Joel Roberts

Rings of invariants of reductive groups acting on regular rings are Cohen-Macaulay

Introduction 115 1. Some Applications of the Main Theorem 117 2. Counterexamples 124 3. Preliminaries and Basic Facts 127 4. Outline of the Proof 130 5. R&me of Cohomology 132 6. Pure Subrings 135 7. The Main Results in Characteristic p 141 8. Generic Freeness 146 9. Generic Uniform Convergence of Koszul Cohomology Along the Fibers 148 10. Resume of Invariant Theory 151 11. Reduction to the Graded Case 154 12. Proof of Proposition A, First Step: Descent 158 13. Proof of Proposition A, Second Step: The Key Argument in Characteristic p 162 14. Arithmetically Cohen-Macaulay Projective Schemes 165 15. Proof of Proposition B, First Step: Descent 168 16. Proof of Proposition B, Second Step: The Final Twist in Characteristic p 170 References 172

The purity of the Frobenius and local cohomology

rZ ring homomorphism (all rings are commutative, with identity, and homomorphisms preserve the identity) R --f S is called pure if for every R-module M, M ---f M RR S via m -+ m @ 1 is injective. Quite generally, if I is a homogeneous ideal of a graded ring R and rad(1) = rad((fo ,..., fi2)R), where f0 ,...,fn are forms of R, then the local cohomology modules H,i(R) can be expressed as direct limits of Koszul cohomology:

A short proof of a theorem of M. Hashimoto

Abstract Recently M. Hashimoto announced that the resolutions of determinantal ideals depend on the characteristic of the base field. The purpose of this note is to give a short proof of this result. One should stress that even though the method we use is different from Hashimotos, his result provided us with the clue where the additional relation exists.