I wanted to just present an explicit example of a nonvanishing higher de Rham cohomology group of the spectrum of an Artinian finite dimensional C-algebra.<\/p>\n
Consider an element f of C[x, y] where C is the complex numbers. Let ω be a 1-form in x, y such that<\/p>\n
d(ω) = f d(x) ∧ d(y)<\/p><\/blockquote>\n
Such a form always exists by the Poincare lemma for C[x, y]. The form ω will give a nonzero cohomology class in the de Rham complex of A = C[x, y]\/(f) unless we can write<\/p>\n
ω = d(h) + gd(f) + f η<\/p><\/blockquote>\n
for some h, g in C[x,y] and 1 form η. Taking d of this relation we find that one needs to have a g and η such that<\/p>\n
f d(x) ∧ d(y) = d(g) ∧ d(f) + f d(η) + d(f) ∧ η<\/p><\/blockquote>\n
This means that with θ = η – d(g) we have<\/p>\n
f d(x) ∧ d(y) = f d(θ) + d(f) ∧ θ = d(fθ)<\/p><\/blockquote>\n
If we write θ = a d(x) + b d(y) then this gives<\/p>\n
(*) f = – ∂(fa)\/∂ y + ∂(fb)\/∂ x<\/p><\/blockquote>\n
Now we consider an example due to Reiffen. It is carefully written out in the second appendix of 2505.03978<\/a> that (*) doesn’t have a solution if f = x^4 + y^5 + x y^4 (see proof of B.8). In fact, the proof shows that there cannot even be a, b in C[x, y] such that (*) holds modulo the maximal ideal (x, y) to the power 6.<\/p>\n
Artinian Example.<\/strong> Let B = C[x, y]\/(x^4 + y^5 + x y^4, x^100, y^100). Then the de Rham complex Ω^*_{B\/C} has cohomology in degree 1. Namely, take the form ω above. If it maps to zero in H^1(Ω^*_{B\/C}) then the reader goes through the arguments above and shows that one gets a solution to (*) modulo (x, y)^6 which is a contradiction.<\/p>\n
I would welcome a reference for examples of this type (please email me; I will edit the post and put it here). We already have some references to related material in Infinite dimensional de Rham cohomology<\/a>.<\/p>\n
Enjoy!<\/p>\n","protected":false},"excerpt":{"rendered":"
I wanted to just present an explicit example of a nonvanishing higher de Rham cohomology group of the spectrum of an Artinian finite dimensional C-algebra. Consider an element f of C[x, y] where C is the complex numbers. Let ω … Continue reading