The two main theorems proved here are as follows: If is a finite dimensional algebra over an algebraically closed field, the identity component of the algebraic group of outer automorphisms of is invariant under derived equivalence. This invariance is obtained as a consequence of the following generalization of a result of Voigt. Namely, given an appropriate geometrization of the family of finite -module complexes with fixed sequence of dimensions and an ``almost projective' complex , there exists a canonical vector space embedding
where is the pertinent product of general linear groups acting on , tangent spaces at are denoted by , and is identified with its image in the derived category .
Several properties of Anick's spaces are established which give a retraction of Anick's off if and . The proof is alternate to and more immediate than the two proofs of Neisendorfer's.
with prescribed data . More specifically, we are interested in the behavior of when the data is of the form for some prescribed function . One of our results asserts that if is sufficiently nice and has sufficiently well-behaved moments, then converges to a limit which can be completely characterized. As an application we identify the limits of certain fundamental interpolatory splines whose knot set is , where is an arbitrary finite subset of the integer lattice , as their degree goes to infinity.
Let be a smooth projective curve over a field . For each closed point of let be the coordinate ring of the affine curve obtained by removing from . Serre has proved that is isomorphic to the fundamental group, , of a graph of groups , where is a tree with at most one non-terminal vertex. Moreover the subgroups of attached to the terminal vertices of are in one-one correspondence with the elements of , the ideal class group of . This extends an earlier result of Nagao for the simplest case .
Serre's proof is based on applying the theory of groups acting on trees to the quotient graph , where is the associated Bruhat-Tits building. To determine he makes extensive use of the theory of vector bundles (of rank 2) over . In this paper we determine using a more elementary approach which involves substantially less algebraic geometry.
The subgroups attached to the edges of are determined (in part) by a set of positive integers , say. In this paper we prove that is bounded, even when Cl is infinite. This leads, for example, to new free product decomposition results for certain principal congruence subgroups of , involving unipotent and elementary matrices.
Let be an -primary ideal in a Gorenstein local ring (, ) with , and assume that contains a parameter ideal in as a reduction. We say that is a good ideal in if is a Gorenstein ring with . The associated graded ring of is a Gorenstein ring with if and only if . Hence good ideals in our sense are good ones next to the parameter ideals in . A basic theory of good ideals is developed in this paper. We have that is a good ideal in if and only if and . First a criterion for finite-dimensional Gorenstein graded algebras over fields to have nonempty sets of good ideals will be given. Second in the case where we will give a correspondence theorem between the set and the set of certain overrings of . A characterization of good ideals in the case where will be given in terms of the goodness in their powers. Thanks to Kato's Riemann-Roch theorem, we are able to classify the good ideals in two-dimensional Gorenstein rational local rings. As a conclusion we will show that the structure of the set of good ideals in heavily depends on . The set may be empty if , while is necessarily infinite if and contains a field. To analyze this phenomenon we shall explore monomial good ideals in the polynomial ring in three variables over a field . Examples are given to illustrate the theorems.
Let be the group of automorphisms of a homogeneous tree and let be the tensor product of two spherical irreducible unitary representations of . We complete the explicit decomposition of commenced in part I of this paper, by describing the discrete series representations of which appear as subrepresentations of .
We study the finite groups for which the set of irreducible complex character degrees consists of the two most extreme possible values, that is, and . We are easily reduced to finite -groups, for which we derive the following group theoretical characterization: they are the -groups such that is a square and whose only normal subgroups are those containing or contained in . By analogy, we also deal with -groups such that is not a square, and we prove that if and only if a similar property holds: for any , either or . The proof of these results requires a detailed analysis of the structure of the -groups with any of the conditions above on normal subgroups, which is interesting for its own sake. It is especially remarkable that these groups have small nilpotency class and that, if the nilpotency class is greater than , then the index of the centre is small, and in some cases we may even bound the order of .
The main result of this paper is that the variety of presentations of a general cubic form in variables as a sum of cubes is isomorphic to the Fano variety of lines of a cubic -fold , in general different from .
A general surface of genus determines uniquely a pair of cubic -folds: the apolar cubic and the dual Pfaffian cubic (or for simplicity and ). As Beauville and Donagi have shown, the Fano variety of lines on the cubic is isomorphic to the Hilbert scheme of length two subschemes of . The first main result of this paper is that parametrizes the variety of presentations of the cubic form , with , as a sum of cubes, which yields an isomorphism between and . Furthermore, we show that sets up a correspondence between and . The main result follows by a deformation argument.
Let be a group with a normal subgroup contained in the upper central subgroup . In this article we study the influence of the quotient group on the lower central subgroup . In particular, for any finite group we give bounds on the order and exponent of . For equal to a dihedral group, or quaternion group, or extra-special group we list all possible groups that can arise as . Our proofs involve: (i) the Baer invariants of , (ii) the Schur multiplier of relative to a normal subgroup , and (iii) the nonabelian tensor product of groups. Some results on the nonabelian tensor product may be of independent interest.
Let be the group of automorphisms of a homogeneous tree , and let be a lattice subgroup of . Let be the tensor product of two spherical irreducible unitary representations of . We give an explicit decomposition of the restriction of to . We also describe the spherical component of explicitly, and this decomposition is interpreted as a multiplication formula for associated orthogonal polynomials.
One way to understand the geometry of the real Grassmann manifold parameterizing oriented -dimensional subspaces of is to understand the volume-minimizing subvarieties in each homology class. Some of these subvarieties can be determined by using a calibration. In previous work, one of the authors calculated the set of -planes calibrated by the first Pontryagin form on for all , and identified a family of mutually congruent round -spheres which are consequently homologically volume-minimizing. In the present work, we associate to the family of calibrated planes a Pfaffian system on the symmetry group , an analysis of which yields a uniqueness result; namely, that any connected submanifold of calibrated by is contained in one of these -spheres. A similar result holds for -calibrated submanifolds of the quotient Grassmannian of non-oriented -planes.
We determine an expression for the virtual Euler characteristics of the moduli spaces of -pointed real ) and complex () algebraic curves. In particular, for the space of real curves of genus with a fixed point free involution, we find that the Euler characteristic is where is the th Bernoulli number. This complements the result of Harer and Zagier that the Euler characteristic of the moduli space of complex algebraic curves is
The proof uses Strebel differentials to triangulate the moduli spaces and some recent techniques for map enumeration to count cells. The approach involves a parameter that permits specialization of the formula to the real and complex cases. This suggests that itself may describe the Euler characteristics of some related moduli spaces, although we do not yet know what these spaces might be.
Kadison has shown that local derivations from a von Neumann algebra into any dual bimodule are derivations. In this paper we extend this result to local derivations from any -algebra into any Banach -bimodule . Most of the work is involved with establishing this result when is a commutative -algebra with one self-adjoint generator. A known result of the author about Jordan derivations then completes the argument. We show that these results do not extend to the algebra of continuously differentiable functions on . We also give an automatic continuity result, that is, we show that local derivations on -algebras are continuous even if not assumed a priori to be so.
Let be an algebraically closed field of characteristic zero. Let be the ring of (-linear) differential operators with coefficients from a regular commutative affine domain of Krull dimension which is the tensor product of two regular commutative affine domains of Krull dimension . Simple holonomic -modules are described. Let a -algebra be a regular affine commutative domain of Krull dimension and be the ring of differential operators with coefficients from . We classify (up to irreducible elements of a certain Euclidean domain) simple -modules (the field is not necessarily algebraically closed).