The local cohomology theory introduced by Grothendieck(1) is a useful tool for attacking problems in commutative algebra and algebraic geometry. Let A denote a local ring with its unique maximal ideal m. For an ideal I ⊂ A and a finitely generated A-module M we consider the local cohomology modules HiI (M), i є ℤ, of M with respect to I, see Grothendieck(1) for the definition. In particular, the vanishing resp. non-vanishing of the local cohomology modules is of a special interest. For more subtle considerations it is necessary to study the cohomological annihilators, i.e. aiI(M): = AnnΔHiI(M), iєℤ. In the case of the maximal ideal I = m these ideals were used by Roberts (6) to prove the ‘New Intersection Theorem’ for local rings of prime characteristic. Furthermore, we used this notion (7) in order to show the amiability of local rings possessing a dualizing complex. Note that the amiability of a system of parameters is the key step for Hochster's construction of big Cohen-Macaulay modules for local rings of prime characteristic, see Hochster(3) and (4).

In this paper the values of m(6)2,9, m(5)2,8 and m(5)2,9 have been determined as well as improved bounds on other arcs, and in particular a general construction for certain large arcs in planes of square order has been given. Table 5 summarizes the known values of m(n)2, q for 2 ≤ n ≤ q and q ≤ 9.

1. It was shown by Barlotti (1) that the number k of points on a (k, n)-arc of a finite projective piane of order q

and that if q ≢ 0 (mod n) then for n ≥ 3

.

In this paper we investigate the local form of the tangent developable of a curve in 3-space, and obtain results relating this to the order of vanishing of the torsion. In particular,

Theorem 1. If at the point γ(t) on the curve γ, the torsion vanishes to order k (0 ≼ k ≼ 4), then the germ of the tangent developable at that point has one curve of self intersection if k is odd, and none if k is even.

Let F: n × r → be a sufficiently small representative, defined near (0,0), of a versal unfolding of a germ F0: (n, 0) → (, 0) with isolated singularity at 0∈n. Then a result of Teissier ((4), p. 338) says that, for u∈r sufficiently close to 0, F acts as a versal unfolding of all the various singularities, close to 0, on the fibre Fu = 0 (where Fu(x) = F(x, u)). Let us fix a small neighbourhood W of 0 in n and restrict u to be so close to 0 that all singularities of Fu lie in W. Suppose that the fibre Fu = 0 has singularities of (contact) type χ1, …, χm, all isolated. Suppose that, for each i, the collection {χij}(1 ≤ j ≤ k(i)) of singularities specializes to χi, that is it occurs on a single fibre arbitrarily close to χi in a versal unfolding of χi. Then Teissier's result shows that there will exist fibres Fv = 0 of F, with v arbitrarily close to u, such that the fibre Fv = 0 carries singularities, all in W, of all the types χij for 1 ≤ i ≤ m, 1 ≤ j ≤ k(i). In other words, if ‘despecializations’ {χij} of the separate singularities χi are possible, then they can all occur together on a single fibre of F, provided F is versal.

Let φ (X) denote a function satisfying limx→∞, φ(X) = ∞. Let ℬ be a sequence of positive integers and let T(X) = T(X, ℬ, φ) be the set of real numbers y, 1 ≤ y ≤ X, such that the interval (y, y + φ(y)] contains no member of ℬ.

In this paper we consider complex projective surfaces V, defined by an equation of the form fn–1 (x, y, z) w + fn (x, y, z) = 0, where fi is homogeneous of degree i, and relate the geometry of the intersections of the piane projective curves fn–1 = 0 and fn = 0 with the singularities of V. The results we obtain clarify and generalize some of those presented by Bruce and Wall (3).

In chapter 7 of (2) conditions were given for a ring to be embeddable in a skew field; in particular, it was shown that any semifir has a universal field of fractions, over which all full matrices can be inverted. This was generalized in two different directions, by Bergman (in a letter to one of the authors in 1971) and by Dicks and Sontag(7). Dicks and Sontag characterized those rings having a field of fractions in which all full matrices are inverted; they showed that this is equivalent to Sylvester's law of nullity, and further showed that this forces the ring to have weak global dimension not exceeding 2 and all finitely generated projective modules to be free. Bergman on the other hand investigated weakly semihereditary rings having a rank function on projective modules which takes values in the natural numbers. He showed that there was a homomorphism from any such ring to a field of fractions in which every full map between finitely generated projective modules is inverted. Weakly semihereditary rings with a rank function to the natural numbers are the analogue of semifirs and so it is natural to look for a characterization of rings with a rank function on projective modules such that all full maps between projective modules become invertible in a suitable field of fractions. We shall find that, as before, this is the case if and only if Sylvester's law of nullity holds with respect to the rank function, for maps between projective modules. Further, the ring must have weak global dimension at most two. This is the content of Sections 2 and 3.

A quadratic complex Q is the set of lines in 3-dimensional projective space 3 given by a single non-trivial quadratic equation in the Plücker coordinates. The lines of the complex which pass through a fixed point of 3 are, in general, the generators of a quadric cone; this cone degenerates for the points of a sub variety K of 3. Thus one can associate with Q a fibration with base 3 - K and fibre isomorphic to 1, and ask whether this fibration is associated with an algebraic vector bundle of rank 2. When the base field is and Q is non-singular, the answer is negative; this was proved some years ago by Narasimhan and Ramanan ((8), proposition 8·1), and has the consequence that there is no universal family of stable vector bundles of rank 2 and degree 0 over a curve of genus 2.

This note consists of some investigations about the condition ht(A) = l(A) where A is an ideal in a local ring and l(A) is the analytic spread of A (9).

In (4) we proved the following: If R is a local ring and P a prime ideal such that R/P is regular then (under some technical assumptions) ht(P) = l(P) is equivalent to the equimultiplicity e(R) = e(RP). Also for a general ideal A (which need not be prime), the condition ht(A) = l(A) can be translated into an equality of certain multiplicities (see Theorem 0).

In this paper we will study the homological properties of the enveloping algebra U = U (Sl2(ℂ)), with particular reference to the homological dimension of simple U-modules and the global dimension of the primitive factor rings of U.

If M is any algebraic structure, and R is any Boolean ring, then a structure called the (bounded) Boolean power of M by R, denoted MR, can be defined. This construction, which is also called a bounded Boolean extension, is a sort of generalized direct power, and was introduced by Foster in the 1950's (as a refinement of his previous notion of a Boolean extension). In this paper we shall study isomorphism types and automorphisms of Boolean powers of groups, and obtain information about their characteristic subgroups: we shall be chiefly concerned with Boolean powers of finite groups.

If F is a (commutative) field let XF denote the class of all groups G such that every irreducible FG module has finite dimension over F. In the first paper (9) of this series we classified finitely generated soluble XF-groups for each field F and in the second (10) we characterized soluble XF-groups for each field F of characteristic zero. Here we consider soluble XF-groups over fields F of positive characteristic.

In (1) it is claimed that the main results of that paper have applications to the representation theory of algebraic groups, of polycyclic groups and of Lie algebras. An application to algebraic groups is given in Corollary 6·4 of (1), the applications to polycyclic groups are given in (2), the purpose of this work is to deal with the outstanding case of enveloping algebras. To make use of the results of (1), in this context, we show that the Hopf algebra dual of the enveloping algebra of a finite dimensional Lie algebra over a field of characteristic zero is quasi-affine (see § 1·5). This is done by an easy field extension argument and a generalization, to the Hopf algebra dual of the smash product of Hopf algebras, of Proposition 1·6·3 of (2) on the dual of the group algebra of a semidirect product of groups. Since this paper is aimed at those readers interested in enveloping algebras, the Hopf theoretic aspects are dealt with at a fairly leisurely pace.

Sandling(6) determined the dimension subgroups of the semidirect product of a normal abelian subgroup and a subgroup; namely if G = NT is the semidirect product of a normal abelian subgroup N and a subgroup T, then the mth dimension subgroup Dm(G) of G is equal to [N, (m – 1) G] · Dm (T) for all m ≧ 1, where

1. Following Lorentz, we suppose throughout that Ω(n) is a non-negati ve non-decreasing function of the non-negative integer n such that Ω(n)→ ∞ as n → ∞. Consider the summability method given by the sequence-to-sequence transformation corresponding to the matrix A = (ank). We say that Ω(n) is a summability function for A (or absolute summability function for A) if the following holds: Any bounded sequence {sn} such that the number of values of ν with ν ≤ n, sν ≠ 0 does not exceed Ω(n) is summable A (or is absolutely summable A, respectively). These definitions are due to Lorentz (4), (6). We shall be concerned with the case in which A is a regular Hausdorff method, say A = H = (H, μn). Then H is given by the matrix (hnk) with

with

X(0) = X(0 + ) = 0, X(1) = 1;(see e.g.(1), chapter XI). We shall suppose throughout that these conditions are satisfied. It is known that H is then necessarily also absolutely regular.

Since their introduction by Fischer(12) and Fischer, Gaschütz and Hartley (13) Fitting classes of soluble groups have attracted attention on two fronts (all groups considered in this paper will be finite and soluble). On the one hand is their important role in the structure of finite soluble groups, a good account of which can be found in Gaschütz (14), and on the other is their intrinsic interest as classes of groups. This paper falls into the second category, and is a continuation and completion of (8). There we proved that a subgroup closed Fitting class is a formation if it consists of groups of nilpotent length at most three. Happily, at last, we can remove this qualification.

Let G be a non-discrete locally compact topological group. A construction is given for a self-adjoint measure μ on G that has independent powers, and is, moreover, in the sub-algebra M0(G) of M(G). From this, the asymmetry of M0(G) (and of M(G) itself) is immediate.

An ordinary differential equation for Hermite-Bell polynomials is derived. Bell's error concerning orthogonality of his polynomials is corrected.

The main properties of the Hausdorff dimension, here denoted by dim, are

In ℝp, in variance under a group Н of homeomorphisms: ∀HεH, dim О H = dim. The definition of H, introduced in (15) and (16), is recalled in § 2.