [formula omitted] containing x(x − α)(x − β) with given index

It is well-known that a connected regular graph is strongly-regular if and only if its adjacency matrix has exactly three eigenvalues. Let B denote an integral square matrix and 〈B〉 denote the subring of the full matrix ring generated by B. Then 〈B〉 is a free Z-module of finite rank, which guarantees that there are only finitely many ideals of 〈B〉 with given finite index. Thus, the formal Dirichlet series ζ〈B〉(s)=∑n≥1ann−s is well-defined where an is the number of ideals of 〈B〉 with index n. In this article we aim to find an explicit form of ζ〈B〉(s) when B has exactly three eigenvalues all of which are integral, e.g., the adjacency matrix of a strongly-regular graph which is not a conference graph with a non-squared number of vertices. By isomorphism theorem for rings, 〈B〉 is isomorphic to Z[x]/m(x)Z[x] where m(x) is the minimal polynomial of B over Q, and Z[x]/m(x)Z[x] is isomorphic to Z[x]/m(x+γ)Z[x] for each γ∈Z. Thus, the problem is reduced to counting the number of ideals of Z[x]/x(x−α)(x−β)Z[x] with given finite index where 0,α and β are distinct integers.

Matrix of ℤ-module1

Summary In this article, we formalize a matrix of ℤ-module and its properties. Specially, we formalize a matrix of a linear transformation of ℤ-module, a bilinear form and a matrix of the bilinear form (Gramian matrix). We formally prove that for a finite-rank free ℤ-module V, determinant of its Gramian matrix is constant regardless of selection of its basis. ℤ-module is necessary for lattice problems, LLL (Lenstra, Lenstra and Lovász) base reduction algorithm and cryptographic systems with lattices [22] and coding theory [14]. Some theorems in this article are described by translating theorems in [24], [26] and [19] into theorems of ℤ-module.

Reduced Gröbner bases and Macaulay–Buchberger Basis Theorem over Noetherian rings

In this paper, we extend the characterization of Z[x]/〈f〉, where f∈Z[x] to be a free Z-module to multivariate polynomial rings over any commutative Noetherian ring, A. The characterization allows us to extend the Gröbner basis method of computing a k-vector space basis of residue class polynomial rings over a field k (Macaulay–Buchberger Basis Theorem) to rings, i.e. A[x1,…,xn]/a, where a⊆A[x1,…,xn] is an ideal. We give some insights into the characterization for two special cases, when A=Z and A=k[θ1,…,θm]. As an application of this characterization, we show that the concept of Border bases can be extended to rings when the corresponding residue class ring is a finitely generated, free A-module.

Submodule of free Z-module

Abstract In this article, we formalize a free Z-module and its property. In particular, we formalize the vector space of rational field corresponding to a free Z-module and prove formally that submodules of a free Z-module are free. Z-module is necassary for lattice problems - LLL (Lenstra, Lenstra and Lov´asz) base reduction algorithm and cryptographic systems with lattice [20]. Some theorems in this article are described by translating theorems in [11] into theorems of Z-module, however their proofs are different.

Preface Issue 2-2013

“Curvature driven interface evolution” is one of the major current themes in the numerics resp. the numerical analysis resp. the analytical theory of partial differential equations. It is particularly attractive due to numerous physical applications on the one hand and due to close relations to differential geometry on the other hand. Harald Garcke gives a survey on the most important topics in this research area. Among these are models for crystal growth of snow flakes, melting ice in water or grain coarsening (aging) in two-phase mixtures: Stefan problem, Mullins-Sekerka-problem and variants. Moreover, the mean curvature and surface diffusion flow are covered as well as phase field models like the Allen-Cahn and Cahn-Hilliard equation. In the latter sharp classical interfaces are replaced by interfacial regions. All of these models are introduced, their physical applicability as well as their main properties are explained and they are illustrated by means of numerical simulations. The present survey will certainly prove to be a very helpful reference work for anybody working on resp. interested in these kinds of models. The integral group ring ZG of a group G consists of elements of the form ∑ g∈G zgg, where only finitely many of the coefficients zg ∈ Z may be different from 0. These elements form a free Z-module with G as a basis, while the multiplication in ZG is the Z-linear extension of that in G. Wolfgang Kimmerle’s survey article gives a brief account of classical results and focusses then on recent achievements and developments concerning the structure of the unit group of ZG. The Zentralblatt fur Mathematik and the Jahresbericht der DMV begin a collaboration to review again classical books and the effect that they have had or could have had on the development of specific mathematical branches from today’s point of view.

Free ℤ-module

Summary In this article we formalize a free ℤ-module and its rank. We formally prove that for a free finite rank ℤ-module V , the number of elements in its basis, that is a rank of the ℤ-module, is constant regardless of the selection of its basis. ℤ-module is necessary for lattice problems, LLL(Lenstra, Lenstra and Lovász) base reduction algorithm and cryptographic systems with lattice [15]. Some theorems in this article are described by translating theorems in [21] and [8] into theorems of Z-module.

On the Mordell-Weil Groups of Jacobians of Hyperelliptic Curves over Certain Elementary Abelian 2-extensions

Abstract. Let J be the Jacobian variety of a hyperelliptic curve over Q. Let M be thefield generated by all square roots of rational integers over a finite number field K. Thenwe prove that the Mordell-Weil group J(M) is the direct sum of a finite torsion groupand a free Z-module of infinite rank. In particular, J(M) is not a divisible group. On theother hand, if Mf is an extension of M which contains all the torsion points of J over Q,then J(Mf sol )/J(Mf ) tors is a divisible group of infinite rank, where Mf sol is the maximalsolvable extension of Mf. 1. IntroductionLet K be a number field. Let A be a nonzero abelian variety defined over K.For an extension M over K, we denote the group of M-rational points by A(M)and its torsion subgroup by A(M) tors . We call A(M) is the Mordell-Weil group ofA over M. In [1], Frey and Jarden have asked whether the Mordell-Weil group ofevery nonzero abelian variety A defined over K has infinite Mordell-Weil rank overthe maximal abelian extension K ab of K. They proved that for elliptic curves Edefined over Q, the Mordell-Weil group E(Q

Specht filtrations and tensor spaces for the Brauer algebra

Let m,n∈ℕ. In this paper we study the right permutation action of the symmetric group \(\mathfrak{S}_{2n}\) on the set of all the Brauer n-diagrams. A new basis for the free ℤ-module \(\mathfrak {B}_{n}\) spanned by these Brauer n-diagrams is constructed, which yields Specht filtrations for \(\mathfrak {B}_{n}\) . For any 2m-dimensional vector space V over a field of arbitrary characteristic, we give an explicit and characteristic-free description of the annihilator of the n-tensor space V ⊗n in the Brauer algebra \(\mathfrak {B}_{n}(-2m)\) . In particular, we show that it is a \(\mathfrak{S}_{2n}\) -submodule of \(\mathfrak {B}_{n}(-2m)\) .

Multiplicative $SK$ Invariants on $Z_n$-Manifolds with Boundary

Let Z n be the cyclic group of order n. In this paper, we study a map T for Z n -manifolds with boundary which takes values in the ring Z and is additive with respect to the disjoint union of Z n -manifolds. We call T a Z n -SK invariant if it is invariant under Z n -cuttings and pastings. Then T induces an additive homomorphism T : SK Z n*(pt,pt) → Z, where SK Z n* (pt, pt) is a cutting and pasting group (SK group) of all Z n -manifolds. First we obtain a basis of a free Z-module I Z n* of all these invariants by using the Euler characteristic X of manifolds with boundary. As a result, we determine the class of all multiplicative invariants, which includes X Z s (and X Z s) in particular.

CALCULATION FOR RINGEL—HALL NUMBERS OF CYCLIC SERIAL ALGEBRAS

C. M. Ringel in [1] introduced the Hall algebra, now also called the Ringel—Hall algebra, of a finitary ring. This kind of algebras turns out to be strongly related to Lie algebras and quantum groups (see, for example, 2-8). Let R be a finitary ring. For any finite (left) R-modules L, M, and N, denote by F LN M the number of submodules U of M such that U ≃ N and M/U ≃ L. Then the Ringel—Hall algebra of R is an associative ring, which is a free Z-module with a basis {u [M]}[M] indexed by the isomorphism classes of all finite R-modules with the multiplication defined by For this reason we call the structural constants F LN M as the Ringel--Hall numbers of R. In general, it is not easy to calculate Ringel—Hall numbers, even for some special finitary rings (see, for example, [5]). For a cyclic serial algebra, Guo [9] gave a calculating formula of F LN M when L or N is semisimple. In this note, we give another algorithm for F LN M with L or N indecomposable. Our proofs are more simple and the algorithm is more direct. Then one can use a general method to calculate any Ringel—Hall number. In addition, we obtain a very simple formula of F LN M when L and N both are indecomposable.

THE MOST REFINED VASSILIEV INVARIANT OF DEGREE ONE OF KNOTS AND LINKS IN ℝ1-FIBRATIONS OVER A SURFACE

As it is well-known, all Vassiliev invariants of degree one of a knot K ⊂ ℝ3 are trivial. There are nontrivial Vassiliev invariants of degree one, when the ambient space is not ℝ3. Recently, T. Fiedler introduced such invariants of a knot in an ℝ1-fibration over a surface F. They take values in the free ℤ-module generated by all the free homotopy classes of loops in F. Here, we generalize them to the most refined Vassiliev invariant of degree one. The ranges of values of all these invariants are explicitly described. We also construct a similar invariant of a two-component link in an ℝ1-fibration. It generalizes the linking number.

Constructing and counting phylogenetic invariants.

The method of invariants is an approach to the problem of reconstructing the phylogenetic tree of a collection of m taxa using nucleotide sequence data. Models for the respective probabilities of the 4m possible vectors of bases at a given site will have unknown parameters that describe the random mechanism by which substitution occurs along the branches of a putative phylogenetic tree. An invariant is a polynomial in these probabilities that, for a given phylogeny, is zero for all choices of the substitution mechanism parameters. If the invariant is typically non-zero for another phylogenetic tree, then estimates of the invariant can be used as evidence to support one phylogeny over another. Previous work of Evans and Speed showed that, for certain commonly used substitution models, the problem of finding a minimal generating set for the ideal of invariants can be reduced to the linear algebra problem of finding a basis for a certain lattice (that is, a free Z-module). They also conjectured that the cardinality of such a generating set can be computed using a simple "degrees of freedom" formula. We verify this conjecture. Along the way, we explain in detail how the observations of Evans and Speed lead to a simple, computationally feasible algorithm for constructing a minimal generating set.

Some Morita Equivalences of Rings of Differential Operators

Let N be a free Z-module of rank r and M its Z-module dual. Let σ be a simplicial cone in N ⊗ZQ and σ∨ its dual cone in M ⊗ZQ. Let Λ be a subsemigroup of the additive semigroup σ ∨ ∩ M such that Λ̃ = σ ∨ ∩ M. Then we find a precise criterion to determine when D(k Λ) is Morita equivalent to D(k[σ ∨ ∩ M]). We also show that there is a Morita equivalence exactly when both kΛ is Cohen-Macaulay (or S2) and the normalisation map Spec k[σ∨M] → Spec kΛ is injective.

Permutation Summands Over Z

A permutation lattice for a finite group G is a free Z-module with a finite basis which is permuted by G; direct summands of these, as Z G-module, are the lattices of the title. Numerical genus invariants are constructed for these lattices by globalizing the "species" of Benson and Parker. These are used to study the Grothendieck ring of the category of permutation summands. The Galois module structure of cyclotomic integers plays a role.

Multiplicative lattices in global fields

Consider the classical problem of packin9 equal non-overlappin9 spheres in IR N. Let L be the set of centers and let A (L) be the density. Set 2(L) = (log 2 A(L))/N and call it the density exponent. Let us restrict ourselves to the case when L is a lattice in R N. It corresponds to a quadratic formf(x) on a free Z-module of rank N, and the problem of finding the least possible 2 is equivalent to the other classical problem of finding a form of discriminant 1 with the maximum value of 7(L) = min f(x),cf. [9]. x ~ ~N -{0} Consider a family of lattices {L N c ~u} with N ~ ~ , and let 2({LN}) = lira sup 2(LN). It is known that 2({LN} ) > 0.599 (the Kabatiansky-Levenshtein

Torus-equivariant vector bundles on projective spaces

For a free Z-module N of rank n, let T = TN be an n-dimensional algebraic torus over an algebraically closed field k defined by N. Let X = TN emb (Δ) be a smooth complete toric variety defined by a fan Δ (cf. [6]). Then T acts algebraically on X. A vector bundle E on X is said to be an equivariant vector bundle, if there exists an isomorphism ft: t*E → E for each k-rational point t in T, where t: X → X is the action of t. Equivariant vector bundles have T-linearizations (see Definition 1.2 and [2], [4]), hence we consider T-linearized vector bundles.

On Certain Zeta Functions Attached to Two Hilbert Modular Forms: I. The Case of Hecke Characters

To make our exposition smooth, let us first introduce some notational conventions. For an algebraic number field K of finite degree, we denote by JK the set of all embeddings of K into C, and by IK the free Z-module generated by the elements of JK. We then put RIK = IK0 R and CIK = IK0 C. If p = EaPaU E IK with a E JK and PO E Z. we put xP = fl,(xa)Pfor 0 / x E K; this is meaningful for p e CIK if xa are all real and positive. Throughout the paper, we denote by D the complex upper half plane and by F a totally real algebraic number field of degree n. Now the zeta function to be studied in this paper, when suitably specialized, has the form

ON THE WITT GROUP OF A 2-ADIC GROUP RING

Z2TT, where -n is a 2-group, endowed with the orientable involution f = g~1. In this paper we explicitly calculate the Witt group W(A) of Hermitian forms over A with respect to this involution. The difficult part of the calculation is the quotient W(A)/W™(A), where W°(A) denotes the subgroup of even Hermitian forms. To obtain W(A), one must observe that Wv(A)ssZ/2 and is a direct summand of W(A). This was pointed out by W. Pardon. The calculation is a generalization of the method of characteristi c elements, which one uses to compute W(Z)/W™(Z)ssZ/8. (See [4].) There one associates to every non-singular bilinear form over Z a characteristic element x in the underlying free Z-module and observes that {x, x) is an invariant of the isomorphism class of the form if reduced mod 8. Moreover, this invariant vanishes on even forms, hence the above isomorphism. In the case of A = Z2ir, we find that we must associate a collection of characteristi c elements Xu • • • > Xk> m tne underlying free A-module, one for each conjugacy class of involutions in n. Furthermore, the elements (xi, xj) are aU invariants in suitable quotient groups of A, and most of the work in this paper is the evaluation of the quotient groups. These invariants detect all but a small subgroup of W(A)/W°(A), i.e., the subgroup of Hermitian forms for which all these invariants vanish is isomorphic to TT^, ® ZJ2, where TI^, is the commutator quotient TT/[TT, IT].