If $f:X\to Y$ is a morphism of schemes of relative dimension $d$, and $\Omega_{X/Y}$ the sheaf of relative Kahler differentials, then we have the associated de Rham complex $$\mathcal{O}_X\to \Omega^1_{X/Y} \to \dots \to \Omega^d_{X/Y}$$
$\bullet$ The terms $\Omega^i_{X/Y}$ vanish for $i>d$, whether or not $X$ is regular or $f$ is smooth? I would appreciate either an explanation or a reference.
I can understand why this is the case when $\Omega^1_{X/Y}$ has $\leq d$ generators, since the wedge powers won't be able to avoid duplicate terms, so I could reduce to that question. But I don't understand why the (relative) Krull dimension bounds the number of generators, especially when singularities can allow the dimensions of some of the cotangent spaces to be strictly larger than the dimension.
$\bullet$ If the higher-dimensional de Rham terms can actually be nonzero, is it still true that the cohomology of the de Rham complex for affine schemes vanishes above dimension $d$? Or for general scheme vanishes above dimension $2d$?
$\bullet$ I really hope I don't need to worry about D-modules or perverse sheaves, but how would I know if that's the right thing to do?