On the group of units and the Picard group of a product

Abstract

Let S be a reduced scheme and let f: X--> S and g: Y-->S be faithfully flat morphisms locally of finite presentation with geometrically connected and geometrically reduced maximal fibers. We discuss the canonical maps G_{m,S}(X)+G_{m,S}(Y)-->G_{m,S}(X x_{S} Y) and Pic X+Pic Y-->Pic(X x_{S} Y) induced by f and g (both sums are direct). Under certain additional conditions on X,Y,S, f and g, we describe the kernel and cokernel of the preceding maps, thereby extending known results when S is the spectrum of a field.