Invariant Subspaces of the Ring of Functions on a Vector Space over a Finite Field Original Research Article

IfFqis the finite field of characteristicpand orderq = ps, let image(q) be the category whose objects are functors from finite dimensionalFq-vector spaces toFq-vector spaces, and with morphisms the natural transformations between such functors. A fundamental object in image(q) is the injectiveIFqdefined byIFq(V) = FqV* = S*(V)/(xq − x).We determine the lattice of subobjects ofIFq. It is the distributive lattice associated to a certain combinatorially defined poset image(p, s) whoseqconnected components are all infinite (with one trivial exception). An analysis of image(p, s) reveals that every proper subobject of an indecomposable summand ofIFqis finite. ThusIFqis Artinian. FilteringIFqand image(p, s) in various ways yields various finite posets, and we recover the main results of papers by Doty, Kovács, and Krop on the structure ofS*(V)/(xq) overFq, andS*(V) overimage.