State morphism MV-algebras

We present a complete characterization of subdirectly irreducible MV-algebras with internal states (SMV-algebras). This allows us to classify subdirectly irreducible state morphism MV-algebras (SMMV-algebras) and describe single generators of the variety of SMMV-algebras, and show that we have a continuum of varieties of SMMV-algebras.     

Hilbert-Samuel functions of modules over Cohen-Macaulay rings

For a finitely generated, non-free module over a CM local ring , it is proved that for the length of is given by a polynomial of degree . The vanishing of is studied, with a view towards answering the question: If there exists a finitely generated -module with such that the projective dimension or the injective dimension of is finite, then is regular? Upper bounds are provided for beyond which the question has an affirmative answer.

     

The Moore-Penrose inverse of a morphism with factorization

Given an m-by-n matrix A of rank r over a field with an involutory automorphism, it is well known that A has a Moore-Penrose inverse if and only if rank A¹A=r= rank AA¹. By use of the full-rank factorization theorem, this result may be restated in the category of finite matrices as follows: if (A₁, r, A₂) is an (epic, monic) factorization of A:m→n through r, then A has a Moore-Penrose inverse if and only if (A¹A₁, r, A₂) and (A₁, r, A₂A¹) are, respectively, (epic, monic) factorizations of A¹A:n→n and AA¹:m→m through r. This characterization of the existence of Moore-Penrose inverses is extended to arbitrary morphisms with (epic, monic) factorizations.     

EXISTENCE OF A NON-STRICT MOPRPHISM OF QUOTIENT BANACH SPACES

Abstract. In this paper we prove the existence of a morphism of quotient Banach spaces that is not strict.     

A simple remark on a flat projective morphism with a Calabi-Yau fiber

If a K3 surface is a fiber of a flat projective morphisms over a connected noetherian scheme over the complex number field,then any smooth connected fiber is also a K3 surface.Observing this,Professor Nam-Hoon Lee asked if the same is true for higher dimensional Calabi-Yau fibers.We shall give an explicit negative answer to his question in each dimension greater than or equal to three as well as a proof of his initial observation.     

Direct images of bundles under Frobenius morphism

Let X be a smooth projective variety of dimension n over an algebraically closed field k with char(k)=p>0 and F:X→X ₁ be the relative Frobenius morphism. For any vector bundle W on X, we prove that instability of F _* W is bounded by instability of W⊗T^ℓ(Ω¹ _X ) (0≤ℓ≤n(p-1)) (Corollary 4.9). When X is a smooth projective curve of genus g≥2, it implies F _* W being stable whenever W is stable. Dedicated to Professor Zhexian Wan on the occasion of his 80th birthday.     

The morphism induced by Frobenius push-forward

Let X be a smooth projective curve of genus g 2 over an algebraically closed field k of characteristic p0,and F:X→X(1)the relative Frobenius morphism.Let M s X(r,d)(resp.M ss X(r,d))be the moduli space of(resp.semi-)stable vector bundles of rank r and degree d on X.We show that the set-theoretic map S ss Frob:M ss X(r,d)→M ss X(1)(rp,d+r(p-1)(g-1))induced by[E]→[F(E)]is a proper morphism.Moreover,the induced morphism S s Frob:M s X(r,d)→M s X(1)(rp,d+r(p-1)(g-1))is a closed immersion.As an application,we obtain that the locus of moduli space M s X(1)(p,d)consisting of stable vector bundles whose Frobenius pull backs have maximal Harder-Narasimhan polygons is isomorphic to the Jacobian variety Jac X of X.     

A transfer morphism for Witt cogroups

Let A be a ring with involution in which 2 is invertible, ε be 1 or −1, and s(∈ A) be a central regular element such that s* = s. A transfer homomorphism _εW′₀(A/s) → _ε W′₁(A) for Witt cogroups is constructed and a projection formula is proved. Bibliography: 11 titles. Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 356, 2008, pp. 149–158.     

Hilbert-Samuel coefficients and postulation numbers of graded components of certain local cohomology modules

Let be a Noetherian homogeneous ring with one-dimensional local base ring . Let be an -primary ideal, let be a finitely generated graded -module and let . Let denote the -th local cohomology module of with respect to the irrelevant ideal 0} R_n$"> of . We show that the first Hilbert-Samuel coefficient of the -th graded component of with respect to is antipolynomial of degree in . In addition, we prove that the postulation numbers of the components with respect to have a common upper bound.