Let $R\_1,R\_2$, and $S$ be commutative rings with maps $R\_1,R\_2 \to S$ and form the fiber product $R = R\_1 \times\_S R\_2$. A well-known theorem of Milnor says that under certain assumptions the category of finitely generated projective $R$-modules is equivalent to a category of "patching data": an object in this category consists of finitely generated projective modules $P\_1$ and $P\_2$ over $R\_1$ and $R\_2$ respectively, together with an $S$-isomorphism $P\_1 \otimes\_{R\_1} S \cong P\_2 \otimes\_{R\_2} S$. This may be applied, for instance, to derive a Meyer-Vietoris sequence for Picard groups of rings.


My question: does this generalize to pushouts of schemes? For instance, if I am computing the Picard group of a projective nodal cubic, which can be obtained from $\mathbb{P}^1$ by identifying two points, can I apply a Meyer-Vietoris sequence similar to the one that works in the affine case?



Milnor patching for schemes
First of all, you have to show that these pushouts exist at all. More precisely, if $A \hookrightarrow X$ is a closed immersion and $A \to Y$ is an arbitrary morphism, then the pushout $X \cup\_A Y$ exists in the category of schemes. Moreover, the underlying ringed space is the pushout of the underlying ringed spaces (thus topologically it's a pushout in the usual sense, and algebraically it is given by a fiber product of structure sheaves); remark that this is not trivial at all. One can prove that the canonical map $Y \to X \cup\_A Y$ is a closed immersion. You can find all that in Karl Dahlke's paper "Gluing schemes and a scheme without closed points".


Milnor's Theorem states that the functor $\mathrm{Vect}(X \cup\_A Y) \to \mathrm{Vect}(X) \times\_{\mathrm{Vect}(A)} \mathrm{Vect}(Y)$ is an equivalence of categories, where the right hand side is the $2$-fiber product and $\mathrm{Vect}(-)$ denotes the category of algebraic vector bundles. At least when everything is affine, but since everything in sight is local, we immediately also get it for arbitrary schemes.


