Lattices of pretorsion classes
Fuente:
arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866918216518860800 |
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| author | Campanini, Federico Fedele, Francesca Yıldırım, Emine |
| author_facet | Campanini, Federico Fedele, Francesca Yıldırım, Emine |
| contents | Since their introduction, torsion theories have played a key role in the study of abelian and pointed categories. In representation theory, torsion theories and lattices of torsion classes of mod$ A$, for $A$ a finite-dimensional algebra, have been widely studied. The more recent definition of pretorsion theories, that can be given for any category, has expanded the theory, giving many more instances of ``non-pointed torsion theories'' in unexpected settings. In this work, we introduce and study the lattice $\mathcal{L}_t(A)$ of pretorsion classes of mod$ A$. These lattices are in close connection with the lattices tors$ A$ of torsion classes of mod$ A$. We fully describe the completely join-irreducible elements of $\mathcal{L}_t(A)$. Moreover, we characterise and give a full classification of when $\mathcal{L}_t(A)$ is distributive and further describe when it can be identified with the \emph{distributive closure} of tors$ A$. Finally, we show how the lattices of pretorsion classes, together with their duals, can be used to build pretorsion theories in mod$ A$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_19223 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Lattices of pretorsion classes Campanini, Federico Fedele, Francesca Yıldırım, Emine Representation Theory Combinatorics Category Theory Rings and Algebras 18E40, 16S90, 18A99, 16G70, 06B23, 06D05 Since their introduction, torsion theories have played a key role in the study of abelian and pointed categories. In representation theory, torsion theories and lattices of torsion classes of mod$ A$, for $A$ a finite-dimensional algebra, have been widely studied. The more recent definition of pretorsion theories, that can be given for any category, has expanded the theory, giving many more instances of ``non-pointed torsion theories'' in unexpected settings. In this work, we introduce and study the lattice $\mathcal{L}_t(A)$ of pretorsion classes of mod$ A$. These lattices are in close connection with the lattices tors$ A$ of torsion classes of mod$ A$. We fully describe the completely join-irreducible elements of $\mathcal{L}_t(A)$. Moreover, we characterise and give a full classification of when $\mathcal{L}_t(A)$ is distributive and further describe when it can be identified with the \emph{distributive closure} of tors$ A$. Finally, we show how the lattices of pretorsion classes, together with their duals, can be used to build pretorsion theories in mod$ A$. |
| title | Lattices of pretorsion classes |
| topic | Representation Theory Combinatorics Category Theory Rings and Algebras 18E40, 16S90, 18A99, 16G70, 06B23, 06D05 |
| url | https://arxiv.org/abs/2511.19223 |