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Let P n \textbf {P}^n be the projective n n -space over the complex numbers. In this note we show that an indecomposable rigid coherent sheaf on P n \textbf {P}^n has a trivial endomorphism algebra. This generalizes a result of Drézet for n = 2. n=2.

This paper attempts to characterize the dicritical set of a rational function which generates a special pencil at a simple point of an algebraic or arithmetical surface.

For a dominant algebraically stable rational self-map of the complex projective plane of degree at least 2, we will consider three different definitions of the Fatou set and show the equivalence of them (Ascoli-Arzela type theorem). As a corollary, it follows that all Fatou components are Stein. This is an improvement of an early result by Fornaess and Sibony.

For a coanalytic–complete or \wideutilde {\Pi }^1_2–complete subspace X X of a Polish space we prove that there exists a continuous bijection of X X onto the Hilbert cube [ 0 , 1 ] N [0,1]^{\mathbb {N}} . This extends results of Pytkeev. As an application of our main theorem we give an answer to some questions of Arkhangelskii and Christensen. Under the assumption of Projective Determinacy we also give some generalizations of these results to higher projective classes.

Burstall classified conformal non-superminimal harmonic two-tori in spheres and complex projective spaces. In this paper, we shall classify conformal non-superminimal harmonic two-tori in a 2or 3-dimensional quaternionic protective space, which are not always covered by primitive harmonic two-tori of finite type.

Local variational min-sup characterizations are presented for the real spectrum of a selfadjoint operator pencil. Instead of minimizing over all subspaces of fixed codimension as in the classical result, the new characterizations minimize over subspaces that are close to extremal subspaces. In this way, the entire real spectrum, including continuous spectrum, can be characterized.

Fix integers n , k , d n,k,d with n ≥ 2 , d ≥ 2 n\ge 2,d\ge 2 and k > 0 k>0 ; if n = 2 n=2 assume d ≥ 3 d\ge 3 . Let P 1 , … , P k P_1,\dotsc ,P_k be general points of the complex projective space P n \mathbf {P}^n and let π : X → P n \pi :X\to \mathbf {P}^n be the blow up of P n \mathbf {P}^n at P 1 , … , P k P_1,\dotsc ,P_k with exceptional divisors E i := π − 1 ( P i ) E_i:=\pi ^{-1}(P_i) , 1 ≤ i ≤ k 1\le i\le k . Set H := π ∗ ( O P n ( 1 ) ) H:=\pi ^*(\mathbf {O}_{\mathbf {P}^n}(1)) . Here we prove that the divisor L := d H − ∑ 1 ≤ i ≤ k E i L:=dH-\sum _{1\le i\le k}E_i is ample if and only if L n > 0 L^n>0 , i.e. if and only if d n > k d^n>k .

We find the maximum number of orthogonal skewsymmetric anticommuting integer matrices of order n for each natural number n and relate this to finding free direct summands of certain generic projective modules. While studying composition of quadratic forms, Hurwitz [4] and Radon [6] considered families of orthogonal matrices {A1,3L , A,} satisfying the conditions (1) Ai= -At, i= 1, ,s (2) AiAj = -AjAi, i $ j. DEFINITION. (1) A family of orthogonal matrices satisfying (1) and (2) above will be called a Hurwitz-Radon (H-R)family. If n is a positive integer and n=2ab, b odd, then we write a=4c+d where O0d

In previous papers we developed a version of homological algebra for Z [ G ] Z[G] modules ( G G a finite group) using summands of permutation modules in place of projective modules. The resulting theory is more discriminating than the usual homological algebra. For example, an f.g. Z Z -torsion free Z [ G ] Z[G] module is either projective or has infinite projective dimension, whereas for G G cyclic, all f.g. Z [ G ] Z[G] modules have permutation projective dimension one. In this paper we apply results of Endo and Miyata on permutation projective modules to characterize groups of dimension two.

The minimum number of triple points of a substantial generic immersion of a projection plane in a lens space of type ( 2 k , q ) (2k,q) depends quadratically on k k .

In the paper [4], the author introduced the notion of m-adic free modules in order to give a differential characterization of regular local rings. Later, more generally, A. Grothendieck introduced the notion of formally projective modules for the similar purpose in his [3]. Comparisons of these two notions in important cases were given in [5]. In the present paper, we intend to study the relationship between the flatness and the formally projectivity. The main purpose is to show the flatness of complete formally projective modules over complete Zariski rings, using results in [5]. As a result of it, we can characterize the formally projectivity of complete modules over complete semilocal rings by the flatness. Meanwhile, we prove the separatedness of finite coefficients extension of complete formally projective modules over complete Zariski rings. Let R be a commutative ring with an identity and m be an ideal. Let M be an R-module. We consider the m-adic topologies both in R and M. By the completeness of R-modules, we always mean the separated completeness. We follow the terminology in [5]. We begin with a supplement to [5]. The following lemma is a part of Proposition 4.1, n04, IV, in [2].

Let A A be a commutative ring, and let X X = projective r r -space over A A . Then we prove that K 1 {K_1} of the category of locally free sheaves of finite type on X X is isomorphic to the direct sum of r + 1 r + 1 copies of K 1 ( A ) {K_1}(A) .

In this article we establish that if a member of the pencil f ( x , y ) + λ f(x,y) + \lambda is analytic irreducible at ∞ \infty then all members are.

We give a more illuminating derivation of a well-known condition (obtained by M. Atiyah and J. A. Todd) for certain vector bundles over complex projective spaces to be stably fiber homotopic trivial, together with a generalization.

Distance spheres in complex projective spaces are counterexamples to the odd-dimensional extension of a lemma of Klingenberg.

THEOREM. Under the conditions stated, (2) represents a constant breadth convex curve which is orthogonal to the line family (1) and has a continuously turning tangent line at each point. Letting c approach supe [Dp(0) +J'p(0)dO] from above, a limiting constant breadth convex curve (or a point if F is a pencil) is obtained. With this inclusion (2) represents all constant breadth convex curves in the plane.

A structure theorem for regular pure projective modules over any associative ring with identity is given. This extends an earlier result for regular projective modules, which in turn extended a theorem of Kaplansky on the structure of projective modules over regular rings.