Invariant subspaces of the ring of functions on a vector space over a finite field
Let \(F_q\) denote the Galois field of order \(q=p^s\), \({\mathcal I}(q)\) the lattice of functors from finite-dimensional \(F_q\)-vector spaces to \(F_q\)-vector spaces with natural transformations as morphisms. A functor \(F\) is called a subobject of a functor \(G\) iff \(F(V)\subseteq G(V)\) for any finite-dimensional \(F_q\)-vector space \(V\). The author considers the injective \(I_{F_q}\in{\mathcal I}(q)\) defined by \(I_{F_q}(V):=F^{V^*}_q\) (where * denotes the dual vector space). The latter is also equal to the quotient of the polynomial algebra over \(V\) by the ideal generated by \(\{x^q-x\mid x\in V\}\). As a main result the author determines the lattice \({\mathcal L}\) of subobjects of \(I_{F_q}\). \({\mathcal L}\) is shown to be isomorphic to the distributive lattice of order ideals of the poset \({\mathcal I}(p,s)\). For \(s=1\) the latter is defined to be \((\mathbb{N},<)\) where \(n<m\) iff \(m=n+(p-1)\). For \(s>1\) he defines \({\mathcal I}(p,s)=(\mathbb{N}^s,<)\) with \(J<I\) iff \(I=J+R_j\) for some \(j\in\{0,1,\dots,s-1\}\). Here \(R_0:=(-1,0,\dots,p)\), and \(R_j:=(r_0,\dots,r_{s-1})\) with \(r_j=p\), \(r_{j+1}=-1\), \(r_k=0\) for all \(j\neq k\neq j+1\) if \(j>0\). Furthermore, the proper subobjects of indecomposable summands of \(I_{F_q}\) are shown to be finite. Filtration methods applied to \(I_{F_q}\) and \({\mathcal I}(p,s)\) yield the main results of the papers of \textit{S. R. Doty} [J. Algebra 95, 373-383 (1985; Zbl 0577.20031)], \textit{L. G. Kovács} [Proc. Symp. Pure Math. 47, 207-218 (1987; Zbl 0652.20045)], and \textit{L. Krop} [J. Algebra 99, 370-421 (1986; Zbl 0588.20039)].
scientific article; zbMATH DE number 1019691
scientific article