Let $X$ be a compact complex n-fold . Then for every coherent sheaf $\mathfrak{F}$ on $X$ , and every holomorphic line bundle $L$ on $X$ , then the dimension of $H^0 (X,\mathfrak{F}\otimes\mathcal{O}_X(L))$ does not depend on $L$ when dim Supp$\mathfrak{F}=0$ .
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1$\begingroup$ Context? Motivation? Some comments to illustrate which cases you can already do, what you know, etc? $\endgroup$– Yemon ChoiCommented May 8, 2011 at 1:47
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1 Answer
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$\dim\mathrm{Supp}\\, \mathfrak F=0$ implies that $\mathfrak F\otimes \mathscr O_X(L)\simeq \mathfrak F$ and hence its cohomology is independent of $L$.
answered May 8, 2011 at 2:49
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$\begingroup$ The support of $\mathfrak{F}$ is just a finite set of closed points. Now choose trivializations of $L$ around each point to get the desired isomorphism. $\endgroup$ Commented May 8, 2011 at 9:26