flatness of coherent analytic sheaf - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-18T19:43:17Z http://mathoverflow.net/feeds/question/65323 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/65323/flatness-of-coherent-analytic-sheaf flatness of coherent analytic sheaf HKSHLZW 2011-05-18T12:49:30Z 2011-05-18T13:07:21Z <p>I meet a problem like this : given a short exact sequence $0\rightarrow E_1\rightarrow E_2\rightarrow E_3\rightarrow 0$ , where $E_i,i=1,2,3$ are coherent sheaves over a compact complex manifold $X$ . Let $L$ be a holomorphic line bundle over $X$ , $\mathcal{O}_X(L)$ be the associated coherent analytic sheaf , can we get $0\rightarrow E_1\otimes\mathbb{O}_X(L) \rightarrow E_2\otimes\mathcal{O}_X(L) \rightarrow E_3\otimes\mathcal{O}_X(L) \rightarrow 0$ ? THen furthermore for any other coherent analytic sheaf $S$ , can we get $0\rightarrow E_1\otimes S \rightarrow E_2\otimes S \rightarrow E_3\otimes S \rightarrow 0$ ?</p> http://mathoverflow.net/questions/65323/flatness-of-coherent-analytic-sheaf/65326#65326 Answer by J.C. Ottem for flatness of coherent analytic sheaf J.C. Ottem 2011-05-18T13:01:39Z 2011-05-18T13:07:21Z <p>Yes, the sequence $0\rightarrow E_1\otimes\mathcal{O}_X(L) \rightarrow E_2\otimes\mathcal{O}_X(L) \rightarrow E_3\otimes\mathcal{O}_X(L) \rightarrow 0\,\,$ is certainly exact since $L$ is locally free (hence flat).</p> <p>For the second question, the answer is negative in general. Take $0 \to I_Y \to O_X \to O_Y \to 0$ and $S=O_Y$, where $X=\mathbb{A}^1$ and $Y=pt$.</p>