Category of finitely generated projective modules

Definition

Let $A$ be a commutative unital ring. The category of finitely generated projective $A$ -modules, denoted $\mathbf{Proj}(A)$ , has finitely generated projective $A$ -modules as objects and $A$ -linear maps as morphisms.

This is a full subcategory of the category of $A$ -modules. It is additive: the zero module is a zero object and finite direct sums are biproducts. Every object is isomorphic to the image of an idempotent matrix $p\in M_n(A)$ , or equivalently to a direct summand of $A^n$ .

Exactness and fixed scalars

The category $\mathbf{Proj}(A)$ is generally not abelian. For example, the cokernel of a map between projective modules need not be projective. It does carry the standard split exact structure, and direct sums and direct summands remain inside the category.

The ring $A$ is fixed. A homomorphism $A\to B$ leads to extension and restriction of scalars between different module categories, but it is not itself a morphism in $\mathbf{Proj}(A)$ .

For a noncommutative ring one must instead specify left or right modules and use the compatible matrix convention. That handed version is not part of the notation on this page.

Geometric example

For a connected finite-dimensional Hausdorff second-countable smooth manifold $M$ , the Serre–Swan theorem gives a covariant equivalence

\mathbf{Vect}_{\mathbb F}(M)\simeq \mathbf{Proj}\bigl(C^\infty(M,\mathbb F)\bigr).

Both sides use morphisms over one fixed base or one fixed ring. This is not the contravariant reconstruction of maps between different manifolds from their smooth-function algebras. No compactness hypothesis is needed here; on a disconnected base the matching bundle category has globally bounded rank.

References

T. Y. Lam, Lectures on Modules and Rings, Springer, 1999. DOI record. Relevant: projective modules, finite generation, and idempotent matrices.
Richard G. Swan, “Vector Bundles and Projective Modules,” Transactions of the American Mathematical Society 105 (1962), 264–277. DOI record. Relevant: finite projective modules and vector bundles.