Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

I have to define a sheaf over $X$ knowing what happens in each open for a basis of the topology of $X$. That is, I have $\mathcal{B} = \{U_i\}$ a basis of $X$, for each $U_i$ I have an $k$-algebra $K_i$ and $\rho_{U_i}^{U_j}: K_j \to K_i$ morphisms such that $\rho_{U_i}^{U_i} = id_{K_i}$ and $\rho_{U_i}^{U_j} \circ \rho_{U_j}^{U_k} = \rho_{U_i}^{U_k}$ for all $U_i \subseteq U_j \subseteq U_k \subseteq X$ and I want a sheaf over $X$ to match with the open sets of $\mathcal{B}$.

Thank you for your help.

share|improve this question
add comment

1 Answer 1

up vote 2 down vote accepted

Here is the link to notes by Ravi Vakil: http://math.stanford.edu/~vakil/216blog/. Look at chapter 3, section 7. It is best to try and do the exercises in this section once. He also explains how to define a sheaf on X when you are given a sheaf on a base of X. If you know why the sheaf of compatible germs is a sheaf (i.e. why sheafification of a presheaf gives you a sheaf), then it should be clear why a sheaf on a base extends to a sheaf on the whole space according to Ravi's definition. Hence, you might also want to look at $3.4$ of the notes. I am also looking at the October 23rd version of the notes, and I don't know if the numbers are different in earlier versions.

share|improve this answer
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.