Let $X$ be a topological space and $$\varepsilon \ :\ 0 \to A \to B \to C \to 0$$ be an exact sequence of sheaves of ${\cal O}_X$-modules. $\varepsilon$ is said to be pure if for each point $x\in X$, the induced sequence $$0\to A_x \to B_x \to C_x \to 0$$ is pure as a sequence of ${\cal O}_{X,x}$-modules.

Is it true to say that any split exact sequence of ${\cal O}_X$-modules is pure?

  • 4
    $\begingroup$ Isn't it obvious from the definitions? $\endgroup$ May 3, 2013 at 10:17
  • $\begingroup$ Tensor products commute with direct sums, and if the sequence is split, $B \cong A \oplus C$. $\endgroup$ May 3, 2013 at 15:24
  • $\begingroup$ A short exact sequence of modules is pure iff it is a filtered colimit of split ones. $\endgroup$ May 3, 2013 at 17:29


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