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- W3018698974 abstract "Let $k$ be a field of characteristic $0$. Using the method of idealization, we show that there is a non-Koszul, quadratic, Artinian, Gorenstein, standard graded $k$-algebra of regularity $3$ and codimension $8$, answering a question of Mastroeni, Schenck, and Stillman. We also show that this example is minimal in the sense that no other idealization that is non-Koszul, quadratic, Artinian, Gorenstein algebra, with regularity $3$ has smaller codimension. We also construct an infinite family of graded, quadratic, Artinian, Gorenstein algebras $A_m$, indexed by an integer $m ge 2$, with the following properties: (1) there are minimal first syzygies of the defining ideal in degree $m+2$, (2) for $m ge 3$, $A_m$ is not Koszul, (3) for $m ge 7$, the Hilbert function of $A_m$ is not unimodal, and thus (4) for $m ge 7$, $A_m$ does not satisfy the weak or strong Lefschetz properties. In particular, the subadditivity property fails for quadratic Gorenstein ideals. Finally, we show that the idealization of a construction of Roos yields non-Koszul quadratic Gorenstein algebras such that the residue field $k$ has a linear resolution for precisely $alpha$ steps for any integer $alpha ge 2$. Thus there is no finite test for the Koszul property even for quadratic Gorenstein algebras." @default.
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- W3018698974 date "2020-04-21" @default.
- W3018698974 modified "2023-09-27" @default.
- W3018698974 title "Quadratic Gorenstein algebras with many surprising properties" @default.
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