# vanishing of higher algebraic de Rham cohomology and sheaves of differentials for singular curves

I'm looking for some results or references about de Rham cohomology of curves in less-than-optimal cases. The two vanishing results I care about are:

• $$H^{k}_{\text{dR}}(X) = 0$$ for $$k>2$$, since $$H^{\bullet}_{\text{dR}}$$ is a Weil cohomology theory (in characteristic zero), which implies the result if $$X$$ is a smooth, projective curve over $$\text{char}(k)=0$$.

• Before taking cohomology, we also have the vanishing $$\Omega^i_{X/k}=0$$ for $$i>1$$.

What I would like to know is under what conditions these vanishing statements remain true when allowing one or more of the following hypothesis relaxations:

• Smooth $$\to$$ singular (or nonsmooth) (even reducible, but still connected)

• Projective $$\to$$ quasiprojective

• Characteristic $$0$$ $$\to$$ arbitrary characteristic.

Unfortunately I've been unsuccesful in finding proofs or references, or in proving statements myself. So I would appreciate some help.

• For $X$ a curve with one ordinary double point $p$, $\Omega ^2_{X/k}$ is nonzero (it is isomorphic to $k(p)$).
– abx
Aug 23, 2019 at 16:16
• @abx ok, thank you. My calculations suggested this, but I was unsure of myself. And how about the cohomology? Is there anywhere I can see these calculations carried out? Aug 23, 2019 at 16:19
• See this MO question. You won't find many references, because the approach in the smooth case clearly does not work in the singular case. For a correct approach, see Hartshorne's paper in Publ. IHES 45 (1975).
– abx
Aug 23, 2019 at 16:38
• I am aware of that MO question and that paper, but so far I haven't been able to extract the desired information from them. Aug 23, 2019 at 17:08