The moduli space of commutative algebras of finite rank
The moduli space of commutative algebras of finite rank
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有限阶交换代数的模空间
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发表时间：
2006
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通讯作者：
B. Poonen
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作者：
B. Poonen
赋有序基的秩元n元交换代数的模空间是$元上的有限型仿射方案，具有几何连通的纤维.它光滑的充要条件是$n le 3$。如果$n ge 8$，则它是可约的(反之亦然，至少如果我们去掉$2$和$3$以上的纤维)。$frakB_n$的相对维度为$frac{2}{27}n^3+O(n^{8/3})$。参数化代数的子格式同构于维度仅为$n^2$的$GL_n/S_n$.对于$n GE 8$，存在不是‘etale代数极限的代数.维度计算还得到了$p^n$阶交换环的个数和$d$-空间中$n$点的Hilbert格式的维数的新的渐近公式.
The moduli space of rank-$n$ commutative algebras equipped with an ordered basis is an affine scheme $frakB_n$ of finite type over $$, with geometrically connected fibers. It is smooth if and only if $n le 3$. It is reducible if $n ge 8$ (and the converse holds, at least if we remove the fibers above $2$ and $3$). The relative dimension of $frakB_n$ is $frac{2}{27} n^3 + O(n^{8/3})$. The subscheme parameterizing 'etale algebras is isomorphic to $GL_n/S_n$, which is of dimension only $n^2$. For $n ge 8$, there exist algebras that are not limits of 'etale algebras. The dimension calculations lead also to new asymptotic formulas for the number of commutative rings of order $p^n$ and the dimension of the Hilbert scheme of $n$ points in $d$-space for $d ge n/2$.