Kernels of modular inclusion maps
Let \(R\) be a field and \(\Omega\) an \(n\)-element set. Consider the vector space \(M_k\) spanned over \(R\) by all \(k\)-element subsets \(\Delta\) of \(\Omega\). The inclusion map \(\partial : M_k\rightarrow M_{k-1}\) is defined by \(\partial(\Delta):=\Gamma_1+\Gamma_2+\cdots+\Gamma_k\), where the \(\Gamma_i\) are the \((k-1)\)-element subsets of \(\Delta\). Kernels of these maps are closely related with various combinatorial problems, such as existence of \(t\)-designs, reconstruction of graphs and \(k\)-orbits of permutation groups, see \textit{V. B. Mnukhin} [Acta Appl. Math. 29, No. 1/2, 83-117 (1992; Zbl 0769.05069)]. Properties of inclusion maps over \(\mathbb{Q}\) have been studied, for example, in \textit{R. L. Graham, S.-Y. R. Li}, and \textit{W.-C. W. Li} [SIAM J. Algebraic Discrete Methods 1, 8-14 (1980; Zbl 0499.05012)]. The purpose of this note is to study inclusion maps over fields of finite characteristic \(p\geq n\) and to show that their properties in this case are essentially the same as over fields of characteristic 0. The more interesting and more complicated case of \(2<p<n\), when a modular homology appears, is subject of the paper [J. Algebra 199, No. 2, 556-580 (1998)] by \textit{S. Bell, P. Jones}, and the author.
scientific article; zbMATH DE number 1110044