MORPHISMS BETWEEN KOSZUL COMPLEXES II

Abstract

Let K. denote the graded Koszul complex associated to the regular sequence (x₀, …, x_n) in the graded polynomial ring A = k[x₀, …, x_n], [mid ]x_i[mid ] = 1 for all i, over an arbitrary field k. Let K′. denote the Koszul complex associated to another regular sequence of homogeneous elements (p₀, …, p_n) in A. In [5] we have studied ranks of graded chain complex morphisms f.[ratio ]K′.→K′. with the property f₀ = id. Let Ω^k (respectively, Ω′^k) denote the kernel of the Koszul differential d[ratio ]K_k→ K_k−1 (respectively, d′[ratio ] K′_k→ K′_k−1), and let f_k[ratio ] Ω′^k→Ω^k denote the restriction of f_k. The main result was that Rank (f_k)>n−k.