TY - JOUR

T1 - Orbits of actions of group superschemes

AU - Bovdi, V. A.

AU - Zubkov, A. N.

N1 - Funding Information:
This work was supported by United Arab Emirates University grants G00004159 and G00003324. The second author was also partially supported in accordance with the state task of the IM, SB RAS, project FWNF-2022-0003. Authors thank for all these supports.
Funding Information:
This work was supported by United Arab Emirates University grants G00004159 and G00003324 . The second author was also partially supported in accordance with the state task of the IM, SB RAS , project FWNF-2022-0003 . Authors thank for all these supports.
Publisher Copyright:
© 2023 Elsevier B.V.

PY - 2023/11

Y1 - 2023/11

N2 - Working over an algebraically closed field k, 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 k-point of X, is closed if and only if Gevx is closed in Xev, or equivalently, if and only if Gresx is closed in Xres. Here Gev is the largest purely even group super-subscheme of G and Gres is Gev regarded as a group scheme. Similarly, Xev is the largest purely even super-subscheme of X and Xres is Xev regarded as a scheme. We also prove that sdim(Gx)=sdim(G)−sdim(Gx), where Gx is the stabilizer of x.

AB - Working over an algebraically closed field k, 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 k-point of X, is closed if and only if Gevx is closed in Xev, or equivalently, if and only if Gresx is closed in Xres. Here Gev is the largest purely even group super-subscheme of G and Gres is Gev regarded as a group scheme. Similarly, Xev is the largest purely even super-subscheme of X and Xres is Xev regarded as a scheme. We also prove that sdim(Gx)=sdim(G)−sdim(Gx), where Gx is the stabilizer of x.

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U2 - 10.1016/j.jpaa.2023.107404

DO - 10.1016/j.jpaa.2023.107404

M3 - Article

AN - SCOPUS:85152711978

SN - 0022-4049

VL - 227

JO - Journal of Pure and Applied Algebra

JF - Journal of Pure and Applied Algebra

IS - 11

M1 - 107404

ER -