Invariant subspaces of the ring of functions on a vector space over a finite field

[摘要] If F-q is the finite field of characteristic p and order q = p(s), let F(q) be the category whose objects are functors from finite dimensional F-q-vector spaces to F-q-vector spaces, and with morphisms the natural transformations between such functors. A fundamental object in F(q) is the injective I-Fq defined by I-Fq(V) = F-q(V*) = S*(V)/(x(q)-x). We determine the lattice of subobjects of I-Fq. It is the distributive lattice associated to a certain combinatorially defined poset F(p, s) whose q connected components are all infinite (with one trivial exception). An analysis of F(p, s) reveals that every proper subobject of an indecomposable summand of I-Fq is finite. Thus I-Fq is Artinian. Filtering I-Fq and F(p, s) in various ways yields various finite posets, and we recover the main results of papers by Doty, Kovacs, and Krop on the structure of S*(V)/(x(q)) over F-q, and S*(V) over (F) over bar(p). (C) 1997 Academic Press.

[发布日期] 1997-05-01