On graded algebras of global dimension 3

Assume that a graded associative algebra $A$ over a field $k$ is minimally presented as the quotient algebra of a free algebra $F$ by the ideal $I$ generated by a set $f$ of homogeneous elements. We study the following two extensions of $A$: the algebra $\overline F=F/I\oplus I/I^2\oplus\dotsb$ associated with $F$ with respect to the $I$-adic filtration, and the homology algebra $H$ of the Shafarevich complex $\operatorname{Sh}(f,F)$ (which is a non-commutative version of the Koszul complex). We obtain several characterizations of algebras of global dimension 3. In particular, the $A$-algebra $H$ in this case is free, and the algebra $\overline F$ is isomorphic to the quotient algebra of a free $A$-algebra by the ideal generated by a so-called strongly free (or inert) set.