Search

We prove that the Newton polygons of Frobenius on the crystalline cohomology of proper smooth varieties satisfy a symmetry that results, in the case of projective smooth varieties, from Poincaré duality and the hard Lefschetz theorem. As a corollary, we deduce that the Betti numbers in odd degrees of any proper smooth variety over a field are even (a consequence of Hodge symmetry in characteristic zero), answering an old question of Serre. Then we give a generalization and a refinement for arbitrary varieties over finite fields, in response to later questions of Serre and of Katz.

In fact, if X is any vector space on which the primitive Banach algebra A acts faithfully and irreducibly, then X can be converted in a Banach space in such a way that the requirements in the theorem are satisfied and even the inclusion A ↪→ BL(X) is contractive. Roughly speaking, the aim of this paper is to prove the appropriate Jordan variant of the above theorem. The notion of primitiveness for Jordan algebras was introduced and developed in 1981 by L. Hogben and K. McCrimmon [10]. Primitive Jordan algebras are relevant particular types of prime nondegenerate Jordan algebras but, although the celebrated Zel’manov prime theorem ([19], 1983) gave a precise description of these last algebras, it has happened only very recently that the appropriate variant of Zel’manov’s theorem for primitive Jordan algebras has been obtained (see [3] and [17]). Also very recently several particular normed versions of Zel’manov’s theorem have been provided (see [8], [6], [16], and [7]). Nevertheless, to obtain a Zel’manov type theorem for primitive Jordan-Banach algebras has remained an open problem in the last years [15]. In fact we have been able to prove such a theorem but only passing through a general normed version of the Zel’manov prime theorem (see Theorem 1) which is in our opinion one of the most important novelties in the paper. Since Theorem 1 will probably have applications different from that in the paper, we have included in its statement and proof some details not strictly needed for our main purpose. The same comment should be made concerning Theorem 2, which is nothing but a fine improvement of Theorem 1 under the additional assumption of completeness. From Theorem 2 and the main results in [3], [18], and [5], the desired Jordan variant of Theorem 0 (Theorem 3) follows easily. Again roughly speaking, it asserts that primitive complex Jordan-Banach algebras, different from the simple exceptional 27-dimensional one and the simple Jordan algebras of a continuous symmetric bilinear form on a complex Banach space, can be continuously regarded as Jordan algebras of bounded linear operators ”acting irreducibly” on a suitable complex Banach space.

For an impurity molecule the translational, rotational, and vibrational coordinates are introduced by means of the Eckart conditions. The kinetic energy operator is transformed from Cartesian to new coordinates mentioned with the help of the Frobenius theorem and the Gaussian algorithm of matrix algebra. [Russian Text Ignored].