# Milnor patching for general modules

The Milnor patching theorem for projective modules is the following statement. Given a pullback diagram of rings $$\begin{array}{} R & \xrightarrow{f_2} & R_2 \\ \downarrow{f_1} & & \downarrow{j_2} \\ R_1 & \xrightarrow{j_1} & R_3 \end{array}$$ with $j_1$ or $j_2$ surjective, and projective modules $P_1$ and $P_2$ over $R_1$ and $R_2$ respectively together with an isomorphism $h : R_3 \otimes_{R_2} P_2 \to R_3 \otimes_{R_1} P_1$, then

1) the $R$-module $P$ as the pullback of $$\begin{array}{} P & \xrightarrow{} & P_2 \\ \downarrow{} & & \downarrow{h(1 \otimes \text{id})} \\ P_1 & \xrightarrow{1 \otimes \text{id}} & R_3 \otimes_{R_1} P_1 \end{array}$$ is projective. If $P_1$ and $P_2$ are finitely generated over $R_1$ and $R_2$ respectively, then $P$ is finitely generated.

2) There are natural isomorphisms $P \otimes_R R_1 \to P_1$ and $P \otimes_R R_2 \to P_2$.

3) All projective $R$-modules arise for appropriately chosen $P_1$, $P_2$ and $h$.

This theorem gives an equivalence of categories of projective modules on $R$, and "patching data" of $P_1$, $P_2$ and $h$ above.

Specifically, the equivalence may be described as follows. If we let $(P_1,P_2,h)$, $(P_1',P_2',h')$ be such patching data, a morphism of such triples are homorphisms $\phi_1 : P_1 \to P_1'$ and $\phi_2 : P_2 \to P_2'$ such that $$\begin{array}{} R_3 \otimes_{R_2} P_2 & \xrightarrow{1 \otimes \phi_2} & R_3 \otimes_{R_2} P_2' \\ \downarrow{h} & & \downarrow{h'}\\ R_3 \otimes_{R_1} P_1 & \xrightarrow{1 \otimes \phi_1} & R_3 \otimes_{R_1} P_1' \end{array}$$ commutes. A projective module $P$ on $R$ yields a triple $(R_1 \otimes_R P, R_2 \otimes_R P, \text{id})$, where $\text{id}$ denotes the identity map $R_3 \otimes_R P \to R_3 \otimes P$. A morphism of modules $P \to P'$ induces a morphism of triples in the obvious way.

My question is whether there exists a similar result for finitely generated modules (and modules in general), that is, not assuming $P_1$ and $P_2$ projective, with reasonable assumptions on $j_1$ and $j_2$. Optimally, no stronger conditions though.

I would like an equivalence of categories of finitely generated $R$-modules and a certain notion of "patching data" in this context. For example, perhaps by allowing $h$ to be any homomorphism would yield the corresponding result for finitely generated modules over $R$?