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

and
τ − n := Ext n Λ(DΛ, −) : mod Λ → mod Λ.
Then τ n = H 0 (ν n −) and τn
− = H 0 (νn
−1 −). Using <strong>the</strong>se functors we can capture <strong>the</strong> notion
<strong>of</strong> n-representation finiteness in <strong>the</strong> following way.
Proposition 1. [3] Let Λ be a finite dimensional K-algebra with gl.dim Λ ≤ n. Then <strong>the</strong>
following conditions are equivalent.
(a) Λ is n-representation finite.
(b) For every indecomposable projective Λ-module P , <strong>the</strong>re is a non-negative integer
l P such that ν −l P
n P is an indecomposable injective Λ-module.
We remark that if condition (b) is satisfied <strong>the</strong>n νn −i P ≃ τn
−i P for all 0 ≤ i ≤ l P and
⊕
P
l P
⊕
i=0
τn
−i P = ⊕ P
l P
⊕
i=0
ν −i
n P
is an n-cluster tilting Λ-module [1]. Fur<strong>the</strong>rmore, since ν −1 sends injectives to projectives
we have
ν −(l P +1)
n P = ν −1 (ν −l P
n P )[n] = P ′ [n] ∈ mod Λ[n]
for some indecomposable projective P ′ . We conclude that knowing <strong>the</strong> τn − -orbits <strong>of</strong> <strong>the</strong>
indecomposable projectives in mod Λ is enough to determine <strong>the</strong>ir νn
−1 -orbits. Comparing
this to <strong>the</strong> classical case n = 1 gives us a hint how to define n-representation infinite
algebras.
2. n-representation infinite algebras
Recall that if n = 1 and Λ is representation infinite, <strong>the</strong>n τ −i P is never injective for
an indecomposable projective Λ-module P . In fact ν1 −i P = τ −i P ∈ mod Λ for all i ≥ 0.
Inspired by this we make <strong>the</strong> following definition.
Definition 2. Let Λ be a finite dimensional K-algebra with gl.dim Λ ≤ n. We say that
Λ is n-representation infinite if
ν −i
n Λ ∈ mod Λ
for all i ≥ 0.
We remark that this condition is equivalent to ν i n(DΛ) ∈ mod Λ for all i ≥ 0. In <strong>the</strong>
classical setting <strong>of</strong> n = 1 every indecomposable module is ei<strong>the</strong>r preprojective, preinjective
or regular. We define higher analogues <strong>of</strong> <strong>the</strong>se classes <strong>of</strong> modules as follows.
Definition 3. Let Λ be an n-representation infinite algebra. The full subcategories <strong>of</strong>
n-preprojective, n-preinjective and n-regular modules are defined as
respectively.
P := add{ν −i
n Λ | i ≥ 0},
I := add{ν i n(DΛ) | i ≥ 0},
R := {X ∈ mod Λ | Ext i Λ(P, X) = 0 = Ext i Λ(X, I) for all i ≥ 0},
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