Tensor product of coherent modules - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-19T11:21:18Z http://mathoverflow.net/feeds/question/86107 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/86107/tensor-product-of-coherent-modules Tensor product of coherent modules Martin Brandenburg 2012-01-19T15:21:42Z 2012-01-19T15:59:14Z <p>Let $X$ be a ringed space. Recall that an $\mathcal{O}_X$-module $M$ is called <em>coherent</em> if it is of finite presentation and for every open $U \subseteq X$ every $\mathcal{O}_U$-submodule of finite type of $M|_U$ is of finite presentation. Coherent modules constitute an abelian category (in contrast to modules of finite presentation or just of finite type). See the Stacks project, <a href="http://www.math.columbia.edu/algebraic_geometry/stacks-git/modules.pdf" rel="nofollow">modules</a>, section 12. In general $\mathcal{O}_X$ might be not coherent.</p> <p><strong>Question.</strong> If $M,N$ are coherent $O_X$-modules, is it true that $M \otimes_{\mathcal{O}_X} N$ is coherent?</p> <p>I guess that this will be false in this generality. So let us restrict to schemes, w.l.o.g. affine schemes. Here a module $M$ over a ring $A$ is coherent if it is of finite presentation and every submodule of finite type is of finite presentation.</p> http://mathoverflow.net/questions/86107/tensor-product-of-coherent-modules/86108#86108 Answer by Francesco Polizzi for Tensor product of coherent modules Francesco Polizzi 2012-01-19T15:59:14Z 2012-01-19T15:59:14Z <p>In the analytic category this is indeed true: if $\mathscr{F}$, $\mathscr{G}$ are coherent analytic sheaves on a complex space $X$, then $\mathscr{F} \otimes_{\mathscr{O}_X} \mathscr{G}$ is also coherent.</p> <p>For a reference, look at [Grauert-Remmert, Coherent Analytic Sheaves], Proposition at the bottom of page 240.</p> <p>It seems to me that their proof works also in the algebraic category, maybe you should check.</p>