$\newcommand{\Hdr}{H\_{\mathrm{dRh}}}$
$\newcommand{\spec}[1]{\mathrm{spec}(#1)}$
$\require{amsmath}$


Let $A = k[x\_1,\ldots,x\_n]$ the polynomial ring over a field $k$ of characteristic zero and $I \subseteq A$ an ideal of $A$. Let 
$Y= V(I) = \spec{A/I}$ and $X = \spec{A}$.


According to R. Hartshorne,
On the de Rham cohomology of algebraic varieties,
Publications mathématiques de l’I.H.É.S., tome 45 (1975), p. 5-99
the algebraic de Rham cohomology of $Y$ is defined as follows:


Let $\hat{X}$ be the ringed space with $Y$ as a topological space and the sheaf derived from $\varprojlim\_r A/I^r = \hat{A}$ as structure sheaf. Additionally let $\widehat{\Omega\_X^\bullet}$ be the completion of $\Omega\_{X|k}^\bullet$ (the exterior powers of the ordinary Kähler-differentials) at $I$, so that $\widehat{\Omega\_X^\bullet} = \Omega\_{X|k}^\bullet \otimes\_A \hat{A}$. The exterior derivative extends to $d^p:\widehat{\Omega\_X^p} \to \widehat{\Omega\_X^{p+1}}$.


Now Hartshorne defines 


$$\Hdr^p(Y) = \mathbb{H}^p(\hat{X}, \widehat{\Omega\_X^\bullet})$$


with $\mathbb{H}$ standing for the hypercohomology.


If $Y$ is a smooth $k$-scheme, then by Hartshorne's Proposition 1.1, one can replace $X$ by $Y$ and get 


$$(\*) \quad \quad \Hdr^p(Y) = \mathbb{H}^p(Y, \Omega\_{Y|k}^\bullet) = H^p(Y, \Omega\_{Y|k}^\bullet)$$


where $\Omega\_{Y|k}^\bullet$ is the complex of exterior powers of the ordinary Kähler-differentials of $Y/k$ and at the right stands ordinary cohomology, because of affine-ness of the situtation.


**QUESTION**: Does this equality (\*) still hold, when $Y$ is no longer a smooth scheme over $k$? 


The answers to 


[algebraic de Rham cohomology of singular varieties](https://mathoverflow.net/questions/128853/algebraic-de-rham-cohomology-of-singular-varieties/)


seem to suppose so, but I could not find a proof by searching with google or on my own.



Definition of algebraic de Rham cohomology of non-smooth affine variety
(Synthesis of answers from comments, posted as community-wiki answer for convenience.)


1. If $k = \mathbb{C}$ then algebraic de Rham cohomology, defined a la Hartshorne using the completion of $X$ along $Y$, is *always* isomorphic to singular cohomology. This is a theorem of Hartshorne, stated as Theorem 1.6 of his 1972 Manuscripta Math survey article *Algebraic de Rham cohomology*, and the proof is in Chapter IV of his 1975 PMIHES paper *On the de Rham cohomology of algebraic varieties*.
2. There are examples, as in [this question](https://mathoverflow.net/questions/256379/naive-de-rham-cohomology-fails-for-singular-varieties), of (non-smooth) affine $\mathbb{C}$-varieties $Y$ for which the "naive de Rham cohomology" $\mathbb{H}^\bullet(Y, \Omega^\bullet\_Y)$ does not agree with singular cohomology. The Arapura--Kang curve $x^5 + y^5 + x^2 y^2 = 0$ is one such example.
3. Hence the Arapura--Kang curve, or any other example as in (2), must also be an example where naive de Rham cohomology disagrees with Hartshorne's definition.


In particular, when [this question](https://mathoverflow.net/questions/128853/algebraic-de-rham-cohomology-of-singular-varieties) uses the notation $H^\bullet\_{\mathrm{dR}}(Y)$ to mean $\mathbb{H}^\bullet(Y, \Omega^\bullet\_Y)$, this is misleading and non-standard, since this definition is not equivalent to the standard definition due to Hartshorne.


