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.
Let be a bounded symmetric domain in a complex vector space with a real form and be the real bounded symmetric domain in the real vector space . We construct the Berezin kernel and consider the Berezin transform on the -space on . The corresponding representation of is then unitarily equivalent to the restriction to of a scalar holomorphic discrete series of holomorphic functions on and is also called the canonical representation. We find the spectral symbol of the Berezin transform under the irreducible decomposition of the -space.
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.
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 .
We show that a simply connected homotopy associative and homotopy commutative mod -space with finitely generated mod cohomology is homotopy equivalent to a finite product of , , the three-connected cover and the homotopy fiber of the map for . Our result also shows that a connected -space in the sense of Sugawara with finitely generated mod cohomology has the homotopy type of a finite product of , and for .
Let be the quotient of a regular coherent commutative ring by a finitely generated ideal. In this paper, we classify all abelian subcategories of finitely presented -modules that are closed under extensions. We also classify abelian subcategories of arbitrary -modules that are closed under extensions and coproducts, when is commutative and Noetherian. The method relies on comparison with the derived category of .
This paper deals with upper bounds on arithmetic discriminants of algebraic points on curves over number fields. It is shown, via a result of Zhang, that the arithmetic discriminants of algebraic points that are not pull-backs of rational points on the projective line are smaller than the arithmetic discriminants of families of linearly equivalent algebraic points. It is also shown that bounds on the arithmetic discriminant yield information about how the fields of definition and differ when is an algebraic point on a curve and is a nonconstant morphism of curves. In particular, it is demonstrated that , with at most finitely many exceptions, whenever the degrees of and are sufficiently small, relative to the difference between the genera and . The paper concludes with a detailed analysis of the arithmetic discriminants of quadratic points on bi-elliptic curves of genus 2.
In a fibration we show that finiteness conditions on force the homology Serre spectral sequence with -coefficients to collapse at some finite term. This in particular implies that as graded vector spaces, is ``almost' isomorphic to . One consequence is the conclusion that is elliptic if and only if and are.
Let be a reduced ring that is essentially of finite type over an excellent regular local ring of prime characteristic. Then it is shown that the test ideal of commutes with localization and, if is local, with completion, under the additional hypothesis that the tight closure of zero in the injective hull of the residue field of every local ring of is equal to the finitistic tight closure of zero in . It is conjectured that this latter condition holds for all local rings of prime characteristic; it is proved here for all Cohen-Macaulay singularities with at most isolated non-Gorenstein singularities, and in general for all isolated singularities. In order to prove the result on the commutation of the test ideal with localization and completion, a ring of Frobenius operators associated to each -module is introduced and studied. This theory gives rise to an ideal of which defines the non-strongly F-regular locus, and which commutes with localization and completion. This ideal is conjectured to be the test ideal of in general, and shown to equal the test ideal under the hypothesis that in every local ring of .
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.
Two other interesting results are included. First, we prove that for any , a -random real computable from a --random real is automatically --random. Second, we give a plain Kolmogorov complexity characterization of -randomness. This characterization is related to our proof that implies .
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).
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 .
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.
Let and be right, full, Hilbert -modules over the algebras and respectively and let be a linear surjective isometry. Then can be extended to an isometry of the linking algebras. then is a sum of two maps: a (bi-)module map (which is completely isometric and preserves the inner product) and a map that reverses the (bi-)module actions. If (or ) is a factor von Neumann algebra, then every isometry is either a (bi-)module map or reverses the (bi-)module actions.
-theorems for fusion systemsThis paper proves that a connected matroid in which a largest circuit and a largest cocircuit have and elements, respectively, has at most elements. It is also shown that if is an element of and and are the sizes of a largest circuit containing and a largest cocircuit containing , then . Both these bounds are sharp and the first is proved using the second. The second inequality is an interesting companion to Lehman's width-length inequality which asserts that the former inequality can be reversed for regular matroids when and are replaced by the sizes of a smallest circuit containing and a smallest cocircuit containing . Moreover, it follows from the second inequality that if and are distinct vertices in a -connected loopless graph , then cannot exceed the product of the length of a longest -path and the size of a largest minimal edge-cut separating from .