论文标题
具有许多令人惊讶特性的二次戈伦斯坦代数
Quadratic Gorenstein algebras with many surprising properties
论文作者
论文摘要
令$ k $为特征$ 0 $的字段。使用理想化方法，我们表明有一个非Koszul，二次，Artinian，Gorenstein，标准分级为$ k $ -k $ - 规律性$ 3 $和CODIMENSION $ 8 $，回答了Mastroeni，Schenck和Stillman的问题。我们还表明，这个示例在某种意义上是最小的，因为没有其他理想化是非koszul，Quadratic，Artinian，Artinian，Gorenstein代数的，$ 3 $ $ 3 $都具有较小的编纂。 我们还建立了一个无限的家族，由整数$ m \ ge 2 $索引，分级，二次，Artinian，Gorenstein代数$ a_m $，具有以下属性：（1）$ m+2 $，（2 $ ge 3 $ ge 3 $ ge 3 $ a $ a gem，kos for $ 2 $，$ a $ a gem，a $ a a $ a gem，a $ a a $ a a $ a gem a a gem a gem，a。 7 $，$ a_m $的希尔伯特功能不是单峰的，因此（4）对于$ m \ ge 7 $，$ a_m $不满足弱或强的lefschetz属性。特别是，二次Gorenstein理想的亚加性特性失败。 最后，我们表明，Roos构造的理想化产生了非koszul二次Gorenstein代数，因此残基Field $ k $具有线性分辨率，用于任何整数$α\ ge 2 $的$α$步骤。因此，即使对于二次Gorenstein代数也没有针对Koszul特性的有限测试。
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 $α$ steps for any integer $α\ge 2$. Thus there is no finite test for the Koszul property even for quadratic Gorenstein algebras.
