One-dimensional affine group schemes

One - Dimensional Affine Group Schemes

The construction of these groups is straightforward; to an algebra B we assign the quotient R,,, G,/G,. The main effort comes in showing that these are the only possibilities. The key to this, and the basic technical idea in the paper, is the use of N&on blow-ups of group schemes over valuation rings. This process has been used before [ 1, 21 as a tool for resolving singularities, but in fact i...

Basic Theory of Affine Group Schemes

Moduli of Affine Schemes with Reductive Group Action

For a connected reductive group G and a finite–dimensional G–module V , we study the invariant Hilbert scheme that parameterizes closed G–stable subschemes of V affording a fixed, multiplicity–finite representation of G in their coordinate ring. We construct an action on this invariant Hilbert scheme of a maximal torus T of G, together with an open T–stable subscheme admitting a good quotient. ...

One Higher Dimensional Analog of Jet Schemes

We develop the theory of truncated wedge schemes, a higher dimensional analog of jet schemes. We prove some basic properties and give an irreducibility criterion for truncated wedge schemes of a locally complete intersection variety analogous to Mustaţǎ’s for jet schemes. We show that the reduced subscheme structure of a truncated wedge scheme of any monomial scheme is itself a monomial scheme ...

Relative Affine Schemes

1 Affine Morphisms Definition 1. Let f : X −→ Y be a morphism of schemes. Then we say f is an affine morphism or that X is affine over Y , if there is a nonempty open cover {Vα}α∈Λ of Y by open affine subsets Vα such that for every α, fVα is also affine. If X is empty (in particular if Y is empty) then f is affine. Any morphism of affine schemes is affine. Any isomorphism is affine, and the aff...