An elementary approach to the homological properties of constant-rank operators
We give a simple and constructive extension of Raiță’s result that every constant-rank operator possesses an exact potential and an exact annihilator. Our construction is completely self-contained and provides an improvement over the order of the operators constructed by Raiță and the order of the e...
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| Format: | Article |
| Language: | English |
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Académie des sciences
2023-01-01
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| Series: | Comptes Rendus. Mathématique |
| Online Access: | https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.388/ |
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| author | Arroyo-Rabasa, Adolfo Simental, José |
| author_facet | Arroyo-Rabasa, Adolfo Simental, José |
| author_sort | Arroyo-Rabasa, Adolfo |
| collection | DOAJ |
| description | We give a simple and constructive extension of Raiță’s result that every constant-rank operator possesses an exact potential and an exact annihilator. Our construction is completely self-contained and provides an improvement over the order of the operators constructed by Raiță and the order of the explicit annihilators for elliptic operators due to Van Schaftingen. We also give an abstract construction of an optimal annihilator for constant-rank operators, which extends the optimal construction of Van Schaftingen for elliptic operators. Lastly, we discuss the homological properties of operators in relation to the homological properties of their associated symbols. We establish that the constant-rank property is a sufficient and necessary condition for the validity of a generalized Poincaré lemma on spaces of homogeneous maps over $\mathbb{R}^d$, and we prove that the existence of potentials on spaces of periodic maps requires a strictly weaker condition than the constant-rank property. |
| format | Article |
| id | doaj-art-410a1477b95e42e986a06feec1fb2235 |
| institution | Kabale University |
| issn | 1778-3569 |
| language | English |
| publishDate | 2023-01-01 |
| publisher | Académie des sciences |
| record_format | Article |
| series | Comptes Rendus. Mathématique |
| spelling | doaj-art-410a1477b95e42e986a06feec1fb22352025-02-07T11:06:07ZengAcadémie des sciencesComptes Rendus. Mathématique1778-35692023-01-01361G1456310.5802/crmath.38810.5802/crmath.388An elementary approach to the homological properties of constant-rank operatorsArroyo-Rabasa, Adolfo0Simental, José1Université catholique de Louvain, Institut de Recherche en Mathématique et Physique, Chemin du Cyclotron 2, 1348 Louvain-la-Neuve, BelgiumInstituto de Matemáticas, Universidad Nacional Autónoma de México. Ciudad Universitaria, Mexico City, MexicoWe give a simple and constructive extension of Raiță’s result that every constant-rank operator possesses an exact potential and an exact annihilator. Our construction is completely self-contained and provides an improvement over the order of the operators constructed by Raiță and the order of the explicit annihilators for elliptic operators due to Van Schaftingen. We also give an abstract construction of an optimal annihilator for constant-rank operators, which extends the optimal construction of Van Schaftingen for elliptic operators. Lastly, we discuss the homological properties of operators in relation to the homological properties of their associated symbols. We establish that the constant-rank property is a sufficient and necessary condition for the validity of a generalized Poincaré lemma on spaces of homogeneous maps over $\mathbb{R}^d$, and we prove that the existence of potentials on spaces of periodic maps requires a strictly weaker condition than the constant-rank property.https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.388/ |
| spellingShingle | Arroyo-Rabasa, Adolfo Simental, José An elementary approach to the homological properties of constant-rank operators Comptes Rendus. Mathématique |
| title | An elementary approach to the homological properties of constant-rank operators |
| title_full | An elementary approach to the homological properties of constant-rank operators |
| title_fullStr | An elementary approach to the homological properties of constant-rank operators |
| title_full_unstemmed | An elementary approach to the homological properties of constant-rank operators |
| title_short | An elementary approach to the homological properties of constant-rank operators |
| title_sort | elementary approach to the homological properties of constant rank operators |
| url | https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.388/ |
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