So-called highest weight theory plays an important role not only in the theory of algebraic groups, but also in various areas of representation theory. This chapter is devoted to reviewing and studying the theory of good filtrations, Schur algebras and the theory of quasi-hereditary algebras over an arbitrary base ring, from the viewpoint of comodules. As the central purpose is to reconstruct S. Donkin's Schur algebra over an arbitrary base, we will not go into the theory of highest weight category originated by Cline–Parshall–Scott [37], but give a more concrete treatment using coalgebras. For highest weight theory over an arbitrary base from a different approach, see [48, 51, 145].

Highest weight theory over a field

This section is devoted to reviewing the highest weight theory over a field. In view of the later characteristic-free treatment, simple comodules, which depend on characteristic, do not appear in the first definition.

Weak split highest weight coalgebras

(1.1.1) For an ordered set P and x,y ∈ P, we use the interval notation such as [x,y] ≔ {z ∈ P | x ≤ z ≤ y} and (−∞, x) ≔ {z ∈ P | z < x}.

A subset Q of an ordered set P is called a poset ideal of P if q ∈ Q, p ∈ P and p ≤ q together imply p ∈ Q. For p ∈ P, the intervals (−∞, p] and (−∞, p) are poset ideals of P.