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Short question

Can we describe a quasi-coherent module on a scheme by usual modules with respect to an affine cover, which satisfy some compatibility conditions, which can be formulated in the language of commutative algebra (actually tensor products of modules)? Thus, restrictions to non-affine subsets are not allwed. This sounds artificial, but it is (I think) necessary to work out Gabber's proof for this question http://mathoverflow.net/questions/39941/does-qcohx-admit-a-generating-set.

Example

If the scheme is separated, the answer is of course yes: Choose an affine cover $U_i$ of our scheme $X$, thus also $U_i \cap U_j$ is affine. If $M_i$ is a $\mathcal{O}(U_i)$-module such that we have isomorphisms $\phi_{ij} : M_i \otimes_{\mathcal{O}(U_i)} \mathcal{O}(U_i \cap U_j) \cong M_j \otimes_{\mathcal{O}(U_j)} \mathcal{O}(U_i \cap U_j)$ of $\mathcal{O}(U_i \cap U_j)$-modules, which satisfy the cocycle conditions, namely $\phi_{ii} = id$ and $\phi_{ijk} = \phi_{kji} \phi_{ikj}$ where $\phi_{ijk} = \phi_{ij} \otimes_{\mathcal{O}(U_i \cap U_j)} \mathcal{O}(U_i \cap U_j \cap U_k)$. Then the quasi-coherent modules $\widetilde{M_i}$ on $U_i$ can be glued to a quasi-coherent module on $X$.

Long question

In the general case, there is an affine cover $U_i \cap U_j = \cup_k W_k$, such that $W_k$ is basic-open in $U_i$ and $U_j$ simulteneously. In this setting we should use isomorphisms $\phi_{ijk} : M_i \otimes_{\mathcal{O}(U_i)} \mathcal{O}(W_k) \cong M_j \otimes_{\mathcal{O}(U_j)} \mathcal{O}(W_k)$. However, in order to formulate the cocycle condition in affine terms, we actually have to cover triple overlaps $U_i \cap U_j \cap U_k$ by affines $R_n$, such that $R_n$ is basic-open in $U_i,U_j,U_j$ and then cover the $U_i \cap U_j$ with these $R_n$, where we vary $k$. Thus we assume isomorphisms $\phi_{ijkn} : M_i \otimes_{\mathcal{O}(U_i)} \mathcal{O}(R_n) \cong M_j \otimes_{\mathcal{O}(U_j)} \mathcal{O}(R_n)$ and may formulate the cocycle condition. Now in order to glue these isomorphisms to an isomorphism on $U_i \cap U_j$ between $\widetilde{M_i}$ and $\widetilde{M_j}$, we also have to ensure that there is some intersection compatibility when we vary $k$. However, this cannot be formulated with the given data.

Before I end up defining refinements of these affine covers by transfinite induction ;-), I thought it would be better to ask here if there is an easy way which I overlook.
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Short question:

Can we describe a quasi-coherent module on a scheme by usual modules with respect to an affine cover, which satisfy some compatibility conditions, which can be formulated in the language of commutative algebra (actually tensor products of modules)? Thus, restrictions to non-affine subsets are not allwed. This sounds artificial, but it is (I think) necessary to work out Gabber's proof for this question http://mathoverflow.net/questions/39941/does-qcohx-admit-a-generating-set.

Example:

If the scheme is separated, the answer is of course yes: Choose an affine cover $U_i$ of our scheme $X$, thus also $U_i \cap U_j$ is affine. If $M_i$ is a $O(U_i)$-module \mathcal{O}(U_i)$-module such that we have isomorphisms $\phi_{ij} : M_i \otimes_{O(U_i)} O(U_i otimes_{\mathcal{O}(U_i)} \mathcal{O}(U_i \cap U_j) \cong M_j \otimes_{O(U_j)} O(U_i otimes_{\mathcal{O}(U_j)} \mathcal{O}(U_i \cap U_j)$ of $O(U_i \mathcal{O}(U_i \cap U_j)$-modules, which satisfy the cocycle conditions, namely $\phi_{ii} = id$ and $\phi_{ijk} = \phi_{kji} \phi_{ikj}$ where $\phi_{ijk} = \phi_{ij} \otimes_{O(U_i otimes_{\mathcal{O}(U_i \cap U_j)} O(U_i \mathcal{O}(U_i \cap U_j \cap U_k)$. Then the quasi-coherent modules $\overline{M_i}$ \widetilde{M_i}$ on $U_i$ can be glued to a quasi-coherent module on $X$.

Long question:

In the general case, there is an affine cover $U_i \cap U_j = \cup_k W_k$, such that $W_k$ is basic-open in $U_i$ and $U_j$ simulteneously. In this setting we should use isomorphisms $\phi_{ijk} : M_i \otimes_{O(U_i)} O(W_k) otimes_{\mathcal{O}(U_i)} \mathcal{O}(W_k) \cong M_j \otimes_{O(U_j)} O(W_k)$. otimes_{\mathcal{O}(U_j)} \mathcal{O}(W_k)$. However, in order to formulate the cocycle condition in affine terms, we actually have to cover triple overlaps $U_i \cap U_j \cap U_k$ by affines $R_n$, such that $R_n$ is basic-open in $U_i,U_j,U_j$ and then cover the $U_i \cap U_j$ with these $R_n$, where we vary $k$. Thus we assume isomorphisms $\phi_{ijkn} : M_i \otimes_{O(U_i)} O(R_n) otimes_{\mathcal{O}(U_i)} \mathcal{O}(R_n) \cong M_j \otimes_{O(U_j)} O(R_n)$ otimes_{\mathcal{O}(U_j)} \mathcal{O}(R_n)$ and may formulate the cocycle condition. Now in order to glue these isomorphisms to an isomorphism on $U_i \cap U_j$ between $\tilde{M_i}$ \widetilde{M_i}$ and $\tilde{M_j}$, \widetilde{M_j}$, we also have to ensure that there is some intersection compatibility when we vary $k$. However, this cannot be formulated with the given data.

Before I end up defining refinements of these affine covers by transfinite induction ;-), I thought it would be better to ask here if there is a an easy way which I overlook.
show/hide this revision's text 1

Quasi-coherent module given by modules and compatibility conditions in the language of commutative algebra

Short question: Can we describe a quasi-coherent module on a scheme by usual modules with respect to an affine cover, which satisfy some compatibility conditions, which can be formulated in the language of commutative algebra? Thus, restrictions to non-affine subsets are not allwed. This sounds artificial, but it is (I think) necessary to work out Gabber's proof for this question http://mathoverflow.net/questions/39941/does-qcohx-admit-a-generating-set.

Example: If the scheme is separated, the answer is of course yes: Choose an affine cover $U_i$ of our scheme $X$, thus also $U_i \cap U_j$ is affine. If $M_i$ is a $O(U_i)$-module such that we have isomorphisms $\phi_{ij} : M_i \otimes_{O(U_i)} O(U_i \cap U_j) \cong M_j \otimes_{O(U_j)} O(U_i \cap U_j)$ of $O(U_i \cap U_j)$-modules, which satisfy the cocycle conditions, namely $\phi_{ii} = id$ and $\phi_{ijk} = \phi_{kji} \phi_{ikj}$ where $\phi_{ijk} = \phi_{ij} \otimes_{O(U_i \cap U_j)} O(U_i \cap U_j \cap U_k)$. Then the quasi-coherent modules $\overline{M_i}$ on $U_i$ can be glued to a quasi-coherent module on $X$.

Long question: In the general case, there is an affine cover $U_i \cap U_j = \cup_k W_k$, such that $W_k$ is basic-open in $U_i$ and $U_j$ simulteneously. In this setting we should use isomorphisms $\phi_{ijk} : M_i \otimes_{O(U_i)} O(W_k) \cong M_j \otimes_{O(U_j)} O(W_k)$. However, in order to formulate the cocycle condition in affine terms, we actually have to cover triple overlaps $U_i \cap U_j \cap U_k$ by affines $R_n$, such that $R_n$ is basic-open in $U_i,U_j,U_j$ and then cover the $U_i \cap U_j$ with these $R_n$, where we vary $k$. Thus we assume isomorphisms $\phi_{ijkn} : M_i \otimes_{O(U_i)} O(R_n) \cong M_j \otimes_{O(U_j)} O(R_n)$ and may formulate the cocycle condition. Now in order to glue these isomorphisms to an isomorphism on $U_i \cap U_j$ between $\tilde{M_i}$ and $\tilde{M_j}$, we also have to ensure that there is some intersection compatibility when we vary $k$. However, this cannot be formulated with the data.

Before I end up defining refinements of these affine covers by transfinite induction ;-), I thought it would be better to ask here if there is a easy way which I overlook.