Milnor patching for schemes

Question

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?

Answer 1

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.