Let M = (ml,... , m.) be a monomial ideal of S = k[xl,... , x?j]. Bayer-Peeva-Sturmfels studied a subcomplex FA of the Taylor resolution, defined by a simplicial complex A C 2r. They proved that if M is generic (i.e., no variable xi appears with the same non-zero exponent in two distinct monomials which are minimal generators), then FAM is the minimal free resolution of S/M, where AM is the Scarf complex of M. In this paper, we prove the following: for a generic (in the above sense) monomial ideal M and each integer depth S/M n. We say that A c 21r} is a simplicial complex, if I c /A and J c I always imply J E A. An element of A is called a face, and the diinension of a face I is defined by dim I = III1. The dimension of the simplicial complex A is Received by the editors May 31, 1997. 1991 Mathematics Subject Classification. Primary 13D02, 13D03, 13H10.