Direct limit of schemes - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-19T20:15:02Z http://mathoverflow.net/feeds/question/48604 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/48604/direct-limit-of-schemes Direct limit of schemes QcH 2010-12-08T01:25:31Z 2010-12-08T04:17:11Z <p>Let $A$ be a ring (say finitely generated algebra over an algebraically closed field). Then, does <code>$\varinjlim \mathrm{Spec} A[t]/(t^m)$</code> exist (in the category of schemes)? And if it does, then is it equal to <code>$\mathrm{Spec} A[[t]]$</code>?</p> <p>Edit based on comment below: The result holds easily when $A=k$ is a field. Then, <code>$\mathrm{Spec} k[t]/(t^n)$</code> only has one point and hence, for any scheme $X$ such that we have a family of morphisms <code>$\mathrm{Spec} k[t]/(t^n) \to X$</code>, we can assume that $X$ is affine and the result follows from the equivalence of the category $Ring^{op}$ and the category of affine schemes.</p>