Acyclic Complexes and Graded Algebras
We already know that the noncommutative <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="double-struck">N</mi></semantics></math></inline-formula>-graded Noether...
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MDPI AG
2023-07-01
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| Series: | Mathematics |
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| Online Access: | https://www.mdpi.com/2227-7390/11/14/3167 |
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| author | Chaoyuan Zhou |
| author_facet | Chaoyuan Zhou |
| author_sort | Chaoyuan Zhou |
| collection | DOAJ |
| description | We already know that the noncommutative <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="double-struck">N</mi></semantics></math></inline-formula>-graded Noetherian algebras resemble commutative local Noetherian rings in many respects. We also know that commutative rings have the important property that every minimal acyclic complex of finitely generated graded free modules is totally acyclic, and we want to generalize such properties to noncommutative <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="double-struck">N</mi></semantics></math></inline-formula>-graded Noetherian algebra. By generalizing the conclusions about commutative rings and combining what we already know about noncommutative graded algebras, we identify a class of noncommutative graded algebras with the property that every minimal acyclic complex of finitely generated graded free modules is totally acyclic. We also discuss how the relationship between AS–Gorenstein algebras and AS–Cohen–Macaulay algebras admits a balanced dualizing complex. We show that AS–Gorenstein algebras and AS–Cohen–Macaulay algebras with a balanced dualizing complex belong to this algebra. |
| first_indexed | 2024-03-11T00:52:13Z |
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| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-03-11T00:52:13Z |
| publishDate | 2023-07-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematics |
| spelling | doaj.art-df8df4573be64f83b44f15dd53fe67022023-11-18T20:21:34ZengMDPI AGMathematics2227-73902023-07-011114316710.3390/math11143167Acyclic Complexes and Graded AlgebrasChaoyuan Zhou0School of Science, Shanghai University, Shanghai 200444, ChinaWe already know that the noncommutative <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="double-struck">N</mi></semantics></math></inline-formula>-graded Noetherian algebras resemble commutative local Noetherian rings in many respects. We also know that commutative rings have the important property that every minimal acyclic complex of finitely generated graded free modules is totally acyclic, and we want to generalize such properties to noncommutative <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="double-struck">N</mi></semantics></math></inline-formula>-graded Noetherian algebra. By generalizing the conclusions about commutative rings and combining what we already know about noncommutative graded algebras, we identify a class of noncommutative graded algebras with the property that every minimal acyclic complex of finitely generated graded free modules is totally acyclic. We also discuss how the relationship between AS–Gorenstein algebras and AS–Cohen–Macaulay algebras admits a balanced dualizing complex. We show that AS–Gorenstein algebras and AS–Cohen–Macaulay algebras with a balanced dualizing complex belong to this algebra.https://www.mdpi.com/2227-7390/11/14/3167AS–Gorenstein algebraAS–Cohen–Macaulay algebraacyclic complextotally acyclic complexbalanced dualizing complex |
| spellingShingle | Chaoyuan Zhou Acyclic Complexes and Graded Algebras Mathematics AS–Gorenstein algebra AS–Cohen–Macaulay algebra acyclic complex totally acyclic complex balanced dualizing complex |
| title | Acyclic Complexes and Graded Algebras |
| title_full | Acyclic Complexes and Graded Algebras |
| title_fullStr | Acyclic Complexes and Graded Algebras |
| title_full_unstemmed | Acyclic Complexes and Graded Algebras |
| title_short | Acyclic Complexes and Graded Algebras |
| title_sort | acyclic complexes and graded algebras |
| topic | AS–Gorenstein algebra AS–Cohen–Macaulay algebra acyclic complex totally acyclic complex balanced dualizing complex |
| url | https://www.mdpi.com/2227-7390/11/14/3167 |
| work_keys_str_mv | AT chaoyuanzhou acycliccomplexesandgradedalgebras |