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Regular embedding
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<p>In <a href="/facts/Algebraic_geometry/rh7RHVp3">algebraic geometry</a>, a <a href="/facts/Closed_immersion/g60V5BK4">closed immersion</a> i : X ↪ Y {\displaystyle i:X\hookrightarrow Y} of schemes is a regular embedding of codimension <i>r</i> if each point <i>x</i> in <i>X</i> has an open affine neighborhood <i>U</i> in <i>Y</i> such that the ideal of X ∩ U {\displaystyle X\cap U} is generated by a <a href="/facts/Regular_sequence/Nxr2uwh7">regular sequence</a> of length <i>r</i>. A regular embedding of codimension one is precisely an <a href="/facts/Effective_Cartier_divisor/V0QDypeo">effective Cartier divisor</a>. </p>