论文标题
组超级化学的轨道
Orbits of actions of group superschemes
论文作者
论文摘要
通过代数关闭的字段$ \ bbbk $工作，我们证明了代数集团Superscheme $ g $ superscheme $ x $有限类型的左动作的所有轨道都在本地关闭。此外，这样的轨道$ gx $，其中$ x $是$ x $的$ x $ - $ x $的点是关闭的，并且仅当$ g_ {ev} x $以$ x_ {ev} $中的$ x_ {ev} $中的$ x_ {ev} $，或仅当$ g_ {res} x $以$ x_}封闭时，$ x_ {ev} $是$ x {ev} $。这里$ g_ {ev} $是$ g $的最大纯粹纯粹的super-subscheme，$ g_ {res} $是$ g_ {ev} $被视为组方案。同样，$ x_ {ev} $是$ x $和$ x_ {res} $的最大纯粹的超级subscheme，是$ x_ {ev} $被视为方案。我们还证明$ \ mathrm {sdim}（gx）= \ mathrm {sdim}（g） - \ mathrm {sdim}（g_x）$，其中$ g_x $是$ x $的稳定器。
Working over an algebraically closed field $\Bbbk$, we prove that all orbits of a left action of an algebraic group superscheme $G$ on a superscheme $X$ of finite type are locally closed. Moreover, such an orbit $Gx$, where $x$ is a $\Bbbk$-point of $X$, is closed if and only if $G_{ev}x$ is closed in $X_{ev}$, or equivalently, if and only if $G_{res}x$ is closed in $X_{res}$. Here $G_{ev}$ is the largest purely even group super-subscheme of $G$ and $G_{res}$ is $G_{ev}$ regarded as a group scheme. Similarly, $X_{ev}$ is the largest purely even super-subscheme of $X$ and $X_{res}$ is $X_{ev}$ regarded as a scheme. We also prove that $\mathrm{sdim}(Gx)=\mathrm{sdim}(G)-\mathrm{sdim}(G_x)$, where $G_x$ is the stabilizer of $x$.
