| Functorial Structure of Units in a Tensor Product (2007) | |||||||||||||
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| . 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. Let A and B be reduced rings containing k, having connected spectra. Let u # A# k B be a unit. Then u = a# b for some units a # A and b # B. Here is a deeper consequence, stated for simplicity in the affine case only. Let k be a field, and let # : R # S be a homomorphism of finitely generated k-algebras such that Spec(#) is dominant. Assume that every irreducible component 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# k A) # /(R# k A) # . Then Q satisfies the following sheaf property: the sequence 0 -# Q(B) -# Q(C) -# Q(C# B C) is exact. This and another result are used to prove (5.2) of [7]. 1 Introduction Let k be a field, and let A be a finitely generated k-algebra. 1 W... | |||||||||||||
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