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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.

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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check out Kontsevich-Rosenberg preprint "noncommutative stack" which gives the answer of this question. Of course, it can be generalized to description of quasi coherent modules on locally affine noncommutative space, for example, noncommutative projective space –  Shizhuo Zhang Dec 13 '10 at 6:11
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1 Answer 1

up vote 3 down vote accepted

I think the explicit description that you suggest can be wrapped up as follows.

For every $i,j$, let $C_{ij}$ be a family of indexes such that $$U_i\cap U_j=\bigcup_{a\in C_{ij}} W_a,$$ with $W_a$ being basic open in $U_i$ and in $U_j$. We may assume that $C_{ii}$ is a singleton set and $C_{ij}=C_{ji}$. Let $\phi_{ij,a}$ be as in your question, $a\in C_{ij}$. Impose the cocycle condition $$\phi_{ij,a}\circ\phi_{jk,b}=\phi_{ik,c}$$ on $W_a\cap W_b\cap W_c$ for all $a\in C_{ij}$, $b\in C_{jk}$, $c\in C_{ik}$. Notice that $W_a\cap W_b\cap W_c$ is a basic open in $W_a$, $W_b$, $W_c$, and also in $U_i$, $U_j$, $U_k$, so this is an identity in algebraic terms, as required.

A more natural description:

A quasicoherent sheaf on $X$ is given by a $\Gamma(U,O)$-module $M_U$ for every affine open $U\subset X$ and an isomorphism $\phi_{U,V}:M_U|_V\to M_V$ whenever $V\subset U$ is a basic open, such that $\phi_{U,V}$ satisfies obvious cocycle condition on triples $W\subset V\subset U$.

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Just a remark: this is discussed (without proving the equivalence with quasi-coherent sheaves) in Enochs and Estrada, "Relative homological algebra in the category of quasi-coherent sheaves," Adv. in Math. 194. –  Thomas Nevins Sep 29 '10 at 17:42
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