Let A be an Artin algebra. Then there are finitely many non-isomorphic simple
A-modules. Suppose S1, S2, …, Sn form a complete list of all non-isomorphic simple
A-modules and we fix this ordering of simple modules. Let Pi and Qi be the projective
cover and the injective envelope of Si respectively. With this order of simple modules
we define for each i the standard module Δ(i) to be the maximal quotient of Pi with
composition factors Sj with j [les ] i. Let Δ be the set of all these standard modules
Δ(i). We denote by [Fscr ](Δ) the subcategory of A-mod whose objects are the modules
M which have a Δ-filtration, namely there is a finite chain
0 = M0 ⊂ M1 ⊂ M2 ⊂ … ⊂ Mt = M
of submodules of M such that Mi/Mi−1 is isomorphic to a module in Δ for all i.
The modules in [Fscr ](Δ) are called Δ-good modules. Dually, we define the costandard
module ∇(i) to be the maximal submodule of Qi with composition factors Sj with
j [les ] i and denote by ∇ the collection of all costandard modules. In this way, we have
also the subcategory [Fscr ](∇) of A-mod whose objects are these modules which have
a ∇-filtration. Of course, we have the notion of ∇-good modules. Note that Δ(n) is
always projective and ∇(n) is always injective.