While working through a proof of [this](https://arxiv.org/pdf/1711.04148.pdf) paper,1 at the middle of page 45, the author's claim of a short exact sequence seems to amount to the following problem:



> 
> Let $A$ be a commutative ring and let $s,t\in A$ which generate the unit ideal. Suppose also that $M,N$ are modules over $A$. 
> 
> 
> Suppose that we have maps between the localizations
>  $$
> f\_s:M\_s\rightarrow N\_s
> $$
> $$f\_t:M\_t\rightarrow N\_t$$
> $$f\_{st}:M\_{st}\rightarrow N\_{st}
> $$
>  such that the diagram
>  $\require{AMScd}$
> $$\begin{CD}
>  M\_s @>f\_s > > N\_s\\
>  @V V V @V V V\\
>  M\_{st} @> >f\_{st} > N\_{st}
> \end{CD}$$
>  and the analogous one for $t$ commute.
> 
> 
> Does there exist a map $f:M\rightarrow N$ making all the relevant diagrams commute?
> 
> 
> 


I thought that applying the theory of the faithfully flat descent for $A\rightarrow A\_{s}\times A\_{t}$ could be a good idea here but led me nowhere.


1Bhargav Bhatt, Jacob Lurie: *A Riemann-Hilbert correspondence in positive characteristic*, <https://arxiv.org/abs/1711.04148>



Map between localizations induces map on underlying modules for Zariski covering
This\* follows immediately from the fact that the presheaf $\mathcal{Hom}(\tilde{M},\tilde{N})$ (Hartshorne's notation) on $\mathrm{Spec}(A)$ given by


$$U\mapsto Hom(\tilde{M}|\_U,\tilde{N}|\_U)$$


[is](https://math.stackexchange.com/questions/294802/prove-that-sheaf-hom-is-a-sheaf) a sheaf in the Zariski topology. In particular, it verifies the gluing condition (given your two commuting square diagrams for $s$ and $t$) for its sections on an open cover $\{U\_s,U\_t\}$ such as the one in the question.


\* I'm talking about the assertion in the question, not the linked paper


