Tilting and untilting for ideals in perfectoid rings
For a perfectoid ring R of characteristic 0 with tilt R ♭ , we introduce and study a tilting map ( - ) ♭ from the set of p -adically closed ideals of R to the set of ideals of R ♭ and an untilting map ( - ) ♯ from the set of radical ideals of R ♭ to the set of ideals of R . The untilting map ( - ) ♯...
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| Veröffentlicht in: | Mathematische Zeitschrift 2024-08, Vol.307 (4), Article 66 |
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| creator | Dine, Dimitri Ishizuka, Ryo |
| description | For a perfectoid ring
R
of characteristic 0 with tilt
R
♭
, we introduce and study a tilting map
(
-
)
♭
from the set of
p
-adically closed ideals of
R
to the set of ideals of
R
♭
and an untilting map
(
-
)
♯
from the set of radical ideals of
R
♭
to the set of ideals of
R
. The untilting map
(
-
)
♯
is defined purely algebraically and generalizes the analytically defined untilting map on closed radical ideals of a perfectoid Tate ring of characteristic
p
introduced in the first author’s previous work. We prove that the two maps
J
↦
J
♭
and
I
↦
I
♯
define an inclusion-preserving bijection between the set of ideals
J
of
R
such that the quotient
R
/
J
is perfectoid and the set of
p
♭
-adically closed radical ideals of
R
♭
, where
p
♭
∈
R
♭
corresponds to a compatible system of
p
-power roots of a unit multiple of
p
in
R
. Finally, we prove that the maps
(
-
)
♭
and
(
-
)
♯
send (closed) prime ideals to prime ideals and thus define a homeomorphism between the subspace of
Spec
(
R
)
consisting of prime ideals
p
of
R
such that
R
/
p
is perfectoid and the subspace of
Spec
(
R
♭
)
consisting of
p
♭
-adically closed prime ideals of
R
♭
. In particular, we obtain a generalization and a new proof of the main result of the first author’s previous work which concerned prime ideals in perfectoid Tate rings. |
| doi_str_mv | 10.1007/s00209-024-03537-1 |
| format | Article |
| fullrecord | <record><control><sourceid>proquest_cross</sourceid><recordid>TN_cdi_proquest_journals_3085661741</recordid><sourceformat>XML</sourceformat><sourcesystem>PC</sourcesystem><sourcerecordid>3085661741</sourcerecordid><originalsourceid>FETCH-LOGICAL-c244t-f0f4e851da144784d3ca33cb0e50b60bdad11ca7290a68ea0269e0a06179c4823</originalsourceid><addsrcrecordid>eNp9kE9LxDAQxYMouK5-AU8Fz9FJMmnaoyzqCgte1nNIk1SyrG1N2oPf3tQuePM0DL_35s8j5JbBPQNQDwmAQ02BIwUhhaLsjKwYCk5ZxcU5WWUuqawUXpKrlA4AGSpcEdyH4xi6j8J0rpi68dS1fSyC8-aYitAVg4-tt2MfXBEzTdfkos3I35zqmrw_P-03W7p7e3ndPO6o5YgjbaFFX0nmDENUFTphjRC2AS-hKaFxxjFmjeI1mLLyBnhZezBQMlVbzHevyd0yd4j91-TTqA_9FLu8UguoZJmFyLKKLyob-5Sib_UQw6eJ35qBntPRSzo6p6N_09GzSSymNMwv-fg3-h_XD6pvZi4</addsrcrecordid><sourcetype>Aggregation Database</sourcetype><iscdi>true</iscdi><recordtype>article</recordtype><pqid>3085661741</pqid></control><display><type>article</type><title>Tilting and untilting for ideals in perfectoid rings</title><source>SpringerLink Journals</source><creator>Dine, Dimitri ; Ishizuka, Ryo</creator><creatorcontrib>Dine, Dimitri ; Ishizuka, Ryo</creatorcontrib><description>For a perfectoid ring
R
of characteristic 0 with tilt
R
♭
, we introduce and study a tilting map
(
-
)
♭
from the set of
p
-adically closed ideals of
R
to the set of ideals of
R
♭
and an untilting map
(
-
)
♯
from the set of radical ideals of
R
♭
to the set of ideals of
R
. The untilting map
(
-
)
♯
is defined purely algebraically and generalizes the analytically defined untilting map on closed radical ideals of a perfectoid Tate ring of characteristic
p
introduced in the first author’s previous work. We prove that the two maps
J
↦
J
♭
and
I
↦
I
♯
define an inclusion-preserving bijection between the set of ideals
J
of
R
such that the quotient
R
/
J
is perfectoid and the set of
p
♭
-adically closed radical ideals of
R
♭
, where
p
♭
∈
R
♭
corresponds to a compatible system of
p
-power roots of a unit multiple of
p
in
R
. Finally, we prove that the maps
(
-
)
♭
and
(
-
)
♯
send (closed) prime ideals to prime ideals and thus define a homeomorphism between the subspace of
Spec
(
R
)
consisting of prime ideals
p
of
R
such that
R
/
p
is perfectoid and the subspace of
Spec
(
R
♭
)
consisting of
p
♭
-adically closed prime ideals of
R
♭
. In particular, we obtain a generalization and a new proof of the main result of the first author’s previous work which concerned prime ideals in perfectoid Tate rings.</description><identifier>ISSN: 0025-5874</identifier><identifier>EISSN: 1432-1823</identifier><identifier>DOI: 10.1007/s00209-024-03537-1</identifier><language>eng</language><publisher>Berlin/Heidelberg: Springer Berlin Heidelberg</publisher><subject>Mathematics ; Mathematics and Statistics ; Rings (mathematics) ; Subspaces</subject><ispartof>Mathematische Zeitschrift, 2024-08, Vol.307 (4), Article 66</ispartof><rights>The Author(s) 2024</rights><rights>The Author(s) 2024. This work is published under http://creativecommons.org/licenses/by/4.0/ (the “License”). Notwithstanding the ProQuest Terms and Conditions, you may use this content in accordance with the terms of the License.</rights><lds50>peer_reviewed</lds50><oa>free_for_read</oa><woscitedreferencessubscribed>false</woscitedreferencessubscribed><cites>FETCH-LOGICAL-c244t-f0f4e851da144784d3ca33cb0e50b60bdad11ca7290a68ea0269e0a06179c4823</cites></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><linktopdf>$$Uhttps://link.springer.com/content/pdf/10.1007/s00209-024-03537-1$$EPDF$$P50$$Gspringer$$Hfree_for_read</linktopdf><linktohtml>$$Uhttps://link.springer.com/10.1007/s00209-024-03537-1$$EHTML$$P50$$Gspringer$$Hfree_for_read</linktohtml><link.rule.ids>314,776,780,27901,27902,41464,42533,51294</link.rule.ids></links><search><creatorcontrib>Dine, Dimitri</creatorcontrib><creatorcontrib>Ishizuka, Ryo</creatorcontrib><title>Tilting and untilting for ideals in perfectoid rings</title><title>Mathematische Zeitschrift</title><addtitle>Math. Z</addtitle><description>For a perfectoid ring
R
of characteristic 0 with tilt
R
♭
, we introduce and study a tilting map
(
-
)
♭
from the set of
p
-adically closed ideals of
R
to the set of ideals of
R
♭
and an untilting map
(
-
)
♯
from the set of radical ideals of
R
♭
to the set of ideals of
R
. The untilting map
(
-
)
♯
is defined purely algebraically and generalizes the analytically defined untilting map on closed radical ideals of a perfectoid Tate ring of characteristic
p
introduced in the first author’s previous work. We prove that the two maps
J
↦
J
♭
and
I
↦
I
♯
define an inclusion-preserving bijection between the set of ideals
J
of
R
such that the quotient
R
/
J
is perfectoid and the set of
p
♭
-adically closed radical ideals of
R
♭
, where
p
♭
∈
R
♭
corresponds to a compatible system of
p
-power roots of a unit multiple of
p
in
R
. Finally, we prove that the maps
(
-
)
♭
and
(
-
)
♯
send (closed) prime ideals to prime ideals and thus define a homeomorphism between the subspace of
Spec
(
R
)
consisting of prime ideals
p
of
R
such that
R
/
p
is perfectoid and the subspace of
Spec
(
R
♭
)
consisting of
p
♭
-adically closed prime ideals of
R
♭
. In particular, we obtain a generalization and a new proof of the main result of the first author’s previous work which concerned prime ideals in perfectoid Tate rings.</description><subject>Mathematics</subject><subject>Mathematics and Statistics</subject><subject>Rings (mathematics)</subject><subject>Subspaces</subject><issn>0025-5874</issn><issn>1432-1823</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2024</creationdate><recordtype>article</recordtype><sourceid>C6C</sourceid><recordid>eNp9kE9LxDAQxYMouK5-AU8Fz9FJMmnaoyzqCgte1nNIk1SyrG1N2oPf3tQuePM0DL_35s8j5JbBPQNQDwmAQ02BIwUhhaLsjKwYCk5ZxcU5WWUuqawUXpKrlA4AGSpcEdyH4xi6j8J0rpi68dS1fSyC8-aYitAVg4-tt2MfXBEzTdfkos3I35zqmrw_P-03W7p7e3ndPO6o5YgjbaFFX0nmDENUFTphjRC2AS-hKaFxxjFmjeI1mLLyBnhZezBQMlVbzHevyd0yd4j91-TTqA_9FLu8UguoZJmFyLKKLyob-5Sib_UQw6eJ35qBntPRSzo6p6N_09GzSSymNMwv-fg3-h_XD6pvZi4</recordid><startdate>20240801</startdate><enddate>20240801</enddate><creator>Dine, Dimitri</creator><creator>Ishizuka, Ryo</creator><general>Springer Berlin Heidelberg</general><general>Springer Nature B.V</general><scope>C6C</scope><scope>AAYXX</scope><scope>CITATION</scope></search><sort><creationdate>20240801</creationdate><title>Tilting and untilting for ideals in perfectoid rings</title><author>Dine, Dimitri ; Ishizuka, Ryo</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-c244t-f0f4e851da144784d3ca33cb0e50b60bdad11ca7290a68ea0269e0a06179c4823</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2024</creationdate><topic>Mathematics</topic><topic>Mathematics and Statistics</topic><topic>Rings (mathematics)</topic><topic>Subspaces</topic><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>Dine, Dimitri</creatorcontrib><creatorcontrib>Ishizuka, Ryo</creatorcontrib><collection>Springer Nature OA Free Journals</collection><collection>CrossRef</collection><jtitle>Mathematische Zeitschrift</jtitle></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>Dine, Dimitri</au><au>Ishizuka, Ryo</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>Tilting and untilting for ideals in perfectoid rings</atitle><jtitle>Mathematische Zeitschrift</jtitle><stitle>Math. Z</stitle><date>2024-08-01</date><risdate>2024</risdate><volume>307</volume><issue>4</issue><artnum>66</artnum><issn>0025-5874</issn><eissn>1432-1823</eissn><abstract>For a perfectoid ring
R
of characteristic 0 with tilt
R
♭
, we introduce and study a tilting map
(
-
)
♭
from the set of
p
-adically closed ideals of
R
to the set of ideals of
R
♭
and an untilting map
(
-
)
♯
from the set of radical ideals of
R
♭
to the set of ideals of
R
. The untilting map
(
-
)
♯
is defined purely algebraically and generalizes the analytically defined untilting map on closed radical ideals of a perfectoid Tate ring of characteristic
p
introduced in the first author’s previous work. We prove that the two maps
J
↦
J
♭
and
I
↦
I
♯
define an inclusion-preserving bijection between the set of ideals
J
of
R
such that the quotient
R
/
J
is perfectoid and the set of
p
♭
-adically closed radical ideals of
R
♭
, where
p
♭
∈
R
♭
corresponds to a compatible system of
p
-power roots of a unit multiple of
p
in
R
. Finally, we prove that the maps
(
-
)
♭
and
(
-
)
♯
send (closed) prime ideals to prime ideals and thus define a homeomorphism between the subspace of
Spec
(
R
)
consisting of prime ideals
p
of
R
such that
R
/
p
is perfectoid and the subspace of
Spec
(
R
♭
)
consisting of
p
♭
-adically closed prime ideals of
R
♭
. In particular, we obtain a generalization and a new proof of the main result of the first author’s previous work which concerned prime ideals in perfectoid Tate rings.</abstract><cop>Berlin/Heidelberg</cop><pub>Springer Berlin Heidelberg</pub><doi>10.1007/s00209-024-03537-1</doi><oa>free_for_read</oa></addata></record> |
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| issn | 0025-5874 1432-1823 |
| language | eng |
| recordid | cdi_proquest_journals_3085661741 |
| source | SpringerLink Journals |
| subjects | Mathematics Mathematics and Statistics Rings (mathematics) Subspaces |
| title | Tilting and untilting for ideals in perfectoid rings |
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