Let be a commutative ring and an ideal in which is locally generated by a regular sequence of length . Then, each f. g. projective -module has an -projective resolution of length . In this paper, we compute the homology of the -th Koszul complex associated with the homomorphism for all , if . This computation yields a new proof of the classical Adams-Riemann-Roch formula for regular closed immersions which does not use the deformation to the normal cone any longer. Furthermore, if , we compute the homology of the complex where and denote the functors occurring in the Dold-Kan correspondence.
An -character of a group is the trace of an -representation of We show that all algebraic relations between -characters of can be visualized as relations between graphs (resembling Feynman diagrams) in any topological space with We also show that all such relations are implied by a single local relation between graphs. In this way, we provide a topological approach to the study of -representations of groups.
The motivation for this paper was our work with J. Przytycki on invariants of links in 3-manifolds which are based on the Kauffman bracket skein relation. These invariants lead to a notion of a skein module of which, by a theorem of Bullock, Przytycki, and the author, is a deformation of the -character variety of This paper provides a generalization of this result to all -character varieties.
Let be a finite group and let be a degree 1, -framed map such that and are simply connected, closed, oriented, smooth manifolds of dimension and such that the dimension of the singular set of the -space is at most . In the previous article, assuming is -connected, we defined the -equivariant surgery obstruction in a certain abelian group. There it was shown that if then is -framed cobordant to a homotopy equivalence . In the present article, we prove that the obstruction is a -framed cobordism invariant. Consequently, the -surgery obstruction is uniquely associated to above even if it is not -connected.
Let be a nilpotent Lie algebra, over a field of characteristic zero, and its universal enveloping algebra. In this paper we study: (1) the prime ideal structure of related to finitely generated -modules , and in particular the set of associated primes for such (note that now is equal to the set of annihilator primes for ); (2) the problem of nontriviality for the modules when is a (maximal) prime of , and in particular when is the augmentation ideal of . We define the support of , as a natural generalization of the same notion from commutative theory, and show that it is the object of primary interest when dealing with (2). We also introduce and study the reduced localization and the reduced support, which enables to better understand the set . We prove the following generalization of a stability result given by W. Casselman and M. S. Osborne in the case when , as in the theorem, are abelian. We also present some of its interesting consequences.
Theorem. Let be a finite-dimensional Lie algebra over a field of characteristic zero, and an ideal of ; denote by the universal enveloping algebra of . Let be a -module which is finitely generated as an -module. Then every annihilator prime of , when is regarded as a -module, is -stable for the adjoint action of on .
Let be a hyperbolic diffeomorphism on a basic set and let be a connected Lie group. Let be Hölder. Assuming that satisfies a natural partial hyperbolicity assumption, we show that if is a measurable solution to a.e., then must in fact be Hölder. Under an additional centre bunching condition on , we show that if assigns `weight' equal to the identity to each periodic orbit of , then for some Hölder . These results extend well-known theorems due to Livsic when is compact or abelian.
Generalized Eilenberg-Borsuk Theorem. Let be a countable CW complex. If is a separable metrizable space and is an absolute extensor of for some CW complex , then for any map , closed in , there is an extension of over an open set such that .
Theorem. Let be countable CW complexes. If is a separable metrizable space and is an absolute extensor of , then there is a subset of such that and .
Theorem. Suppose are countable, non-trivial, abelian groups and 0$">. For any separable metrizable space of finite dimension 0$">, there is a closed subset of with for .
Theorem. Suppose is a separable metrizable space of finite dimension and is a compactum of finite dimension. Then, for any , , there is a closed subset of such that and .
Theorem. Suppose is a metrizable space of finite dimension and is a compactum of finite dimension. If and are connected CW complexes, then
We prove that for the set of Cauchy problems of dimension which have a global solution is -complete and that the set of ordinary differential equations which have a global solution for every initial condition is -complete. The first result still holds if we restrict ourselves to second order equations (in dimension one). We also prove that for the set of Cauchy problems of dimension which have a global solution even if we perturb a bit the initial condition is -complete.
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.
J. Zemánek is obtained.
Fix integers with k>0$"> and . Let be an integral projective curve with and a rank torsion free sheaf on which is a flat limit of a family of locally free sheaves on . Here we prove the existence of a rank subsheaf of such that . We show that for every there is an integral projective curve not Gorenstein, and a rank 2 torsion free sheaf on with no rank 1 subsheaf with . We show the existence of torsion free sheaves on non-Gorenstein projective curves with other pathological properties.
We find a direct condition on the quadratic map that characterizes when the extension is Bockstein closed. Using this characterization, we show for example that quadratic maps induced from the fundamental quadratic map given by yield Bockstein closed extensions.
On the other hand, it is well known that an extension is Bockstein closed if and only if it lifts to an extension for some -lattice . In this situation, one may write for a ``binding matrix' with entries in . We find a direct way to calculate the module structure of in terms of . Using this, we study extensions where the lattice is diagonalizable/triangulable and find interesting equivalent conditions to these properties.
We prove that is sufficient to construct a model in which is measurable and is a closed and unbounded subset of containing only inaccessible cardinals of . Gitik proved that is necessary.
We also calculate the consistency strength of the existence of such a set together with the assumption that is Mahlo, weakly compact, or Ramsey. In addition we consider the possibility of having the set generate the closed unbounded ultrafilter of while remains measurable, and show that Radin forcing, which requires a weak repeat point, cannot be improved on.
We apply our methods for orientable Poincaré duality groups of dimension 3 and show that the pro- completion of is a pro- Poincaré duality group of dimension 3 if and only if every subgroup of finite index in has deficiency 0 and is infinite. Furthermore if is infinite but not a Poincaré duality pro- group, then either there is a subgroup of finite index in of arbitrary large deficiency or is virtually . Finally we show that if every normal subgroup of finite index in has finite abelianization and the profinite completion of has an infinite Sylow -subgroup, then is a profinite Poincaré duality group of dimension 3 at the prime .