So, I think my first comment is just exactly what you explained in general (thanks for this).

My problem was that I was searching for an embedding of $S(V)$ into a general graded Hecke algebra and this cannot work. In case of rational Cherednik algebras I'm only embedding 'half' of this algebra and I didn't realize this. Anyways, the problem remains if I have to choose a section as above to get my vector space decomposition or if this can be done canonically (it's still very pedantic but I'm still not sure if I miss a point here).

Well, I started too general and this generality was source of my confusion. Above I mentioned (if it's not wrong!) that for any section I have a vector space isomorphism $S(V) \otimes \mathbb{C}G \rightarrow A_\kappa$ (equivalent to PBW for $A_\kappa$). Now, in the case of a rational Cherednik algebra I have as vector space $V \oplus V^*$ and the above gives a vector space isomorphism $S(V) \otimes \mathbb{C}G \otimes S(V^*) \rightarrow A_\kappa$ (the triangular decomposition!?). AND, this also gives an algebra embedding of $S(V)$ and $S(V^*)$ into $A_\kappa$ because $[V,V^*]=0$ in $A_\kappa$

At this point I wasn't referring to your answer but to the nonsense I was writing (the isomorphism). As I didn't want to leave something wrong there, I removed this. But you're right; I will rewrite this...

I missed a point in my first comment: Is $\xi$ in the case of the Weyl algebra still an isomorphism? If so, then is the restriction to the particular $\kappa$ just something to make life simpler when considering deformations of $S(V) \sharp \mathbb{C} G$? As for the pedantic point: Perhaps I could have just summarized this in the question "what precisely is the vector space isomorphism $\mathrm{S}(V) \otimes \mathbb{C}G \rightarrow A$ the PBW-property induces"? Is it unique?

Your remark about perfect rings is interesting! Thanks.

Hopkins gave a talk on this at Atiyah's birthday conference in 2009. Perhaps you meant this talk or you already know its contents, but anyways, you can find a video and the slides of this talk here: maths.ed.ac.uk/~aar/atiyah80.htm (the title is "Applications of algebra to a problem in topology")

@Donu, Karl: Okay, as I don't know exactly what I want (the reason for asking this question), I am not qualified to comment on that. At least in Liu Quing's book the canonical sheaf on a smooth variety $X$ is defined as $\mathrm{det}\Omega_X^1$ (6.4.2). But I'll try to put things together...

@t3suji: 'semi-separated' was perhaps what I was looking for. But except for separated schemes I don't know of any semi-separated ones. Is there some condition that for a morphism $f:X \rightarrow Y$ of varieties into a semi-separated/separated/affine variety ensures that $X$ is also semi-separated?

@t3suji: Of course! I was somewhere else... @Donu: I think it is $j_*\Omega_X^{\dim X}$ as long as $X$ is irreducible!? @all others: I will see if I can understand your comments...