Locally free sheaf

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In sheaf theory, a field of mathematics, a sheaf of ${\mathcal {O}}_{X}$ -modules ${\mathcal {F}}$ on a ringed space $X$ is called locally free if for each point $p\in X$ , there is an open neighborhood $U$ of $p$ such that ${\mathcal {F}}|_{U}$ is free as an ${\mathcal {O}}_{X}|_{U}$ -module. This implies that ${\mathcal {F}}_{p}$ , the stalk of ${\mathcal {F}}$ at $p$ , is free as a $({\mathcal {O}}_{X})_{p}$ -module for all $p$ . The converse is true if ${\mathcal {F}}$ is moreover coherent. If ${\mathcal {F}}_{p}$ is of finite rank $n$ for every $p\in X$ , then ${\mathcal {F}}$ is said to be of rank $n.$

References

Sections 0.5.3 and 0.5.4 of Grothendieck, Alexandre; Dieudonné, Jean (1960). "Éléments de géométrie algébrique: I. Le langage des schémas". Publications Mathématiques de l'IHÉS 4. MR 0217083.

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Locally free sheaf

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References

External links