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Let us do the case of an affine scheme $X$ first. This is easy. If $M$ is an $O_X$-module, we define $\tilde{M}$ as the quasicoherent $O_X$-module defined by the global sections $M(X)$. Notice that the restriction gives the canonical map $\tilde{M}->M$.

As step 2, we extend to the case of quasiaffine $X$. Quasiaffinity means that $X-> Spec O_X(X)$ is an open embedding. Equivalently, all its quasicoherent modules are generated by global section. Hence, the coherator functor $M |-> \tilde{M}$ is defined in the same way. BTW, it is clear how the coherator works on maps too.

Finally, we have everything ready as a general scheme admits an affine open cover $X=\cup_i U_i$. The quasiaffine case is useful as double and triple intersection $U_{i,j}$, $U_{i,j,k}$ are all quasiaffine. Given an $O_X$-module $M$, we use its open pieces $M_i=M|_{U_i}$ and gluing maps $\phi_{i,j}$ from ${M_i}|_{U(i,j)}$ to ${M_j}|{U(i,j)}$ ({M_j}|_{U(i,j)}$, where$U(i,j)=U{i,j}$U(i,j)=U_{i,j}$ (because tex translator does not like me) is finding it difficult to comprehend it too without each formula starting in a new line), that satisfy the cocycle conditions on triple intersection.

Now the coherator is glued from open pieces $\tilde{M}i$ \tilde{M}_i$using isomorphisms$\tilde{\phi}{i,j}$\tilde{\phi}_{i,j}$ which inherit the cocycle conditions. And That's All Folks!

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Let us do the case of an affine scheme $X$ first. This is easy. If $M$ is an $O_X$-module, we define $\tilde{M}$ as the quasicoherent $O_X$-module defined by the global sections $M(X)$. Notice that the restriction gives the canonical map $\tilde{M}->M$.

As step 2, we extend to the case of quasiaffine $X$. Quasiaffinity means that $X-> Spec O_X(X)$ is an open embedding. Equivalently, all its quasicoherent modules are generated by global section. Hence, the coherator functor $M |-> \tilde{M}$ is defined in the same way. BTW, it is clear how the coherator works on maps too.

Finally, we have everything ready as a general scheme admits an affine open cover $X=\cup_i U_i$. The quasiaffine case is useful as double and triple intersection $U_{i,j}$, $U_{i,j,k}$ are all quasiaffine. Given an $O_X$-module $M$, we use its open pieces $M_i=M|_{U_i}$ and gluing maps $\phi_{i,j}:{M_i}|{U{i,j}}->{M_j}|\phi_{i,j}$ from ${M_i}|_{U(i,j)}$ to ${M_j}|{UU(i,j)}$ (where $U(i,j)=U{i,j}}$ i,j}$because tex translator does not like me) that satisfy the cocycle conditions on triple intersection. Now the coherator is glued from open pieces$\tilde{M}i$using isomorphisms$\tilde{\phi}{i,j}$which inherit the cocycle conditions. And That's All Folks! 1 Let us do the case of an affine scheme$X$first. This is easy. If$M$is an$O_X$-module, we define$\tilde{M}$as the quasicoherent$O_X$-module defined by the global sections$M(X)$. Notice that the restriction gives the canonical map$\tilde{M}->M$. As step 2, we extend to the case of quasiaffine$X$. Quasiaffinity means that$X-> Spec O_X(X)$is an open embedding. Equivalently, all its quasicoherent modules are generated by global section. Hence, the coherator functor$M |-> \tilde{M}$is defined in the same way. BTW, it is clear how the coherator works on maps too. Finally, we have everything ready as a general scheme admits an affine open cover$X=\cup_i U_i$. The quasiaffine case is useful as double and triple intersection$U_{i,j}$,$U_{i,j,k}$are all quasiaffine. Given an$O_X$-module$M$, we use its open pieces$M_i=M|_{U_i}$and gluing maps$\phi_{i,j}:{M_i}|{U{i,j}}->{M_j}|{U{i,j}}$that satisfy the cocycle conditions on triple intersection. Now the coherator is glued from$\tilde{M}i$using isomorphisms$\tilde{\phi}{i,j}\$ which inherit the cocycle conditions. And That's All Folks!