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

TL;DR

This paper investigates the structure of units and the Picard group of a product of schemes over a base scheme, extending known results from fields to more general settings with additional conditions.

Contribution

It provides a detailed description of the kernel and cokernel of natural maps on units and Picard groups for products over a base scheme, generalizing previous results.

Findings

Describes the kernel and cokernel of the maps on units and Picard groups.

Extends known results from fields to more general schemes.

Provides conditions under which the maps are isomorphisms or have specific kernels and cokernels.

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.