Gorenstein‐projective modules over short local algebras

Following the well-established terminology in commutative algebra, any (not necessarily commutative) finite-dimensional local algebra A $A$ with radical J $J$ will be said to short provided 3 = 0 $J^3 0$ . As case, we show: If a has an indecomposable non-projective Gorenstein-projective module M $M$ , then either is self-injective (so that all modules are Gorenstein-projective) and then, of course, | 2 ⩽ 1 $|J^2| \leqslant 1$ or else / − |J/J^2| - $|JM| |J^2||M/JM|$ More generally, focus attention semi-Gorenstein-projective ∞ $\infty$ -torsionfree modules, even ℧ $\mho$ -paths length 2, 4. In particular, show existence reflexive implies < |J/J^2|$ further restrictions. addition, consider exact complexes projective image. Again, as see if such complex exists, satisfies condition Also, Ext ( ) ≠ $\operatorname{Ext}^1(M,M) \ne this way, prove Auslander-Reiten conjecture (one classical homological conjectures) for arbitrary algebras. Many arguments used case actually work general, but there interesting differences some our results may new also case.