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Let k be an algebraically closed field. We understand by a Del Pezzo surface X over k a non-singular rational surface on which the anti-canonical sheaf ―wx is ample. We call the self-intersectionnumber d=a)x of wx the degree of X, then we get that 1^J^E9. It is well known that X is isomorphic to PlxP\ which has degree 8, or an image of P2 under a monoidal transformation with center the union of r―9―d points which satisfiesthe following conditions: (a) no three of them lie on a line; (b) no six of them lie on a conic; (c) there are no cubics which pass through seven of them and have a double point at the eighth point. Conversely any surface described above is a Del Pezzo surface of the corresponding degree ([8,in, Theorem 1]). It is also well known that ―o)x is very ample when d^3 and that ample divisors on X of degree 3, which is a cubic surface, are very ample too. In this paper we will get that ample divisorson X of degree d^3 are very ample and that ample divisors on X of degree 2 [resp. 1] other than ―o>x [resp. ―o)x nor ―2^x] ore very ample. A closed subscheme V in PN is said to be projectively Cohen-Macaulay if its affine cone is Cohen-Macaulay. It is equivalent to that H1(PN,Jv(m))=0 for every meZ and H\V, Ov(m))-0 for every meZ and 0

Here, P1 is the complex projective line. The purpose of this paper is to give affirmative answers to the conjecture under various conditions. It is separated into 3 parts. In Part 1, we assume that n=p is a prime number and obtain a result. Recently, Kato [2] has improved this result extensively. In Part 2, we assume that f(x) is a polynomial of degree p with p a prime number, and obtain a result. In Part 3, we assume

Introduction Let G= be a cyclic group with w = #G>l. For each divisor d of n, let %d denote the character of G of the irreducible representation over Q, the field of rational numbers, whose kernel is equal to (,od). Let k be an algebraicallyclosed ground field and X be a complete non-singular curve over k of genus Q^Z.2.The main purpose of the present paper is to prove the following theorems, under the assumption that G^=Aut(X), the automorphism group of X; in this situation we denote by Tx(G\H1(X, QO) tne character of the natural representation of G on the first/-adic cohomolqgy .fiPCX,Qz) of X, where I is any prime number differentfrom the characteristicof k fcf. Notation, 5 2 and also [41)

of modules closed under cyclicsubmodules, homomorphic images and extensions. Applying Theorem 2 for the Goldie and the Lambek torsion theories, we obtain Corollaries5 and 6. We considerin Corollary 5 a ring R (calleda right V(G)-ring) for which every singular simple right R-module isinjective,and in Corollary 6 a right V(L)-ring for which every dense right idealis an intersection of maximal right ideals. We characterize V-rings in terms of V(G)rings or V(L)-rings in Proposition 8 which is closelyrelatedto Theorem 8 in [7]. In Theorem 9 itis proved that commutative V(G)-rings turn out to be V-rings. In this connection two examples are given to show that neither commutative V(L)-rings nor V(G)-rings are V-rings. Throughout this paper R is a ring with a unit,every right R-module is unitaland Mod-j? is the category of right i?-modules. For a right i?-module M, Z(M), E{M) and J{M) denote the singular submodule of M, the injective hullof M and theintersection of allmaximal submodules of M. A right Rmodule M is called2"-injectivefor a subclass 2"of Mod-i? if Homfi(―, M) preserves the exactness for every exact sequence of right i?-modules Q-*A^B―>C ―0 with Ce2＼

In this paper, the concept of the approximate section extension property (ASEP) is introduced. It is shown that hereditary shape equivalences are exactly the maps with the hereditary ASEP and every $UV^{n-1}$-map with $n$-dimentional range has the ASEP.

The main purpose of the present paper is to estimate id-A a, the injective dimension of A a, in terms of those of RR, MR, and Ss. In fact,if we assume that fd-sM, the flatdimension of SM, is finite,then there hold the inequalities max(id-RR, id-MR, id-Ss-fd-sM)^id-AA^ max (max (id-RR, id-MR)+fd-sM, id-S5-l)+l. In this connection, we investigate the case when the left-hand or the righthand side equality holds under the condition that SM is flat. In [7], Zaks shows that the injective dimension of an nXn lower triangular matrix ring over a semiprimary ring R is just equal to id-RR-}-l. An example is constructed to show that the condition on R benig semiprimary is redundant in his theorem. The author wishes to express his hearty thanks to Professors H. Tachikawa and T. Kato for their useful suggestions and remarks. Let ,=[J

Let R be a legt noetherian ring with left Krull demension α. For a let R-module M which has Krull dimension, we denote its Krull dimension by K-dim M in this note. In the previous paper [6], ...