Menu
Home
People
Places
Arts
History
Plants & Animals
Science
Life & Culture
Technology
Reference.org
Direct image functor
open-in-new
<p>In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, the direct image functor is a construction in <a href="/facts/Sheaf_theory/MM3hcRw4">sheaf theory</a> that generalizes the <a href="/facts/Section_of_a_sheaf/MM3hcRw4">global sections functor</a> to the relative case. It is of fundamental importance in <a href="/facts/Topology/2EqW47cU">topology</a> and <a href="/facts/Algebraic_geometry/rh7RHVp3">algebraic geometry</a>. Given a sheaf <i>F</i> defined on a <a href="/facts/Topological_space/v2ut7CbK">topological space</a> <i>X</i> and a <a href="/facts/Continuous_map/sKbl02pB">continuous map</a> <i>f</i>: <i>X</i> → <i>Y</i>, we can define a new sheaf <i>f</i>∗<i>F</i> on <i>Y</i>, called the direct image sheaf or the pushforward sheaf of <i>F</i> along <i>f</i>, such that the global sections of <i>f</i>∗<i>F</i> is given by the global sections of <i>F</i>. This assignment gives rise to a <a href="/facts/Functor/l2u3rIBE">functor</a> <i>f</i>∗ from the <a href="/facts/Category_(mathematics)/7WzxUThf">category</a> of sheaves on <i>X</i> to the category of sheaves on <i>Y</i>, which is known as the direct image functor. Similar constructions exist in many other algebraic and geometric contexts, including that of <a href="/facts/Quasi-coherent_sheaves/h2NoUN7e">quasi-coherent sheaves</a> and <a href="/facts/%C3%89tale_sheaf/U67cR3JU">étale sheaves</a> on a <a href="/facts/Scheme_(mathematics)/c7uSqGqL">scheme</a>. </p>