I'm reading Algebraic Groups: The theory of Group Schemes of Finite Type over a Field written by J.S.Milne. On p.6 the first example of group variety he gives is $$SL_n=\operatorname{Spm}\left((k[T_{11},\dots,T_{nn}]/(\mathrm{det}(T_{ij}-1)\right).$$
But I can't see why this maximal spectrum is a scheme. Note that on p.3, he defines an algebraic variety as a geometrically reduce and separated algebraic scheme. I know that it's a ringed space, but is it a scheme?
Formally speaking, the maximal spectrum is not a scheme. But there is a construction making the category of reduced schemes of finite type over an algebraically closed field (this can be weakened) equivalent to the category of locally ringed spaces which are locally maximal spectra as in your example, with sheaf of functions being the sheaf of regular functions. Therefore, when people want to talk about schemes of finite type over a field more concretely, they can choose to speak in that language. One can have variants, for example having all schemes of finite type over a field, not only reduced ones, and then on the other side you need the sheaf of rings to be with nilpotents, so not as concrete - not simply the sheaf of regular functions.