Automorphism group functors of algebraic superschemes
The famous theorem of Matsumura–Oort states that if X is a proper scheme, then the automorphism group functor Aut ( X ) of X is a locally algebraic group scheme. In this paper we generalize this theorem to the category of superschemes, that is if X is a proper superscheme, then the automorphism grou...
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| Published in | Mathematische Zeitschrift Vol. 308; no. 1 |
|---|---|
| Main Author | |
| Format | Journal Article |
| Language | English |
| Published |
Berlin/Heidelberg
Springer Berlin Heidelberg
01.09.2024
Springer Nature B.V |
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| Abstract | The famous theorem of Matsumura–Oort states that if
X
is a proper scheme, then the automorphism group functor
Aut
(
X
)
of
X
is a locally algebraic group scheme. In this paper we generalize this theorem to the category of superschemes, that is if
X
is a proper superscheme, then the automorphism group functor
Aut
(
X
)
of
X
is a locally algebraic group superscheme. Moreover, we also show that if
H
1
(
X
,
T
X
)
=
0
, where
X
is the geometric counterpart of
X
and
T
X
is the tangent sheaf of
X
, then
Aut
(
X
)
is a smooth group superscheme. |
|---|---|
| AbstractList | The famous theorem of Matsumura–Oort states that if
X
is a proper scheme, then the automorphism group functor
Aut
(
X
)
of
X
is a locally algebraic group scheme. In this paper we generalize this theorem to the category of superschemes, that is if
X
is a proper superscheme, then the automorphism group functor
Aut
(
X
)
of
X
is a locally algebraic group superscheme. Moreover, we also show that if
H
1
(
X
,
T
X
)
=
0
, where
X
is the geometric counterpart of
X
and
T
X
is the tangent sheaf of
X
, then
Aut
(
X
)
is a smooth group superscheme. The famous theorem of Matsumura–Oort states that if X is a proper scheme, then the automorphism group functor Aut(X) of X is a locally algebraic group scheme. In this paper we generalize this theorem to the category of superschemes, that is if X is a proper superscheme, then the automorphism group functor Aut(X) of X is a locally algebraic group superscheme. Moreover, we also show that if H1(X,TX)=0, where X is the geometric counterpart of X and TX is the tangent sheaf of X, then Aut(X) is a smooth group superscheme. |
| ArticleNumber | 4 |
| Author | Zubkov, A. N. |
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| Cites_doi | 10.1007/s00031-009-9055-z 10.1016/j.jpaa.2023.107404 10.1090/pspum/094/04 10.1007/BF01404615 10.1007/978-3-662-07386-5 10.1016/j.jalgebra.2022.04.027 10.1080/00927872.2022.2085286 10.1016/j.jalgebra.2020.06.019 10.1016/j.jpaa.2016.06.012 10.1016/j.aim.2023.108890 10.1007/BF01404578 10.1007/s10469-018-9485-6 10.1016/j.jalgebra.2018.02.002 10.1007/BF02684309 10.1016/j.jalgebra.2023.12.033 10.4171/097 10.1016/j.jalgebra.2011.08.038 10.2140/pjm.2018.295.385 10.1007/978-1-4757-3849-0 10.1016/j.jpaa.2019.106245 |
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| References_xml | – volume-title: Algebra, Graduate Texts in Mathematics year: 2002 ident: 3572_CR15 contributor: fullname: S Lang – volume: 14 start-page: 713 issue: 3 year: 2009 ident: 3572_CR28 publication-title: Transform. Groups doi: 10.1007/s00031-009-9055-z contributor: fullname: AN Zubkov – volume: 227 year: 2023 ident: 3572_CR2 publication-title: J. Pure Appl. Algebra doi: 10.1016/j.jpaa.2023.107404 contributor: fullname: VA Bovdi – ident: 3572_CR4 doi: 10.1090/pspum/094/04 – ident: 3572_CR1 – ident: 3572_CR14 – volume: 8 start-page: 114 year: 1969 ident: 3572_CR16 publication-title: Invent. Math. doi: 10.1007/BF01404615 contributor: fullname: AHM Levelt – ident: 3572_CR17 doi: 10.1007/978-3-662-07386-5 – volume: 605 start-page: 89 year: 2022 ident: 3572_CR22 publication-title: J. Algebra doi: 10.1016/j.jalgebra.2022.04.027 contributor: fullname: A Masuoka – volume: 50 start-page: 5387 issue: 12 year: 2022 ident: 3572_CR27 publication-title: Commun. Algebra doi: 10.1080/00927872.2022.2085286 contributor: fullname: AN Zubkov – volume: 562 start-page: 28 year: 2020 ident: 3572_CR13 publication-title: J. Algebra doi: 10.1016/j.jalgebra.2020.06.019 contributor: fullname: M Hoshi – volume: 221 start-page: 339 issue: 2 year: 2017 ident: 3572_CR23 publication-title: J. Pure Appl. Algebra doi: 10.1016/j.jpaa.2016.06.012 contributor: fullname: A Masuoka – volume: 415 year: 2023 ident: 3572_CR3 publication-title: Adv. Math. doi: 10.1016/j.aim.2023.108890 contributor: fullname: U Bruzzo – volume: 4 start-page: 1 year: 1967 ident: 3572_CR19 publication-title: Invent. Math. doi: 10.1007/BF01404578 contributor: fullname: Hideyuki Matsumura – ident: 3572_CR29 doi: 10.1007/s10469-018-9485-6 – ident: 3572_CR9 – volume: 503 start-page: 534 year: 2018 ident: 3572_CR20 publication-title: J. Algebra doi: 10.1016/j.jalgebra.2018.02.002 contributor: fullname: A Masuoka – volume-title: Revetements etales et groupe fondamental (SGA1) year: 2003 ident: 3572_CR10 contributor: fullname: A Grothendieck – volume: 23 start-page: 5 year: 1964 ident: 3572_CR18 publication-title: Inst. Hautes Études Sci. Publ. Math. No. doi: 10.1007/BF02684309 contributor: fullname: JP Murre – ident: 3572_CR11 – volume: 644 start-page: 232 year: 2024 ident: 3572_CR26 publication-title: J. Algebra doi: 10.1016/j.jalgebra.2023.12.033 contributor: fullname: E Vishnyakova – ident: 3572_CR5 doi: 10.4171/097 – volume: 348 start-page: 135 year: 2011 ident: 3572_CR21 publication-title: J. Algebra doi: 10.1016/j.jalgebra.2011.08.038 contributor: fullname: A Masuoka – volume: 295 start-page: 385 year: 2018 ident: 3572_CR7 publication-title: Pac. J. Math. doi: 10.2140/pjm.2018.295.385 contributor: fullname: R Fioresi – ident: 3572_CR12 doi: 10.1007/978-1-4757-3849-0 – ident: 3572_CR6 – ident: 3572_CR8 – volume: 224 start-page: 106245 issue: 5 year: 2020 ident: 3572_CR24 publication-title: J. Pure Appl. Algebra doi: 10.1016/j.jpaa.2019.106245 contributor: fullname: A Masuoka – ident: 3572_CR25 |
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| Snippet | The famous theorem of Matsumura–Oort states that if
X
is a proper scheme, then the automorphism group functor
Aut
(
X
)
of
X
is a locally algebraic group... The famous theorem of Matsumura–Oort states that if X is a proper scheme, then the automorphism group functor Aut(X) of X is a locally algebraic group scheme.... |
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| Title | Automorphism group functors of algebraic superschemes |
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