Equivariant class group. III. Almost principal fibrer bundles

Abstract

As a formulation of 'codimension-two arguments' in invariant theory, we define a (rational) almost principal bundle. It is a principal bundle off closed subsets of codimension two or more. We discuss the behavior of the category of reflexive modules over locally Krull schemes, the category of the coherent sheaves which satisfy Serre's condition $(S'_2)$ over Noetherian $(S_2)$ schemes with dualizing complexes, the class group, the canonical module, the Frobenius pushforwards, and global $F$-regularity, of a rational almost principal bundle. We give examples of finite group schemes, multisection rings, surjectively graded rings, and determinantal rings, and give unified treatment and new proofs to known results in invariant theory, algebraic geometry, and commutative algebra, and generalize some of them. In particular, we generalize the result on the canonical module of the multisection ring of a sequence of divisors by Kurano and the author. We also give a new proof of a generalization of Thomsen's result on the Frobenius pushforwards of the structure sheaf of a toric variety.