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In the mathematical fields of topology and K-theory, the Serre–Swan theorem, also called Swan's theorem, relates the geometric notion of vector bundles to the algebraic concept of projective modules and gives rise to a common intuition throughout mathematics: "projective modules over commutative rings are like vector bundles on compact spaces".
This can be made precise for the ring of continuous real-valued functions on a compact Hausdorff space, as well as for the ring of smooth functions on a smooth manifold (see Serre–Swan theorem that says a finitely generated projective module over the space of smooth functions on a compact manifold is the space of smooth sections of a smooth ...
This is in fact a special case of the previous example by the Serre–Swan theorem and conversely this theorem can be proved by first proving both these facts, the observation that the global sections functor is an equivalence between trivial vector bundles over and free modules over () and then using the universal property of the Karoubi envelope.
Richard Gordon Swan (/ s w ɑː n /; born 1933) is an American mathematician who is known for the Serre–Swan theorem relating the geometric notion of vector bundles to the algebraic concept of projective modules, [1] and for the Swan representation, an l-adic projective representation of a Galois group. [2]
Shows there is a 1−1 correspondence between topological vector bundles over a compact Hausdorff space X and finitely generated projective modules over the ring C(X) of continuous functions on X (Serre–Swan theorem) 1963: Frank Adams–Saunders Mac Lane: PROP categories and PACT categories for higher homotopies. PROPs are categories for ...
Due to the above-mentioned Serre–Swan theorem, odd classical fields on a smooth manifold are described in terms of graded manifolds. Extended to graded manifolds, the variational bicomplex provides the strict mathematical formulation of Lagrangian classical field theory and Lagrangian BRST theory.
Seifert–Van Kampen theorem; Serre–Swan theorem; Simplicial approximation theorem; Snaith's theorem; Stallings–Zeeman theorem; U. Universal coefficient theorem; V.
A geometrical and global counterpart to this is the Serre–Swan theorem, relating projective modules and coherent sheaves. More generally, one has [ 10 ] Let R {\displaystyle R} be a local ring and M {\displaystyle M} a finitely generated module over R {\displaystyle R} .