Geodesic
On graded algebras of global dimension~3
Izvestiya. Mathematics , Tome 65 (2001) no. 3, pp. 557-568

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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. 
@article{IM2_2001_65_3_a6,
     author = {D. I. Piontkovskii},
     title = {On graded algebras of global dimension~3},
     journal = {Izvestiya. Mathematics },
     pages = {557--568},
     publisher = {mathdoc},
     volume = {65},
     number = {3},
     year = {2001},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/IM2_2001_65_3_a6/}
}
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D. I. Piontkovskii. On graded algebras of global dimension~3. Izvestiya. Mathematics , Tome 65 (2001) no. 3, pp. 557-568. http://geodesic.mathdoc.fr/item/IM2_2001_65_3_a6/