The dominant dimension
Definition (Tachikawa 1964). Let (Ii) be the minimal injective
resolution of AA. The dominant dimension of A is at least n,
provided the modules Ii with i < n are projective,
thus if and only of there exists an exact sequence
0 → AA → M1 → M2 →
... → Mn
with projective-injective modules Mi.
Example: Let A be an indecomposable Nakayama algebra with radical-square zero and
finite global dimension. Then dom.dim. A = n-1, where n is the number of
simple A-modules.
Warning. One should be aware that this concept behaves quite differently
as compared to usual "dimensions" such as
- vectorspace dimension
- global dimension, Gelfand-Kirillov dimension,
- representation dimension, ....
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Usually, an increase of a dimension indicates
an increase of complexity. In contrast, an increase of the dominant dimension
reduces the complexity.
Dealing with algebras say with fixed global dimension d, one will expect that the
decisive properties hold also for the algebras with global dimension at most d.
On the contrary, dealing with properties of algebras with dominant dimension d,
one should expect that they hold also for algebras with dominant dimension larger
than d.