Comparison between singular and etale cohomology in Batyrev's paper on Birational Calabi-Yau varieties - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-21T15:56:32Z http://mathoverflow.net/feeds/question/90837 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/90837/comparison-between-singular-and-etale-cohomology-in-batyrevs-paper-on-birational Comparison between singular and etale cohomology in Batyrev's paper on Birational Calabi-Yau varieties Tom Lovering 2012-03-10T17:59:38Z 2012-03-10T18:05:12Z <p>My question refers to the paper <a href="http://arxiv.org/pdf/alg-geom/9710020.pdf" rel="nofollow">http://arxiv.org/pdf/alg-geom/9710020.pdf</a> where Batyrev proves that birational Calabi-Yau algebraic varieties have equal Betti numbers by counting points over finite fields using p-adic integration and so computes the Betti numbers using the Weil conjectures.</p> <p>It seems that he is doing the following. Given a variety $X$ over $\mathbb{C}$, we can actually write it (and all the associated data we care about) as a variety $\mathcal{X}$ over $\mathcal{R}$, some finite-type $\mathbb{Z}$-algebra: i.e. such that $\mathcal{X} \otimes_\mathcal{R} \mathbb{C} = X$. We then fix an approriate maximal ideal $J(\pi)$ of $\mathcal{R}$ which lies above $p \in \mathbb{Z}$. I think we then turn our attention to the variety $\mathcal{X}\otimes_\mathcal{R} (\mathcal{R}/J(\pi))$, and using a ring of integers $R$ of a local number field with this as special fibre, we can count the number of points this variety has over every finite field extension of $\mathbb{F}_q = \mathcal{R}/J(\pi)$ using p-adic integration.</p> <p>So the Weil conjectures give us the Betti numbers of this variety, and by proper smooth base change these Betti numbers are the same as those of $\mathcal{X} \otimes_\mathcal{R} \mathbb{C}$, but where the map $\mathcal{R} \rightarrow \mathbb{C}$ is not the natural inclusion but rather $\mathcal{R} \rightarrow R \hookrightarrow \mathbb{C}$. Since he is trying to compute the cohomology of the former, this doesn't make sense to me.</p> <p>Can anyone see if I'm making a mistake somewhere? (or how my issue can be resolved?)</p> <p>Thanks, Tom.</p>