FUNCTORIAL STRUCTURE OF UNITS IN A TENSOR PRODUCT

The behavior of units in a tensor product of rings is studied, as one factor varies. For example, let k be an algebraically closed field. Le t A and B be reduced rings containing Ic, having connected spectra. Le t u is an element of A x(k), B be a unit. Then u = a x b for some unit s a is an element of A and b is an element of B. Here is a deeper cons equence, stated for simplicity in the affine case only. Let k be a fie ld, and let phi : R --> S be a homomorphism of finitely generated k-al gebras such that Spec(phi is dominant. Assume that every irreducible c omponent of Spec(R(red)) or Spec(S-red) is geometrically integral and has a rational point. Let B --> C be a faithfully flat homomorphism of reduced k-algebras. For A a k-algebra, define Q(A) to be (S x(k) A)/ (R x(k) A). Then Q satisfies the following sheaf property: the sequen ce O --> Q(B) --> Q(C) --> Q(C x(B) C) is exact. This and another resu lt are used to prove (5.2) of [7].