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Compatible local units glue uniquely
PDF p. 9: Surjectivity of the determinant on K1
Statement
Let X be a ringed space with structure sheaf O and let {Uᵢ} cover X. Units uᵢ∈O(Uᵢ)× that agree on every overlap Uᵢ∩Uⱼ glue to a unique u∈O(X)×. The resulting operation is compatible with multiplication.
A ringed space is a topological space equipped with a sheaf of commutative unital rings; a unit is a section having a multiplicative inverse.
Facts & Assumptions
Given: A ringed space, an open cover, and compatible local units as in the statement.
Goal: Glue the units and prove uniqueness and multiplicativity.
Proof
The sheaf gluing axiom gives a unique section u∈O(X) restricting to uᵢ. On overlaps the inverse sections agree, since each is the inverse of the same section. They therefore glue to v∈O(X).
Both uv and 1 restrict to 1 on each Uᵢ. The uniqueness part of the sheaf axiom gives uv=1; commutativity also gives vu=1. Thus u is a unit. If another global unit restricts to the same local units, uniqueness for sections makes it equal to u.
The product of two glued sections restricts to the products of the local sections, so uniqueness identifies it with their glued product. This proves multiplicativity.