this question might seem a bit special, but it came up at a crucial point of a proof I read and so I would be very obliged if someone could explain this to me:
given is a smooth projective variety $X$ with canonical sheaf $\omega_X$ and a closed point $x$ on $X$ with residue field $k(x)$. I want now denote with $k(x)$ also the skyskrapersheaf concentrated in $x$ with the field $k(x)$ as stalk.
Now in the proof occurs that one has an isomorphism
$k(x)\simeq k(x)\otimes \omega_X$ in the bounded derived category of coherent sheaves on X, i.e. in $D^{b}(X)$ in the usual notation.
I don't see where this Iso comes from.
Thank you very much!