The aim of this paper is to study a property of a special kind of exact functors and give some applications to projective modules and injective modules.

In section 1 we introduce the notion of faithfully exact functors [Definition 1] as a generalization of the functor T(X) = X⊗M, where M is a faithfully flat module, and give a property of this class of functors [Theorem 1.1]. Next, applying this general theory to functors ⊗ and Horn, we define the notion of faithfully projective modules [Definition 2] and faithfully injective modules [Definition 3]. In the commutative case “faithfully projective” means, however, simply “projective and faithfully flat” [Proposition 2.3]. In section 2, equivalent conditions for a projective module P to be faithfully projective are given [Theorem 2.2, Proposition 2. 3 and 2.4]. And a simpler proof is given to Y. Hinohara’s result [6] asserting that projective modules over an indecomposable weakly noetherian ring are faithfully flat [Proposition 2.5]. In section 3, we consider faithfully injective modules.

Let $(R,\mathfrak{m})$ be a Noetherian local ring of characteristic $p>0$. We introduce and study $F$-full and $F$-anti-nilpotent singularities, both are defined in terms of the Frobenius actions on the local cohomology modules of $R$ supported at the maximal ideal. We prove that if $R/(x)$ is $F$-full or $F$-anti-nilpotent for a nonzero divisor $x\in R$, then so is $R$. We use these results to obtain new cases on the deformation of $F$-injectivity.

0. Let p be a prime number or zero and let g be a non-negative integer. Then there is a coarse moduli space Mg for complete non-singular irreducible curves of genus g defined over fields of characteristic p, which is an irreducible variety over the algebraic closure F̅p of the prime field Fp. (Especially, F0 is also denoted by Q as usual.) ([8], [2]). The curve corresponding to a generic point of Mg over F̅p is called a generic curve of genus g.

In 1975, Li and Yorke [3] found the following fact. Let f: I→ I be a continuous map of the compact interval I of the real line R into itself. If f has a periodic point of minimal period three, then f exhibits chaotic behavior. The above result is generalized by F.R. Marotto [4] in 1978 for the multi-dimensional case as follows. Let f: Rn → Rn be a differentiate map of the n-dimensional Euclidean space Rn (n ≧ 1) into itself. If f has a snap-back repeller, then f exhibits chaotic behavior.

In this paper, we give a generalization of the above theorem of Marotto. Our theorem can also be regarded as a generalization of the Smale’s results on the transversal homoclinic point of a diffeomorphism.

A nonnegative locally Holder continuous function P(z) on 0<|z |≤1 will be referred to as a density on Ω 0 < |z | < 1 with singularity at δ : z = 0, removable or genuine. A density P on Ω is said to be finite if it is integrable over Ω :

(1) ∫ Ω P(z)dxdy < ∞.

Bourbaki has established the following theorem which we state without proof as

THEOREM A ([3, Theorem 6, §4]). Let R be a noetherian integrally closed domain and M a finitely generated torsion free R-module. Then there exists a free submodule F of M such that M/F is isomorphic to an ideal in R.

It is our purpose in this note to present a few consequences of this theorem. Before giving these results we briefly review some terminology and known results.

Let k be an algebraic number field of finite degree and be a prime ideal of k, lying above a rational prime p. We denote by G () the multiplicative group of residue classes modulo (N ≧ 0) which are relatively prime to . The structure of G () is well-known, when N = 0, or k is the rational number field Q. If k is a quadratic number field, then the direct decomposition of G () is determined by A. Ranum [6] and F.H-Koch [4] who gives a basis of a group of principal units in the local quadratic number field according to H. Hasse [2]. In [5, Theorem 6.2], W. Narkiewicz obtains necessary and sufficient conditions so that G () is cyclic, in connection with a group of units in the -adic completion of k.

Let us consider a finite closed interval [− a, a] which will be thought of as being a medium capable of transporting particles. These particles may move only to the right or to the left with the constant speed c, and each particle changes the moving-direction during the time Δ with probability kΔ + o(Δ). If a right- (left-) moving particle hits the boundary point a (— a), then either it turns to the left (right) with probability 1 — q1(1 − q-1) or dies with probability q1(q-1).

For an abelian totally real number field F and an odd prime number p which splits totally in F, we present a functorial approach to special “p-units” previously built by D. Solomon using “wild” Euler systems. This allows us to prove a conjecture of Solomon on the annihilation of the p-class group of F (in the particular context here), as well as related annihilation results and index formulae.

By means of simple exact sequences in commutative algebra, we can derive some effective criteria for Cohen-Macaulay property of finite partially ordered sets.

A uniform upper bound for the Diederich-Fornaess index is given for weakly pseudoconvex domains whose Levi form of the boundary vanishes in l-directions everywhere.

We study the existence of L2 holomorphic sections of invariant line bundles over Galois coverings. We show that the von Neumann dimension of the space of L2 holomorphic sections is bounded below under weak curvature conditions. We also give criteria for a compact complex space with isolated singularities and some related strongly pseudoconcave manifolds to be Moishezon. As applications we prove the stability of the previous Moishezon pseudoconcave manifolds under perturbation of complex structures as well as weak Lefschetz theorems.

Using an induction formula of local densities by Kitaoka, we give an explicit formula for the Fourier coefficients of Siegel Eisenstein series of degree

Polylogarithmic functions satisfy functional equations. The most famous equation is of course the functional equation of the logarithm

log x + log y = log(x · y).

The other well known equation is the Abel equation of the dilogarithm

In this paper, we shall be concerned with the finiteness property of certain holomorphic mappings into a compact quotient of symmetric bounded domain.

Let be a symmetric bounded domain in n-dimensional complex Euclidean space Cn and Γ\ be a compact quotient of S by a torsion free discrete subgroup Γ of automorphism group of . Further, we denote by l() the maximum value of dimension of proper boundary component of , which is less than n (=dim).

Let be any increasing sequence of integers and M> 1; we connect to them in a very simply way, an increasing unbounded function φ: → R+. Let also X1, X2, · · · be a sequence of i.i.d. random vectors with value in euclidian space Rm. We prove that the cluster set of the sequence almost surely coincides with the unit ball of Rm, if, and only if, the covariance matrix of X1 is the identity matrix of Rm and EX1 is the zero vector of Rm. We define a functional A on the set of increasing sequences of integers as follows:

.

The main aim of this paper is to classify Ulrich ideals and Ulrich modules over two-dimensional Gorenstein rational singularities (rational double points) from a geometric point of view. To achieve this purpose, we introduce the notion of (weakly) special Cohen–Macaulay modules with respect to ideals, and study the relationship between those modules and Ulrich modules with respect to good ideals.

The commutators of convolution operators are considered. By localization and Fourier transform estimates, a sufficient condition such that these commutators are bounded on L2(ℝn) is given. As applications, some new results about the L2(ℝn) boundedness for the commutators of homogeneous singular integral operators are established.

Some results for the Bergman functions in unbounded domains are shown. In particular, a class of unbounded Hartogs domains, which are Bergman complete and Bergman exhaustive, is given.

In this paper, we shall discuss the smoothness of solutions of stochastic evolution equations, which has been investigated in N. V. Krylov and B. L. Rozovskii [2] [3], to establish the existence of a filtering transition density.