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There are two statements about the grassmannian (of complex k-planes in n-space embedded via Plucker coordinates) that I have encountered in several places never accompanied with a proof or reference.

  • The topology inherited from projective space coincides with the Zariski topology.

  • The map from the Chow ring to the cohomology ring is an isomorphism.

I'm looking for nice explanations of these two facts.

EDIT: by the topology inherited from projective space I do not mean the Zariski topology, but the classical topology (given by balls in the Euclidean metric on the affine space from which we construct projective space, say). Since I have no intuition for the Zariski topology in this embedding, I'm not even sure I've got the statement correct: one place where it occurs is on page 147 in Fulton's Young tableaux book.

EDIT 2: I have gotten it wrong; Fulton claims only that the Zariski closures of the open cells equal their classical closures. But he also calls it a general fact. I guess the proper version of my first question should be: why are the Zariski closures of the open schubert cells equal to their classical closures in the Pluecker embedding, and is there a general fact from which this follows?

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The topology inherited from what topology on the projective space? – Mariano Suárez-Alvarez Apr 3 '14 at 18:43
The classical topology on projective space does not induce the Zariski topology on the Grassmannian; it induces the classical topology on the Grassmannian. Many topological statements about the Grassmannian are true for both choices of topology because in practice, most of arguments involving open sets in the classical topology can be made using only open sets from the Zariski topology. – Michael Joyce Apr 3 '14 at 19:41
I updated my answer. – Dan Petersen Apr 3 '14 at 19:50
up vote 4 down vote accepted

$\!$Hej Erik! Your first question asks why an open Schubert cell has the same closure both in the Zariski and in the classical topology. This is the same as asking why the closure of a cell is a closed subvariety. But each cell is defined by a collection of polynomial equations ($f = 0$) and inequations ($f \neq 0$), and it's easy to check that the closure in the classical topology is obtained by just discarding the inequations. Then it's clear that the closure is a subvariety.

For your second question, this follows because Grassmannians admit an algebraic cell decomposition (i.e. they are the disjoint union of locally closed subvarieties isomorphic to some $\mathbf C^d$), the Schubert cell decomposition. See this question: For which varieties is the natural map from the Chow ring to integral cohomology an isomorphism? and the reference to Fulton's book in Donu Arapura's answer.

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Probably the OP is asking why the Pluecker map is a closed immersion, in particular, why it is a homeomorphism from the domain to the image. – Jason Starr Apr 3 '14 at 19:09

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