The homological conjectures

  1. Finitistic conjecture. There exists a natural number n such that any A-module of finite projective dimension has projective dimension at most n.
  2. Nunke's conjecture. For any A-module X, there is i ≥ 0 such that Exti(DA,X) ≠ 0.
  3. Generalized Nakayama Conjecture (Auslander-Reiten) For any simple A-simple S, there is i ≥ 0 such that Exti(DA,S) ≠ 0.
  4. Nakayama Conjecture. If dom.dim.A = ∞, then A is a QF-algebra
  5. First Tachikawa conjecture. If Extie(A)(A,e(A)) = 0 for all i > 0, then A is a QF-algebra.
    (Here, e(A) denotes the enveloping algebra of A: this is the tensor product of A with the opposite of A. Note that Extie(A)(A,M) = Hi(M) is the Hochschild cohomology, for any A-A-bimodule M.)
  6. Second Tachikawa conjecture. If A is a QF-algebra and M is an A-module with Exti(M,M) = 0, for all i ≥ 1, then M is projective.

Here, (1) implies (2),
(2) implies (3) trivially,
(3) implies (4), according to Auslander-Reiten,
(4) implies both (5) and (6), according to Tachikawa.

Remark. Nakayama formulated his conjecture 1958 in a different way, with a deviating version of "dominant dimension" - using projective coresolutions of A as a bimodule,
not proj-inj coresolutions of A as a one-sided module,