Let X be an affine variety over a field k of characteristic 0. Then we have the algebraic de Rham complex Omega* of X over k. The algebraic de Rham cohomology H^*_{dR}(X\/k) of X over k is then the cohomology of the complex of global sections of Omega* (here we are using that X is affine). So if X is the spectrum of the finite type k-algebra A (which is a domain as X is a variety), then we’re looking at the cohomology of the de Rham complex of A over k.<\/p>\n
If X is smooth over k, then Grothendieck proved that each H^i_{dR}(X\/k) has finite dimension. The point of this blog post is to clearly state that this doesn’t hold for all singular affine varieties X over k. I could not find a paper literally stating this fact with an explicit example, so I decided to find one myself and present it to you. If someone emails me a reference I would be thankful and would add that here.<\/p>\n
A slightly different issue is that if k = C is the complex numbers and X is smooth, then H^i_{dR}(X\/C) computes the cohomology of the manifold X(C) with C-coefficients and this is finite dimensional. Googling one easily finds examples where this is false for singular X. Indeed from the work in the references given below it becomes clear that there are many (explicit) examples.<\/p>\n
Our example.<\/strong> Let k be any characteristic 0 field, let n > 6 be an integer, and let A be the k-algebra The idea for this came from an example by Br\u00fcske mentioned at the very end of Bloom and Herrera, De Rham cohomology of an analytic space<\/em>, Invent. Math. 7 (1969), 275\u2013296. The purported example is of a singular analytic space where a stalk of the de Rham cohomology sheaf in the analytic topology is infinite dimensional; I didn’t check the example.<\/p>\n In Reiffen Das Lemma von Poincar\u00e9 f\u00fcr holomorphe Differential-formen auf komplexen R\u00e4umen<\/em>, Math. Z. 101 (1967), 269\u2013284 we find a method to calculate a quotient (!) of the degree 3 de Rham cohomology of X = Spec(A). This quotient itself is the quotient of the k-vector space (f) = fA by the elements bf in (f) of the form Let X be an affine variety over a field k of characteristic 0. Then we have the algebraic de Rham complex Omega* of X over k. The algebraic de Rham cohomology H^*_{dR}(X\/k) of X over k is then the cohomology … Continue reading
\n A = k[x, y, s, t, 1\/(st - 1)]\/(x^n + y^n + sx^{n - 2}y + t xy^{n - 2}) <\/code>
\nLet us denote f the polynomial we’re dividing by.<\/p>\n
\n bf = ∂(g_1f)\/∂x + ∂(g_2f)\/∂y + ∂(g_3f)\/∂s + ∂(g_4f)\/∂t. <\/code>
\nHere is a link to a pdf with a short, but hopefully readable, proof that this quotient has infinite dimension: infinite-de-rham<\/a><\/p>\n","protected":false},"excerpt":{"rendered":"