Ulrich modules and weakly lim Ulrich sequences do not always exist
The existence of Ulrich modules for (complete) local domains has been a difficult and elusive open question. For over thirty years, it was unknown whether complete local domains always have Ulrich modules. In this paper, we answer the question of existence for both Ulrich modules and weakly lim Ulri...
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| Format: | Article |
| Language: | English |
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Cambridge University Press
2023-01-01
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| Series: | Forum of Mathematics, Sigma |
| Subjects: | |
| Online Access: | https://www.cambridge.org/core/product/identifier/S2050509423000683/type/journal_article |
| Summary: | The existence of Ulrich modules for (complete) local domains has been a difficult and elusive open question. For over thirty years, it was unknown whether complete local domains always have Ulrich modules. In this paper, we answer the question of existence for both Ulrich modules and weakly lim Ulrich sequences – a weaker notion recently introduced by Ma – in the negative. We construct many local domains in all dimensions
$d \geq 2$
that do not have any Ulrich modules. Moreover, we show that when
$d = 2$
, these local domains do not have weakly lim Ulrich sequences. |
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| ISSN: | 2050-5094 |