The p-Rank of (0, 1)-matrices

Abstract

In the paper “On the p-Rank of Incidence Matrices and a Question of E. S. Lander” (A. A. Bruen and U. Ott, to appear), the authors proved under certain conditions that (rank p ( M) − 1) 2 ⩾ ( υ − ∥ a∥)(∥ a∥ − 1), where M is the incidence matrix of a linear space, υ is the number of points, and ∥ a∥ denotes the number of points on any line a. In this paper we prove a more general theorem: Let Γ be a semilinear space. Let V 0, V 1, and V be the Z-module generated by the set of points, lines, and chambers, respectively. Let δ 2 V→V 0×V 1 {A,a} (A,−a) Be the usual boundary map, where { A, a} is a chamber, e.g., the point A is incident with the line a. Let St ≔ ker( ∂ 2) denote the Steinberg module of all closed circuits in Γ. Let k be any field; then(rank k(M)⩾ dim k(St) ,where dim k ( St) = dim k ( St ⊗ k). Let ∥ A∥ denote the number of lines through the point A, ∥ a∥ the number of points on the line a. Under the hypothesis that ∥ a∥ ≡ X + 1 (mod p) and ∥ A∥ ≡ Y + 1 (mod p) for all a ∈ L and A ∈ P of the linear space Γ with v points over a field of char( k) = p, we show that rank k(M)⩽ 1+ dim k(St) if>X+1≢0( modp) v ifY(Y+1)≢0( modp) v−1 ifY+1≡0( modp) dim k(St) otherwise The dimension dim k ( St) of a linear space is also calculated. Let a be any fixed line and { a j ∥ j ϵJ} be the set of lines which pass a, then dim k ( St) = ( v − ∥ a∥)(∥ a∥ − 1) + ∑ j ∈ J (∥ a j ∥ − 1). In fact, we give an explicit basis of St consisting of elements of the standard module induced by certain ordinary triangles and quadrangles on a.