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A [[Sheaf|sheaf]] of modules that is locally isomorphic to the direct sum of several copies of the structure sheaf. More precisely, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060450/l0604501.png" /> be a [[Ringed space|ringed space]]. A sheaf of modules <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060450/l0604502.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060450/l0604503.png" /> is said to be locally free if for every point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060450/l0604504.png" /> there is an open neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060450/l0604505.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060450/l0604506.png" />, such that the restriction <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060450/l0604507.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060450/l0604508.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060450/l0604509.png" /> is a free sheaf of modules over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060450/l06045010.png" />, that is, it is isomorphic to the direct sum of a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060450/l06045011.png" /> of copies of the structure sheaf <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060450/l06045012.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060450/l06045013.png" /> is connected and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060450/l06045014.png" /> is finite, for example consisting of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060450/l06045015.png" /> elements, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060450/l06045016.png" /> does not depend on the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060450/l06045017.png" /> and is called the rank of the locally free sheaf <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060450/l06045018.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060450/l06045019.png" /> be a vector bundle of rank <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060450/l06045020.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060450/l06045021.png" /> and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060450/l06045022.png" /> be the sheaf of germs of its sections. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060450/l06045023.png" /> is a locally free sheaf of rank <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060450/l06045024.png" />. Conversely, for every locally free sheaf <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060450/l06045025.png" /> of rank <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060450/l06045026.png" /> there is a vector bundle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060450/l06045027.png" /> of rank <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060450/l06045028.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060450/l06045029.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060450/l06045030.png" /> is the sheaf of germs of its sections (see [[#References|[1]]], [[#References|[2]]]); hence there is a natural one-to-one correspondence between the isomorphy classes of locally free sheaves of rank <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060450/l06045031.png" /> and the isomorphy classes of vector bundles of rank <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060450/l06045032.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060450/l06045033.png" />.
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Example. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060450/l06045034.png" /> be a smooth connected algebraic variety of dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060450/l06045035.png" />. Then the sheaf of regular differential forms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060450/l06045036.png" /> is a locally free sheaf of rank <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060450/l06045037.png" />.
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Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060450/l06045038.png" />, a connected [[Affine scheme|affine scheme]], be the spectrum of the commutative ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060450/l06045039.png" /> (cf. [[Spectrum of a ring|Spectrum of a ring]]), let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060450/l06045040.png" /> be a locally free sheaf of rank <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060450/l06045041.png" /> and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060450/l06045042.png" /> be the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060450/l06045043.png" />-module of its global sections. Then the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060450/l06045044.png" />-module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060450/l06045045.png" /> is projective and the mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060450/l06045046.png" /> establishes a one-to-one correspondence between the set of classes (up to isomorphisms) of locally free sheaves of rank <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060450/l06045047.png" /> and the set of classes (up to isomorphisms) of projective <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060450/l06045048.png" />-modules of rank <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060450/l06045049.png" /> (see [[#References|[2]]]).
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A [[Sheaf|sheaf]] of modules that is locally isomorphic to the direct sum of several copies of the structure sheaf. More precisely, let  $  ( X , {\mathcal O} _ {X} ) $
 +
be a [[Ringed space|ringed space]]. A sheaf of modules  $  {\mathcal F} $
 +
over  $  {\mathcal O} _ {X} $
 +
is said to be locally free if for every point  $  x \in X $
 +
there is an open neighbourhood  $  U \subset  X $,
 +
$  x \in U $,
 +
such that the restriction  $  {\mathcal F} \mid  _ {U} $
 +
of  $  {\mathcal F} $
 +
to  $  U $
 +
is a free sheaf of modules over  $  {\mathcal O} _ {X }  \mid  _ {U }  $,
 +
that is, it is isomorphic to the direct sum of a set  $  I ( x) $
 +
of copies of the structure sheaf  $  {\mathcal O} _ {X} \mid  _ {U }  $.
 +
If  $  X $
 +
is connected and  $  I ( x) $
 +
is finite, for example consisting of  $  n $
 +
elements, then  $  n $
 +
does not depend on the point  $  x $
 +
and is called the rank of the locally free sheaf  $  {\mathcal F} $.
 +
Let $  V $
 +
be a vector bundle of rank  $  n $
 +
on  $  X $
 +
and let  $  {\mathcal F} $
 +
be the sheaf of germs of its sections. Then  $  {\mathcal F} $
 +
is a locally free sheaf of rank  $  n $.
 +
Conversely, for every locally free sheaf  $  {\mathcal F} $
 +
of rank  $  n $
 +
there is a vector bundle  $  V $
 +
of rank  $  n $
 +
on  $  X $
 +
such that  $  {\mathcal F} $
 +
is the sheaf of germs of its sections (see [[#References|[1]]], [[#References|[2]]]); hence there is a natural one-to-one correspondence between the isomorphy classes of locally free sheaves of rank  $  n $
 +
and the isomorphy classes of vector bundles of rank  $  n $
 +
on  $  X $.
 +
 
 +
Example. Let  $  X $
 +
be a smooth connected algebraic variety of dimension  $  n $.  
 +
Then the sheaf of regular differential forms  $  \Omega _ {X}  ^ {1} $
 +
is a locally free sheaf of rank  $  n $.
 +
 
 +
Let  $  X =  \mathop{\rm Spec}  A $,  
 +
a connected [[Affine scheme|affine scheme]], be the spectrum of the commutative ring $  A $(
 +
cf. [[Spectrum of a ring|Spectrum of a ring]]), let $  {\mathcal F} $
 +
be a locally free sheaf of rank $  n $
 +
and let $  M = \Gamma ( X , {\mathcal F} ) $
 +
be the $  A $-
 +
module of its global sections. Then the $  A $-
 +
module $  M $
 +
is projective and the mapping $  {\mathcal F} \mapsto \Gamma ( X , {\mathcal F} ) $
 +
establishes a one-to-one correspondence between the set of classes (up to isomorphisms) of locally free sheaves of rank $  n $
 +
and the set of classes (up to isomorphisms) of projective $  A $-
 +
modules of rank $  n $(
 +
see [[#References|[2]]]).
  
 
====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> R. Godement,   "Topologie algébrique et théorie des faisceaux" , Hermann (1958)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> R. Hartshorne,   "Algebraic geometry" , Springer (1977)</TD></TR></table>
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<table><TR><TD valign="top">[1]</TD> <TD valign="top"> R. Godement, "Topologie algébrique et théorie des faisceaux" , Hermann (1958) {{MR|0102797}} {{ZBL|0080.16201}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> R. Hartshorne, "Algebraic geometry" , Springer (1977) {{MR|0463157}} {{ZBL|0367.14001}} </TD></TR></table>

Latest revision as of 22:17, 5 June 2020


A sheaf of modules that is locally isomorphic to the direct sum of several copies of the structure sheaf. More precisely, let $ ( X , {\mathcal O} _ {X} ) $ be a ringed space. A sheaf of modules $ {\mathcal F} $ over $ {\mathcal O} _ {X} $ is said to be locally free if for every point $ x \in X $ there is an open neighbourhood $ U \subset X $, $ x \in U $, such that the restriction $ {\mathcal F} \mid _ {U} $ of $ {\mathcal F} $ to $ U $ is a free sheaf of modules over $ {\mathcal O} _ {X } \mid _ {U } $, that is, it is isomorphic to the direct sum of a set $ I ( x) $ of copies of the structure sheaf $ {\mathcal O} _ {X} \mid _ {U } $. If $ X $ is connected and $ I ( x) $ is finite, for example consisting of $ n $ elements, then $ n $ does not depend on the point $ x $ and is called the rank of the locally free sheaf $ {\mathcal F} $. Let $ V $ be a vector bundle of rank $ n $ on $ X $ and let $ {\mathcal F} $ be the sheaf of germs of its sections. Then $ {\mathcal F} $ is a locally free sheaf of rank $ n $. Conversely, for every locally free sheaf $ {\mathcal F} $ of rank $ n $ there is a vector bundle $ V $ of rank $ n $ on $ X $ such that $ {\mathcal F} $ is the sheaf of germs of its sections (see [1], [2]); hence there is a natural one-to-one correspondence between the isomorphy classes of locally free sheaves of rank $ n $ and the isomorphy classes of vector bundles of rank $ n $ on $ X $.

Example. Let $ X $ be a smooth connected algebraic variety of dimension $ n $. Then the sheaf of regular differential forms $ \Omega _ {X} ^ {1} $ is a locally free sheaf of rank $ n $.

Let $ X = \mathop{\rm Spec} A $, a connected affine scheme, be the spectrum of the commutative ring $ A $( cf. Spectrum of a ring), let $ {\mathcal F} $ be a locally free sheaf of rank $ n $ and let $ M = \Gamma ( X , {\mathcal F} ) $ be the $ A $- module of its global sections. Then the $ A $- module $ M $ is projective and the mapping $ {\mathcal F} \mapsto \Gamma ( X , {\mathcal F} ) $ establishes a one-to-one correspondence between the set of classes (up to isomorphisms) of locally free sheaves of rank $ n $ and the set of classes (up to isomorphisms) of projective $ A $- modules of rank $ n $( see [2]).

References

[1] R. Godement, "Topologie algébrique et théorie des faisceaux" , Hermann (1958) MR0102797 Zbl 0080.16201
[2] R. Hartshorne, "Algebraic geometry" , Springer (1977) MR0463157 Zbl 0367.14001
How to Cite This Entry:
Locally free sheaf. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Locally_free_sheaf&oldid=15760
This article was adapted from an original article by V.A. Iskovskikh (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article