What's the definition of "geometrically injective"? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T20:10:01Z http://mathoverflow.net/feeds/question/78143 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/78143/whats-the-definition-of-geometrically-injective What's the definition of "geometrically injective"? google 2011-10-14T16:13:07Z 2011-10-17T05:13:35Z <p>I don't know the meaning of geometrically injective morphism f of schemes. </p> <p>What's the definition of "geometrically injective"?</p> <p>I can't find it. I hope your answer.</p> <p>Thanks.</p> http://mathoverflow.net/questions/78143/whats-the-definition-of-geometrically-injective/78144#78144 Answer by Leo Alonso for What's the definition of "geometrically injective"? Leo Alonso 2011-10-14T16:25:22Z 2011-10-14T16:25:22Z <p>A map of schemes $f \colon X \to Y$ is <em>geometrically injective</em> if it is injective on <em>geometric points</em>, i.e. points with values in an algebraic closed field. In more detail, let $K$ be an algebraically closed field. For all pairs of maps ($K$-valued points) $x, y \colon \operatorname{Spec}(K) \to X$ such that they have the same image on $Y$, i.e $f \circ x = f \circ y$ then $x = y$. </p> <p>In other words the map $$\operatorname{Hom}(\operatorname{Spec}(K), X) \longrightarrow \operatorname{Hom}(\operatorname{Spec}(K), Y)$$ given by composition with $f$, is injective for every algebraically closed field $K$.</p> http://mathoverflow.net/questions/78143/whats-the-definition-of-geometrically-injective/78310#78310 Answer by Sanjay for What's the definition of "geometrically injective"? Sanjay 2011-10-17T05:13:35Z 2011-10-17T05:13:35Z <p>I don't find link to add comment. You can find the various equivalent condition for radicial morphism and its proof in "Altman &amp; Kleiman, Introduction to Grothendieck Duality Theory" on page 119.</p>