Proceedings of the 44th Symposium on Ring Theory and ...

Definition 14. CM L f
(R f )
Ob(CM L f
(R f )) = Ob(CM L f
(R f )),
CM L f
(R f )(M, N) := Hom gr
L f -R f
(M, N)/P(M, N).
g ∈ P(M, N) P g ′ : M → P , g ′′ : P → N
g = g ′′ ◦ g ′
CM L f
(R f ) CM L f
(R f )
Proposition 15 (Happel[7]). CM L f
(R f )
□
f
Proposition 16. CM L f
(R f )
∑
dim k CM L f
(R fW )(M, T i N) < ∞,
i
□
CM L f
(R f )
Proposition 17 (Auslander-Reiten[2]). CM L f
(R f ) S = T n−2 ◦ (−⃗ɛ f ) Serre
CM L f
(R fW )(M, N) ≃ Hom k (CM L f
(R fW )(N, SM), k)
R f M ∈ CM L f
(R f )
gr L f -S
0 → F 1
f 1→ F0 → M → 0
f M 0
f 0 : F 0 → F 1
f 1 f 0 = f · id F0 ,
f 0 f 1 = f · id F1
Eisenbud matrix factorization
Definition 18 (Eisenbud[4]). F 0 , F 1 f 0 : F 0 → F 1 , f 1 : F 1 →
F 0 f 1 f 0 = f · id F0 , f 0 f 1 = f · id F1 S-
(F 0 , F 1 , f 0 , f 1 ) f
(
f 0
F := F 0
f 1
–200–
F 1
)
□