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 be a 1-form in x, y such that

d(ω) = f d(x) ∧ d(y)

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

ω = d(h) + gd(f) + f η

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

f d(x) ∧ d(y) = d(g) ∧ d(f) + f d(η) + d(f) ∧ η

This means that with θ = η – d(g) we have

f d(x) ∧ d(y) = f d(θ) + d(f) ∧ θ = d(fθ)

If we write θ = a d(x) + b d(y) then this gives

(*) f = – ∂(fa)/∂ y + ∂(fb)/∂ x

Now we consider an example due to Reiffen. It is carefully written out in the second appendix of 2505.03978 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.

Artinian Example. 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.

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.

Enjoy!

Recording 2 examples here.

The first is to consider for n > 1 the map

A^n —> P^n, (x_1, …, x_n) maps to (x_1x_2…x_n : x_1 – 1 : … : x_n – 1)

This map is quasi-finite and flat, but it is not surjective as the points (1:1:0…0), (1:0:1:0…0), …, (1:0…0:1) are missing in the image. If we take as homogeneous coordinates on P^n the variables T_0, …, T_n then the inverse image of T_1 + … + T_n = 0 is the hyperplane x_1 + … + x_n = n in A^n. Thus we see

There is a surjective quasi-finite flat morphism A^{n – 1} —> P^{n – 1}.

The map we constructed has degree n and that is also the minimum possible.

The second example is to consider for n > 1 the map

A^n —> A^n – {0}, (x_1,…,x_n) maps to (x_1, …, x_{n – 2}, x_{n – 1}x_n – 1, f)

where

f = x_1x_{n – 1}^{n – 1} + … + x_{n – 3}x_{n – 1}^3 + x_{n – 2}x_{n – 1}^2 + x_{n – 1}(x_{n – 1}x_n – 1) + x_n

This map is surjective, quasi-finite flat of degree n (and again that’s minimal).

Enjoy!