On the symmetric algebra of certain first syzygy modules
Let ( R , m ) be a standard graded K -algebra over a field K . Then R can be written as S / I , where I ⊆ ( x 1 ,…, x n ) 2 is a graded ideal of a polynomial ring S = K [ x 1 ,…, x n ]. Assume that n ⩽ 3 and I is a strongly stable monomial ideal. We study the symmetric algebra Sym R (Syz 1 ( m )) of...
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| Veröffentlicht in: | Czechoslovak Mathematical Journal 2022, Vol.72 (2), p.391-409 |
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| container_title | Czechoslovak Mathematical Journal |
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| creator | Restuccia, Gaetana Tang, Zhongming Utano, Rosanna |
| description | Let (
R
,
m
) be a standard graded
K
-algebra over a field
K
. Then
R
can be written as
S
/
I
, where
I
⊆ (
x
1
,…,
x
n
)
2
is a graded ideal of a polynomial ring
S
=
K
[
x
1
,…,
x
n
]. Assume that
n
⩽ 3 and
I
is a strongly stable monomial ideal. We study the symmetric algebra Sym
R
(Syz
1
(
m
)) of the first syzygy module Syz
1
(
m
) of
m
. When the minimal generators of
I
are all of degree 2, the dimension of Sym
R
(Syz
1
(
m
)) is calculated and a lower bound for its depth is obtained. Under suitable conditions, this lower bound is reached. |
| doi_str_mv | 10.21136/CMJ.2021.0508-20 |
| format | Article |
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R
,
m
) be a standard graded
K
-algebra over a field
K
. Then
R
can be written as
S
/
I
, where
I
⊆ (
x
1
,…,
x
n
)
2
is a graded ideal of a polynomial ring
S
=
K
[
x
1
,…,
x
n
]. Assume that
n
⩽ 3 and
I
is a strongly stable monomial ideal. We study the symmetric algebra Sym
R
(Syz
1
(
m
)) of the first syzygy module Syz
1
(
m
) of
m
. When the minimal generators of
I
are all of degree 2, the dimension of Sym
R
(Syz
1
(
m
)) is calculated and a lower bound for its depth is obtained. Under suitable conditions, this lower bound is reached.</description><identifier>ISSN: 0011-4642</identifier><identifier>EISSN: 1572-9141</identifier><identifier>DOI: 10.21136/CMJ.2021.0508-20</identifier><language>eng</language><publisher>Berlin/Heidelberg: Springer Berlin Heidelberg</publisher><subject>Algebra ; Analysis ; Convex and Discrete Geometry ; Lower bounds ; Mathematical Modeling and Industrial Mathematics ; Mathematics ; Mathematics and Statistics ; Modules ; Ordinary Differential Equations ; Polynomials ; Rings (mathematics)</subject><ispartof>Czechoslovak Mathematical Journal, 2022, Vol.72 (2), p.391-409</ispartof><rights>Institute of Mathematics, Czech Academy of Sciences 2021</rights><rights>Institute of Mathematics, Czech Academy of Sciences 2021.</rights><lds50>peer_reviewed</lds50><woscitedreferencessubscribed>false</woscitedreferencessubscribed></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><linktopdf>$$Uhttps://link.springer.com/content/pdf/10.21136/CMJ.2021.0508-20$$EPDF$$P50$$Gspringer$$H</linktopdf><linktohtml>$$Uhttps://link.springer.com/10.21136/CMJ.2021.0508-20$$EHTML$$P50$$Gspringer$$H</linktohtml><link.rule.ids>314,778,782,27907,27908,41471,42540,51302</link.rule.ids></links><search><creatorcontrib>Restuccia, Gaetana</creatorcontrib><creatorcontrib>Tang, Zhongming</creatorcontrib><creatorcontrib>Utano, Rosanna</creatorcontrib><title>On the symmetric algebra of certain first syzygy modules</title><title>Czechoslovak Mathematical Journal</title><addtitle>Czech Math J</addtitle><description>Let (
R
,
m
) be a standard graded
K
-algebra over a field
K
. Then
R
can be written as
S
/
I
, where
I
⊆ (
x
1
,…,
x
n
)
2
is a graded ideal of a polynomial ring
S
=
K
[
x
1
,…,
x
n
]. Assume that
n
⩽ 3 and
I
is a strongly stable monomial ideal. We study the symmetric algebra Sym
R
(Syz
1
(
m
)) of the first syzygy module Syz
1
(
m
) of
m
. When the minimal generators of
I
are all of degree 2, the dimension of Sym
R
(Syz
1
(
m
)) is calculated and a lower bound for its depth is obtained. Under suitable conditions, this lower bound is reached.</description><subject>Algebra</subject><subject>Analysis</subject><subject>Convex and Discrete Geometry</subject><subject>Lower bounds</subject><subject>Mathematical Modeling and Industrial Mathematics</subject><subject>Mathematics</subject><subject>Mathematics and Statistics</subject><subject>Modules</subject><subject>Ordinary Differential Equations</subject><subject>Polynomials</subject><subject>Rings (mathematics)</subject><issn>0011-4642</issn><issn>1572-9141</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2022</creationdate><recordtype>article</recordtype><recordid>eNp1kMtOwzAQRS0EEqXwAewssU7x-BVniSqeKuoG1pbtOCFVHsVOFuHrcSkSK1Yzi3PujC5C10BWFIDJ2_Xry4oSCisiiMooOUELEDnNCuBwihaEAGRccnqOLmLcEUIYcLVAatvj8cPjOHedH0PjsGlrb4PBQ4WdD6Npelw1IY4J-ZrrGXdDObU-XqKzyrTRX_3OJXp_uH9bP2Wb7ePz-m6TOSrVmEnlFEs_MOcscdaSnFGWF7YsHZOiAmlVepqXibBCiKJUwLy0xrBcVpRxtkQ3x9x9GD4nH0e9G6bQp5OaymSoQnCVKDhSLgwxBl_pfWg6E2YNRP8UpFNB-lCQPhSUtuTQoxMT29c-_CX_L30DSw5mxg</recordid><startdate>2022</startdate><enddate>2022</enddate><creator>Restuccia, Gaetana</creator><creator>Tang, Zhongming</creator><creator>Utano, Rosanna</creator><general>Springer Berlin Heidelberg</general><general>Springer Nature B.V</general><scope>AAYXX</scope><scope>CITATION</scope></search><sort><creationdate>2022</creationdate><title>On the symmetric algebra of certain first syzygy modules</title><author>Restuccia, Gaetana ; Tang, Zhongming ; Utano, Rosanna</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-c268t-68c839143ccb0cbb0732379bddc365f16b80214d143b5559d813e6baa376f2343</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2022</creationdate><topic>Algebra</topic><topic>Analysis</topic><topic>Convex and Discrete Geometry</topic><topic>Lower bounds</topic><topic>Mathematical Modeling and Industrial Mathematics</topic><topic>Mathematics</topic><topic>Mathematics and Statistics</topic><topic>Modules</topic><topic>Ordinary Differential Equations</topic><topic>Polynomials</topic><topic>Rings (mathematics)</topic><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>Restuccia, Gaetana</creatorcontrib><creatorcontrib>Tang, Zhongming</creatorcontrib><creatorcontrib>Utano, Rosanna</creatorcontrib><collection>CrossRef</collection><jtitle>Czechoslovak Mathematical Journal</jtitle></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>Restuccia, Gaetana</au><au>Tang, Zhongming</au><au>Utano, Rosanna</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>On the symmetric algebra of certain first syzygy modules</atitle><jtitle>Czechoslovak Mathematical Journal</jtitle><stitle>Czech Math J</stitle><date>2022</date><risdate>2022</risdate><volume>72</volume><issue>2</issue><spage>391</spage><epage>409</epage><pages>391-409</pages><issn>0011-4642</issn><eissn>1572-9141</eissn><abstract>Let (
R
,
m
) be a standard graded
K
-algebra over a field
K
. Then
R
can be written as
S
/
I
, where
I
⊆ (
x
1
,…,
x
n
)
2
is a graded ideal of a polynomial ring
S
=
K
[
x
1
,…,
x
n
]. Assume that
n
⩽ 3 and
I
is a strongly stable monomial ideal. We study the symmetric algebra Sym
R
(Syz
1
(
m
)) of the first syzygy module Syz
1
(
m
) of
m
. When the minimal generators of
I
are all of degree 2, the dimension of Sym
R
(Syz
1
(
m
)) is calculated and a lower bound for its depth is obtained. Under suitable conditions, this lower bound is reached.</abstract><cop>Berlin/Heidelberg</cop><pub>Springer Berlin Heidelberg</pub><doi>10.21136/CMJ.2021.0508-20</doi><tpages>19</tpages></addata></record> |
| fulltext | fulltext |
| identifier | ISSN: 0011-4642 |
| ispartof | Czechoslovak Mathematical Journal, 2022, Vol.72 (2), p.391-409 |
| issn | 0011-4642 1572-9141 |
| language | eng |
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| source | Elektronische Zeitschriftenbibliothek - Frei zugängliche E-Journals; Alma/SFX Local Collection; SpringerLink Journals - AutoHoldings |
| subjects | Algebra Analysis Convex and Discrete Geometry Lower bounds Mathematical Modeling and Industrial Mathematics Mathematics Mathematics and Statistics Modules Ordinary Differential Equations Polynomials Rings (mathematics) |
| title | On the symmetric algebra of certain first syzygy modules |
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