Return to Answer

replaced http://mathoverflow.net/ with https://mathoverflow.net/

Source Link

edited Apr 13, 2017 at 12:58

1
2
3

Have a look at Serre's definition of a sheaf [FAC], namely via étale spaces. This gives a geometric picture of a sheaf. Over every point of our topological space $X$, there sits the fiber of the sheaf. Of course we make some compatibility conditions on these fibers, namely that they vary continuously. Topologists call this a bundle on $X$. Now if $X$ has some extra structure, it is reasonable to study the bundles with some appropriate extra structure. Namely, if $X$ is a ringed space, then the fibers should be modules. The corresponding sheafes are called module sheaves. If $X$ is a scheme, then we restrict to quasi-coherent sheaves in order to involve the local affine charts. In every case, you get a special type of a bundle over $X$.

In the same way as the structure of a ring $A$ may be studied by means of the modules over $A$, the structure of a scheme $X$ may be studied by means of quasi-coherent sheaves on $X$. Actually the Reconstruction Theorem Reconstruction Theorem by Rosenberg justifies this. Even in the affine case this helps to enlighten some concepts of module theory. An example is the support of a module.

Have a look at Serre's definition of a sheaf [FAC], namely via étale spaces. This gives a geometric picture of a sheaf. Over every point of our topological space $X$, there sits the fiber of the sheaf. Of course we make some compatibility conditions on these fibers, namely that they vary continuously. Topologists call this a bundle on $X$. Now if $X$ has some extra structure, it is reasonable to study the bundles with some appropriate extra structure. Namely, if $X$ is a ringed space, then the fibers should be modules. The corresponding sheafes are called module sheaves. If $X$ is a scheme, then we restrict to quasi-coherent sheaves in order to involve the local affine charts. In every case, you get a special type of a bundle over $X$.

In the same way as the structure of a ring $A$ may be studied by means of the modules over $A$, the structure of a scheme $X$ may be studied by means of quasi-coherent sheaves on $X$. Actually the Reconstruction Theorem by Rosenberg justifies this. Even in the affine case this helps to enlighten some concepts of module theory. An example is the support of a module.

Have a look at Serre's definition of a sheaf [FAC], namely via étale spaces. This gives a geometric picture of a sheaf. Over every point of our topological space $X$, there sits the fiber of the sheaf. Of course we make some compatibility conditions on these fibers, namely that they vary continuously. Topologists call this a bundle on $X$. Now if $X$ has some extra structure, it is reasonable to study the bundles with some appropriate extra structure. Namely, if $X$ is a ringed space, then the fibers should be modules. The corresponding sheafes are called module sheaves. If $X$ is a scheme, then we restrict to quasi-coherent sheaves in order to involve the local affine charts. In every case, you get a special type of a bundle over $X$.

In the same way as the structure of a ring $A$ may be studied by means of the modules over $A$, the structure of a scheme $X$ may be studied by means of quasi-coherent sheaves on $X$. Actually the Reconstruction Theorem by Rosenberg justifies this. Even in the affine case this helps to enlighten some concepts of module theory. An example is the support of a module.

Source Link

created Apr 14, 2011 at 12:50

Martin Brandenburg

63.1k
13
207
424

Have a look at Serre's definition of a sheaf [FAC], namely via étale spaces. This gives a geometric picture of a sheaf. Over every point of our topological space $X$, there sits the fiber of the sheaf. Of course we make some compatibility conditions on these fibers, namely that they vary continuously. Topologists call this a bundle on $X$. Now if $X$ has some extra structure, it is reasonable to study the bundles with some appropriate extra structure. Namely, if $X$ is a ringed space, then the fibers should be modules. The corresponding sheafes are called module sheaves. If $X$ is a scheme, then we restrict to quasi-coherent sheaves in order to involve the local affine charts. In every case, you get a special type of a bundle over $X$.

In the same way as the structure of a ring $A$ may be studied by means of the modules over $A$, the structure of a scheme $X$ may be studied by means of quasi-coherent sheaves on $X$. Actually the Reconstruction Theorem by Rosenberg justifies this. Even in the affine case this helps to enlighten some concepts of module theory. An example is the support of a module.