The left and the right parts of a module category - Université de ...

LEFT AND RIGHT PARTS 5
(b) L A consists of the modules N ∈ indA such that, if there exists
a path from an indecomposable injective module to N, then this
path can be refined to a path of irreducible morphisms, and any
such path of irreducible morphisms is sectional.
1.5. Lemma. Let A be an artin algebra, and Γ be a component of
Γ(modA).
(a) If Γ contains projective modules, then R A ∩ Γ is directed.
(b) If Γ contains injective modules, then L A ∩ Γ is directed.
Proof. (a) Assume M ∈R A ∩ Γ, and that M = M 0 −→ · · · −→ M t =
M is a cycle. By (1.2), there exists a path M = N 0 −→ ··· −→
N s = P ,whereP is projective. By (1.3)(a), the composed path M =
M 0 −→ ··· −→ M t = M = N 0 −→ ··· −→ N s = P is refinable to
a sectional path of irreducible morphisms. But this contradicts the
non-sectionality of cycles [10, 11, 18].
□
1.6. Corollary. Let A be a representation-finite artin algebra. Then
L A and R A are directed.
2. The left and right support algebras.
[ ]
B 0
2.1. Proposition. Let A =
be an artin algebra written
M C
in triangular matrix form. Then L A ⊆L B if and only if, for each
primitive idempotent e c ∈ C, the corresponding projective A-module P c
does not lie in L A .
Proof. Sufficiency. We first observe that L A ⊆ indB. Indeed, if
X ∈L A and does not lie in indB, then there exists an idempotent
e c ∈ C such that there exists a non-zero morphism P c −→ X. However,
P c /∈L A and X ∈L A contradict the fact that L A is closed under
predecessors.
Let now X ∈L A . We claim that the full subcategories Pred A X and
Pred B X consisting of the predecessors of X in modA and modB respectively,
coincide. It is clear that Pred B X ⊆ Pred A X. Let Y ∈
Pred A X, then there exists a path Y = Y 0 −→ Y 1 −→ ··· −→ Y t = X
in indA. However, X ∈L A , hence each Y i ∈L A . Since L A ⊆ indB,
this means that each Y i is a B-module. Therefore, Y ∈ Pred B X.This
establishes our claim.
In order to show that X ∈L B , we assume that Y is a predecessor of X.
Since Y precedes X in L A , there exists a minimal projective resolution
in modA
0 −→ P 1 −→ P 0 −→ Y −→ 0