Descent of modules on a finite principal cover

Both $P$ and $Q$ are $A$-modules, the maps $u$ and $v$ are $A$-linear, and $M=\ker (u-v)$ sits in the exact sequence $0\to M\to P\stackrel{u-v}{\to }Q$; every $M_k$ is an $A$-module through $A\to A_{f_k}$ and every $(M_k{)}_{f_l}$ is an $A$-module through $A\to A_{f_k{f}_l}$, so all localisations below are localisations of $A$-modules.

For an $A$-module $N$ on which $f_i$ acts invertibly the localisation map $\lambda :N\to N_{f_i}$ is an isomorphism, with inverse $n\mathrm{/}{f}_i^t\mapsto f_i^{-t}n$: if $f_i^u(f_i^{s}n-f_i^t{n}^{\mathrm{\prime }})=0$ in $N$ with $u,s,t\ge 0$, then multiplying by $f_i^{-(u+s+t)}$ gives $f_i^{-t}n=f_i^{-s}{n}^{\mathrm{\prime }}$, so the inverse is well defined, and it is $A_{f_i}$-linear and inverts $\lambda$ on the image of $N$; in particular $\lambda :(M_k{)}_{f_i}\to (M_k{)}_{f_i{f}_i}=(M_k{)}_{f_i}$ is the identity.

Localising the exact sequence of step 1.1 at $f_i$, [F1] makes $0\to M_{f_i}\to P_{f_i}\stackrel{(u-v{)}_{f_i}}{\to }{Q}_{f_i}$ exact, and [F2] identifies the finite products componentwise, $P_{f_i}\cong {\prod }_k(M_k{)}_{f_i}$ and $Q_{f_i}\cong {\prod }_{k,l}(M_k{)}_{f_i{f}_l}$; under these identifications the $(k,l)$-component of $u_{f_i}$ is $(m_k)\mapsto (m_k){\mathrm{\mid }}_{f_l}$ and the $(k,l)$-component of $v_{f_i}$ is $(m_k)\mapsto {\phi }_{kl}^{-1}((m_l){\mathrm{\mid }}_{f_k})$, so $M_{f_i}$ is the set of families $(m_k)\in {\prod }_k(M_k{)}_{f_i}$ with $(m_k){\mathrm{\mid }}_{f_l}={\phi }_{kl}^{-1}((m_l){\mathrm{\mid }}_{f_k})$ for all $k,l$.

The projection ${\mathrm{p}\mathrm{r}}_i:M\to M_i$ is $A$-linear and hence induces an $A_{f_i}$-linear map ${\theta }_i:M_{f_i}\to (M_i{)}_{f_i}=M_i$, the identification $(M_i{)}_{f_i}=M_i$ being the case $N=M_i$ of step 1.2; on the description of step 2.1, ${\theta }_i$ sends a compatible family $(m_k)$ to its $i$-th component $m_i$, viewed in $(M_i{)}_{f_i}=M_i$.

The map ${\theta }_i$ is surjective: given $m\in M_i$ set $m_k:={\phi }_{ik}(m)\in (M_k{)}_{f_i}$ for every $k$, where ${\phi }_{ik}:(M_i{)}_{f_k}\to (M_k{)}_{f_i}$ and $m$ is regarded in $(M_i{)}_{f_k}$ through the inverse of the isomorphism $(M_i{)}_{f_k}\to (M_i{)}_{f_i{f}_k}=(M_i{)}_{f_k}$ of step 1.2; for all $k,l$ the cocycle condition identifies the localisations of ${\phi }_{il}$ and of ${\phi }_{kl}\circ {\phi }_{ik}$ to $A_{f_i{f}_k{f}_l}$, so $(m_k){\mathrm{\mid }}_{f_l}={\phi }_{ik}(m){\mathrm{\mid }}_{f_i{f}_k{f}_l}={\phi }_{kl}^{-1}({\phi }_{il}(m){\mathrm{\mid }}_{f_i{f}_k{f}_l})={\phi }_{kl}^{-1}((m_l){\mathrm{\mid }}_{f_k})$, and $(m_k)$ is a compatible family with $i$-th component $m$.

The map ${\theta }_i$ is injective: if $(m_k)$ is a compatible family with $m_i=0$, then for every $l$ the $(i,l)$-component of the compatibility reads $(m_i){\mathrm{\mid }}_{f_l}={\phi }_{il}^{-1}((m_l){\mathrm{\mid }}_{f_i})$, whose left side is $m_i=0$ by step 1.2, so $(m_l){\mathrm{\mid }}_{f_i}=0$, and since localising at $f_i$ is an isomorphism on $(M_l{)}_{f_i}$ by step 1.2, $m_l=0$; hence $(m_k)=0$.

Consequently every ${\theta }_i$ is an isomorphism; for the compatibility with the overlap data, let $(m_k)\in M_{f_i{f}_j}$ be a compatible family, whose image under the map induced by ${\theta }_i$ followed by ${\phi }_{ij}$ is ${\phi }_{ij}((m_i){\mathrm{\mid }}_{f_j})$ in $(M_j{)}_{f_i}$ and whose image under the map induced by ${\theta }_j$ is $(m_j){\mathrm{\mid }}_{f_i}$, and the $(i,j)$-component of the compatibility in step 2.1 states exactly that these agree; every construction used only the given modules, the given isomorphisms and universal constructions of kernels and localisations, so no choice principle is invoked.