Detecting Rational Cohomology of Algebraic Groups
نویسندگان
چکیده
Let G be a connected, semisimple algebraic group defined over an algebraically closed field k of positive characteristic p. Assume that G is defined and split over the prime field k0 = GF (p), and for q = p , let G(q) be the subgroup of GF (g)-rational points. Let V be a rational G-module, and, for a non-negative integer r, let V(r) be the rational G-module obtained by 'twisting' the original G-action on V by the r-th power of the Frobenius endomorphism a of G. In [5] we showed (together with W. van der Kallen) that, if V is finite dimensional and n is a non-negative integer, for sufficiently large m and r (depending on V and n), there are isomorphisms
منابع مشابه
Cohomology of classical algebraic groups from the functorial viewpoint
We prove that extension groups in strict polynomial functor categories compute the rational cohomology of classical algebraic groups. This result was previously known only for general linear groups. We give several applications to the study of classical algebraic groups, such as a cohomological stabilization property, the injectivity of external cup products, and the existence of Hopf algebra s...
متن کاملHow to calculate homology groups of spaces of nonsingular algebraic projective hypersurfaces
A general method of computing cohomology groups of the space of non-singular algebraic hypersurfaces of degree d in CP n is described. Using this method, rational cohomology groups of such spaces with n = 2; d 4 and n = 3 = d are calculated.
متن کاملRational Isomorphisms between K-theories and Cohomology Theories
The well known isomorphism relating the rational algebraic K-theory groups and the rational motivic cohomology groups of a smooth variety over a field of characteristic 0 is shown to be realized by a map (the “Segre map”) of infinite loop spaces. Moreover, the associated Chern character map on rational homotopy groups is shown to be a ring isomorphism. A technique is introduced which establishe...
متن کاملPrym-Tjurin Constructions on Cubic Hypersurfaces
In this paper, we give a Prym-Tjurin construction for the cohomology and the Chow groups of a cubic hypersurface. On the space of lines meeting a given rational curve, there is the incidence correspondence. This correspondence induces an action on the primitive cohomology and the primitive Chow groups. We first show that this action satisfies a quadratic equation. Then the Abel-Jacobi mapping i...
متن کاملRational S 1 - Equivariant Elliptic Cohomology
We give a functorial construction of a rational S 1-equivariant cohomology theory from an elliptic curve equipped with suitable coordinate data. The elliptic curve may be recovered from the cohomology theory; indeed, the value of the cohomology theory on the compactification of an S 1-representation is given by the sheaf cohomology of a suitable line bundle on the curve. The construction is eas...
متن کامل