Theorem. Let M a module. Let Q(M) be the cokernels of monomorphisms M' → M'', where M' and M'' belong to add M. Then: the finitistic dimension is bounded on Q(M) (by some quite explicit bound defined by M).
Corollary. Every algebra with representation dimension at most 3 has finite finitistic dimension.
This note (just 4 pages) contains several interesting results for example it shows that the projective dimension of the modules of Loewy length two and finite projective dimension is bounded, but this does not seem to be surprizing. The really important result of this note seems to be the following (Corollary 0.8): If B is an algebra of global dimesnion at most 3 and P is a projective B-module with endomorphism ring A, then the finitistic dimension of A is finite.
In particular we see: Every algebra with representation dimension at most 3 has finite finitistic dimension. Note that the representation dimension of any artin algebra is finite (see Iyama), and no algebra is known with representation dimension greater than 3. One may ask whether any artin algebra has rep dim at most 3.
| During a conference in November 2002, Raphael Rouqier has annonunced a proof of the following result: The representation dimension of the exterior algebra of a 3-dimensional vector space is 4. |