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I have been reading etale cohomology. The book says that it is algebraic analogue of singular cohomology.

My question is that why can’t we compute the singular cohomology of schemes/varieties over the Zariski topology?

JS Milne states many reasons for the inadequacy of Zariski topology in computing sheaf cohomology. But I want to know the short comings of the Zariski topology in calculating singular cohomology (as in algebraic topology).

KReiser
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    You can compute singular cohomology of things endowed with the Zariski topology, but you don't usually get useful answers. For instance, every irreducible scheme is contractible. This is pretty silly. – KReiser Dec 15 '21 at 08:37
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    Say your algebraic variety is actually a complex manifold. We would expect the singular cohomology of the variety to be closely related to its singular cohomology as as a complex manifold. The Zariski topolgy is too weak to be connected to the Euclidean topology. – Just a user Dec 15 '21 at 08:55
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    Another silly issue is that all curves over a given field are homeomorphic in the Zariski topology, which indicates that it does not see something as fundamental as the genus. Although as of a few years ago I should note that this particular issue is a low-dimensional phenomenon. Recent results of Kollár-Lieblich-Olsson-Sawin (building on earlier work by Lieblich-Olsson) indicate that for dimensions $4$ and higher, the Zariski topological space of a smooth scheme is actually a fairly strong invariant (in fact complete when paired with certain codimension $1$ data). – Tabes Bridges Dec 15 '21 at 09:04
  • Here is the paper: On the de Rham cohomology of algebraic varieties Grothendieck, Alexander Publications Mathématiques de l'IHÉS, Tome 29 (1966), pp. 95-103.

    http://www.numdam.org/item/PMIHES_1966__29__95_0.pdf

     – hm2020 Dec 15 '21 at 11:01
  • @ABC- in the above linked paper it is proved that the algebraic deRham complex of a non-singular affine algebraic variety $X:=Spec(A)$ over the complex numbers calculates the singular cohomology groups with complex coefficients. – hm2020 Dec 15 '21 at 11:46

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