By Michel Brion (auth.), H. E. A. Campbell, Aloysius G. Helminck, Hanspeter Kraft, David Wehlau (eds.)
A workforce of Gerry Schwarz’s colleagues and collaborators accrued on the Fields Institute in Toronto for a mathematical festschrift in honor of his sixtieth birthday. This quantity is an outgrowth of that occasion, protecting the wide variety of arithmetic to which Gerry Schwarz has both made primary contributions or encouraged others to pursue. The articles are a sampling of recent day algebraic geometry with linked workforce activities from its major specialists, with a specific specialize in attribute zero and modular invariant theory.
Contributors:
M. Brion
A. Broer
D. Daigle
J. Elmer
P. Fleischmann
G. Freudenberg
D. Greb
P. Heinzner
A. Helminck
B. Kostant
H. Kraft
R. J. Shank
W. Traves
N. R. Wallach
D. Wehlau
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4]). In the first section we show that the direct summand property is inherited by point-stabilizers. In the second section we recall Kemper–Malle’s classification of irreducible pseudo-reflection groups that are not coregular, and describe the other tools used in the proof of the main theorem. In the last section we give the details of the calculations. 1 The direct summand property and point-stabilizers For elementary facts on the invariant theory of finite groups we refer to [2], for a discussion of the direct summand property and the different see [3].
Let θG/H be the part of θG involving the powers of linear forms xα , such that xα ∈ P; then θG = θG/H · θH . So θH and θG/H are relatively prime, and more important θG/H ∈ P The homogeneous element θG is a G-semi-invariant for some character χ : G → k× . Similarly θH is an H-semi-invariant. The quotient θG/H = θG /θH is an element e that is an absolute of A, and is also an H-semi-invariant. , θG/H ∈ C), but e θG/H ∈ P ∩C = q. 24 A. Broer Assume now that B is a direct summand of A as a graded B-module; hence there exists a homogeneous θ˜ ∈ A such that TrG (θ˜ /θG ) = 1.
If ΩB/A is a free B-module, then B[n] = A[m+n] . In view of the Quillen–Suslin Theorem, there follows this corollary. 1. Consider A ⊂ B where A is a Noetherian ring and B is a polynomial ring over a field. If B is an Am -fibration over A, then B[n] = A[m+n] for some n ≥ 0. It was proved by Hamann [9] that if A is a noetherian ring containing Q then the conditions A ⊂ B and B[n] = A[n+1] imply B = A[1] . Combining this with the above result of Asanuma gives the following theorem. 4). Let A be a noetherian ring containing a field of characteristic zero, and let B be an A1 -fibration over A.