AI 中文总结
研究幂零矩阵簇坐标环的性质，用初等证明其是\(F -\)正则的，证明其除子类群是\(\mathbb{Z}/n\mathbb{Z}\)，还探讨对称幂零矩阵坐标环情况，如\(n\geqslant 2\)时不正规，\(K\)为特定域时\(n\)为奇数时是整环。
AI 中文摘要
我们给出一个初等证明，即幂零矩阵簇的坐标环是\(F -\)正则的；在无限域\(K\)上，这个环也是一般线性群\(\textrm{GL}_n(K)\)对多项式环\(K[X]\)共轭作用的幂零锥，其中\(X\)是\(n\times n\)的不定元矩阵。我们证明坐标环的除子类群是循环群\(\mathbb{Z}/n\mathbb{Z}\)。接着研究对称幂零矩阵的情况，情形完全不同：当\(n\geqslant 2\)时坐标环不是正规的；对于特征不为二的代数闭域\(K\)，我们证明当且仅当\(n\)为奇数时坐标环是整环。
英文摘要
We give an elementary proof that the coordinate ring of the variety of nilpotent matrices is $F$-regular; over an infinite field $K$, this ring also arises as the nullcone for the conjugation action of the general linear group $\textrm{GL}_n(K)$ on the polynomial ring $K[X]$, where $X$ is an $n\times n$ matrix of indeterminates. We prove that the divisor class group of the coordinate ring is the cyclic group $\mathbb{Z}/n\mathbb{Z}$. We then study the case of symmetric nilpotent matrices, where the picture is completely different: the coordinate ring is not normal for $n\geqslant 2$; for $K$ algebraically closed of characteristic other than two, we prove that the coordinate ring is an integral domain precisely when $n$ is odd.
Comments12 pages
