Proposition 7

The set $P(A)$ of classes of projective A-modules of rank 1, with the law of composition (6), is a commutative group. If M is a projective A-module of rank 1 and $M^*$ is its dual, then

(7)

The associativity and commutativity of the tensor product show that the law of composition (6) is associative and commutative; the isomorphism between $A \otimes_A M$ and M prove that $\mathrm{cl}(A)$ is the identity element under this law and, by virtue of Theorem 3, $\mathrm{cl}(M) + \mathrm{cl}(M^*) = \mathrm{cl}(A)$, whence the proposition.

Let B be a commutative A-algebra and M a projective A-module of rank 1; then $M_{(B)} = B \otimes_A M$ is a projective B-module of rank 1 (no. 3, Proposition 4). Then there exists a mapping called canonical $\phi : P(A) \to P(B)$ such that

(8)

The equation $M_{(B)} \otimes_B N_{(B)} = (M \otimes_A N)_{(B)}$ for two A-modules M, N proves that the mapping $\phi$ is a commutative group homomorphism.

Remark (2). Condition (e) of Theorem 1 (equivalent to the fact that P is projective and finitely generated) may also be expressed by saying that the sheaf of modules $\tilde{P}$ over $X = \mathrm{Spec}(A)$ associated() with P is locally free and of finite type* and may therefore be interpreted as the sheaf of sections of a vector bundle over X. Conversely, every vector bundle over X arises from a finitely generated projective module, which is determined to within a unique isomorphism; the projective modules of rank $n$ thus correspond to the vector bundles all of whose fibres have dimension $n$. In particular, the vector bundles of rank 1 correspond to the projective modules of rank 1. If we denote by $\mathscr{O}_X$ the structure sheaf $\tilde{A}$ and by $\mathscr{O}_X^*$ the sheaf of units of $\mathscr{O}_X$ (whose sections over an open set U of X are the invertible elements of the ring of sections of $\mathscr{O}_X$ over U), it follows that the group $P(A)$ is isomorphic to the first cohomology group $H^1(X, \mathscr{O}_X^*)$.*

(* See A. Grothendieck, Éléments de géométrie algébrique, I (§ 1) (Publ. Math. I.H.E.S., no. 4, 1960).