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