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Feb. 27th, 2026 08:14 amhttps://www.academia.edu/164706514/ALMOST_ALL_QUADRATIC_ALGEBRAS_ARE_KOSZUL
Abstract. We argue that the Koszul property is generic among
quadratic algebras, with particular emphasis on the noncommu-
tative setting. For a vector space V of dimension n over an al-
gebraically closed field, the space of (noncommutative) quadratic
algebras T (V )/(R) with dim R = m is parameterized by the Grass-
mannian Gr(m, V ⊗ V ). We show that the locus of Koszul alge-
bras within this Grassmannian contains a countable intersection
of dense Zariski-open subsets and is therefore dense. Over R or C,
this locus is comeager in the sense of Baire category and has full
Lebesgue measure. <...>
Example 8.10 (Low-dimensional analysis). For n = 2 generators, ev-
ery noncommutative quadratic algebra is Koszul for m = 0, 1, 2, 3, 4 (all
possible values, since n^2 = 4). This can be verified by direct computa-
tion or by observing that every such algebra admits a PBW basis [15].
Non-Koszul noncommutative examples require at least n ≥ 3 genera-
tors, and non-Koszul commutative monomial examples require n ≥ 4
(by Fr¨oberg’s theorem). This illustrates that non-Koszulity requires
sufficient “room” for pathological interactions between relations.
([15] -- это моя книжка Quadratic Algebras.)
P.S. PDF файл имеется в распоряжении редакции.
Abstract. We argue that the Koszul property is generic among
quadratic algebras, with particular emphasis on the noncommu-
tative setting. For a vector space V of dimension n over an al-
gebraically closed field, the space of (noncommutative) quadratic
algebras T (V )/(R) with dim R = m is parameterized by the Grass-
mannian Gr(m, V ⊗ V ). We show that the locus of Koszul alge-
bras within this Grassmannian contains a countable intersection
of dense Zariski-open subsets and is therefore dense. Over R or C,
this locus is comeager in the sense of Baire category and has full
Lebesgue measure. <...>
Example 8.10 (Low-dimensional analysis). For n = 2 generators, ev-
ery noncommutative quadratic algebra is Koszul for m = 0, 1, 2, 3, 4 (all
possible values, since n^2 = 4). This can be verified by direct computa-
tion or by observing that every such algebra admits a PBW basis [15].
Non-Koszul noncommutative examples require at least n ≥ 3 genera-
tors, and non-Koszul commutative monomial examples require n ≥ 4
(by Fr¨oberg’s theorem). This illustrates that non-Koszulity requires
sufficient “room” for pathological interactions between relations.
([15] -- это моя книжка Quadratic Algebras.)
P.S. PDF файл имеется в распоряжении редакции.
