# Differential Galois Theory by M. van der Put, M. Singer

By M. van der Put, M. Singer

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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 [107]. 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.