By M. van der Put, M. Singer
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Worthwhile illustrations and workouts integrated all through this lucid insurance of team concept, Galois conception and classical perfect conception stressing evidence of vital theorems. comprises many old notes. Mathematical evidence is emphasised. contains 24 tables and figures. Reprint of the 1971 variation.
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Extra resources for Differential Galois Theory
Indeed, (V − S)/GL(n) is the Grassmannian Gr(N, N + n) of N -dimensional linear subspaces of AN+n , which we can view as the homogeneous space GL(N + n)/((GL(N ) × GL(n)) Hom(An , AN )). Therefore (V − S)/G is the homogeneous space GL(N + n)/((GL(N ) × G) Hom(An , AN )). Any quotient of a linear algebraic group by a closed subgroup scheme exists as a quasi-projective scheme [147, pp. 122–123]. So (V − S)/G is a quasi-projective variety. Moreover, the natural map GL(N + n)/(GL(N ) × G) → (V − S)/G is an affine bundle.
More generally, Becker-Gottlieb transfer can be viewed as a stable map in Morel-Voevodsky’s A1 -homotopy category . 17 to fields of any characteristic. (The proof as written requires a smooth G-equivariant compactification of G/N(T ) by a divisor with simple normal crossings, for G a reductive group over a field and T a maximal torus in G. ) By definition, a torus over a field k is a k-group scheme that becomes isomorphic to (Gm )n over the algebraic closure of k, for some natural number n.
Next, a proper morphism f : X → Y of smooth G-schemes over a field k determines a pushforward map on equivariant Chow groups, f∗ : CHGi (X) → CHGi+dim(Y )−dim(X) (Y ). Any morphism f : X → Y of smooth G-schemes determines a pullback map, f ∗ : CHGi Y → CHGi X, and f ∗ is a ring homomorphism. Both types of homomorphism occur in the basic exact sequence for equivariant Chow groups, as follows. 9 Let G be an affine group scheme of finite type over a field k that acts on a smooth k-scheme X and preserves a smooth closed k-subscheme Y of codimension r.