Given a variety $X$ over a field $k$, Huybrechts's book "Fourier-Mukai transforms in algebraic geometry" deals mainly with the bounded derived category $\mathcal{D}^b(X)$ of $X$, which is defined as the bounded derived category of the abelian category $\text{Coh}(X)$ of coherent sheaves on $X$.
Q.: Which definition of variety is Huybrechts using? Neither did I find a definition in his book, nor here on mathstacksexchange or other online sources that I came across. Since I am reading chapter 4 at the moment, I also checked the original publications of Bondal and Orlov (the primary content of chapter 4), but they do also not define varieties.
In chapter 4, Huybrechts mostly assumes that $X$ is smooth and projective to have Serre duality. By projectiveness, we certainly have a dualizing sheaf $\omega_X$ and that having the actual isomorphisms of Serre duality is equivalent to being locally Cohen-Macaulay (all local rings are Cohen-Macaulay) and being equidimensional. Smoothness gives us Cohen-Macaulay-ness, which means that we should definitely assume the notion of a variety to contain being equidimensional (as per usual, of course). So he is not just talking about general schemes and just avoiding the word "scheme" as some authors do.
I guess that Huybrechts actually uses one of the more standard definitions like
- integral separated scheme of finite type
or just
- reduced separated scheme of finite type
but which assumptions does one actually need?