Alphabeta MathHow statement and proof provenance work
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Affine bundle classes are actual projective-module classes
Statement
Assume AC. For a commutative unital ring , the maps and induce inverse bijections between rank- bundle isomorphism classes and constant-rank- finite projective module isomorphism classes. In particular two bundles are isomorphic if and only if their modules of sections are isomorphic.
Facts & Assumptions
Given: The data and hypotheses in the statement.
Goal: Establish the stated conclusion.
The affine equivalence preserves rank and is quasi-inverse. Affine bundles and finite projective modules
Proof
The equivalence sends rank- objects into rank- objects. A functor sends an invertible morphism to an invertible morphism, so both assignments descend to isomorphism classes.
The unit and counit are isomorphisms. Thus the composites induce the identity on classes, and a module isomorphism sheafifies to a bundle isomorphism and conversely.
Depends on
Dependency tree · 3 levels
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