Huah Chu

A note on projective normality

Let G be any finite group, G → GL(V) be a representation of G, where V is a finite-dimensional vector space over an algebraically closed field k. Theorem. Assume that either char k = 0 or char k = p > 0 with p |G|. Then the quotient variety P(V)/G is projectively normal with respect to the line bundle L, where L is the descent of O(1) ⊗m from P(V) with m = |G|!. This partially solves a question raised in the paper of Kannan, Pattanayak and Sardar [Proc. Amer. Math. Soc. 137 (2009), 863-867].

POLYNOMIAL INVARIANTS OF FOUR-DIMENSIONAL ORTHOGONAL GROUPS

Let Q be a nondegenerate quadratic form over the finite field F q and O n (F q ) be the associated orthogonal group. Let O n (F q ) act linearly on the polynomial ring F q [x 1, …, x n ]. We find the invariant subring of O 4(F q ) with explicit generators.

A variant of the Reynolds operator

Let G be a linearly reductive group over a field k, and let R be a k-algebra with a rational action of G. Given rational R-G-modules M and N, we define for the induced G-action on Hom R (M,N) a generalized Reynolds operator, which exists even if the action on Hom R (M,N) is not rational. Given an R-module homomorphism M → N, it produces, in a natural way, an R-module homomorphism which is G-equivariaut. We use this generalized Reynolds operator to study properties of rational R-G modules. In particular, we prove that if M is invariantly generated (i.e. M = R.M G ), then M G is a projective (resp. flat) R G -module provided that M is a projective (resp. flat) R-module. We also give a criterion whether an R-projective (or R-flat) rational R-G-module is extended from an R G -module.

Quartic Fields and Radical Extensions

Let K be a field and K(?) be an extension field ofK . If K(?) : K ] = 3, charK?= 3, and the minimal polynomial of ? over K isT3?uT?v?KT ], it is proved in Kang (2000, Am. Math. Monthly, 107, 254?256) thatK (?) is a radical extension of K if and only if, for somew?K, 81 v2? 12u3=w2if char K?= 2, or u3/v2=w2+w if char K= 2. In this paper, we prove a similar result when K(?) : K ] = 4, charK?= 2, and the minimal polynomial of ? over K isT4?uT2?vT?w?KT ] with v?= 0 :K (?) is a radical extension of K if and only if the following system of polynomial equations is solvable in K, 64 X3? 32uX2+ (4 u2+ 16w )X?v2= 0 and 64wX2? (32 uw? 3v2 )X+ (4 u2w+ 16w2?uv2) ?Y2= 0. The situation when v= 0 will also be solved.

On the integral dependence of power series rings

We prove the following results: (1) Let R ⊂ S be two commutative rings. Suppose that dim R = 0.If f(X) ∈ S[[X]]is integral over R[[X]], then every coefficient of f(X) is integral over R. (2) Let dim R ≥ 1. There exists a ring S containing R and a power series f(X) ∈ S[[X]]such that f(X) is integral over R[[X]], but not all coefficients of f(X) are integral over R. (3) Let k ⊂ R. Suppose that R is algebraic over the field k. Then R[[X]] is integral over k[[X]] if and only if the nilradical of R is nilpotent and the separable degree and the inseparable exponent of R red over k are finite.