What are the relationship between various definitions for quasi coherent sheaves? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-26T02:08:49Z http://mathoverflow.net/feeds/question/15223 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/15223/what-are-the-relationship-between-various-definitions-for-quasi-coherent-sheaves What are the relationship between various definitions for quasi coherent sheaves? Shizhuo Zhang 2010-02-14T01:54:29Z 2010-02-15T04:19:36Z <p>It seems that there are many definitions of quasi coherent sheaves(modules). There is a nice page on nLab <a href="http://ncatlab.org/nlab/show/quasicoherent+sheaf#quasicoherent%5Fsheaves%5Fover%5Faffine%5Fschemes%5F12" rel="nofollow">quasi coherent sheaves</a></p> <p>My questions are:</p> <ol> <li><p>Are there any other definitions of quasi coherent sheaves? For what?(I mean in which situation, they should be redefined to satisfy the working environment)</p></li> <li><p>How are they related?</p></li> </ol> <p>For question 2, nLab says these definitions are equivalent. There are four definitions in nLab. It seems except first definitions, the other three are equivalent, just restate in different language: i.e presheaves &lt;---> pseudo functors &lt;---> fibered categories. </p> <p>But are these definitions <strong>exactly the same as the classical definitions?</strong> (as locally presentable modules)</p>