Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

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$\mathfrak{F}=0$ .

share|improve this question
1  
Context? Motivation? Some comments to illustrate which cases you can already do, what you know, etc? –  Yemon Choi May 8 '11 at 1:47
add comment

1 Answer

up vote 10 down vote accepted

$\dim\mathrm{Supp}\\, \mathfrak F=0$ implies that $\mathfrak F\otimes \mathscr O_X(L)\simeq \mathfrak F$ and hence its cohomology is independent of $L$.

share|improve this answer
    
how to get this isomorphism ? –  HKSHLZW May 8 '11 at 9:06
    
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. –  Martin Brandenburg May 8 '11 at 9:26
    
Thank you very much ! –  HKSHLZW May 8 '11 at 9:31
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.