Free summands of stably free modules
Let R be a commutative ring. One may ask when a general R-module P that satisfies $P \oplus R \cong R^n$ has a free summand of a given rank. M. Raynaud translated this question into one about sections of certain maps between Stiefel varieties: if $V_r(\mathbb {A}^n)$ denotes the variet...
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| Main Authors: | , |
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| Format: | Article |
| Language: | English |
| Published: |
Cambridge University Press
2025-01-01
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| Series: | Forum of Mathematics, Sigma |
| Subjects: | |
| Online Access: | https://www.cambridge.org/core/product/identifier/S2050509425000398/type/journal_article |
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| Summary: | Let R be a commutative ring. One may ask when a general R-module P that satisfies
$P \oplus R \cong R^n$
has a free summand of a given rank. M. Raynaud translated this question into one about sections of certain maps between Stiefel varieties: if
$V_r(\mathbb {A}^n)$
denotes the variety
$\operatorname {GL}(n) / \operatorname {GL}(n-r)$
over a field k, then the projection
$V_r(\mathbb {A}^n) \to V_1(\mathbb {A}^n)$
has a section if and only if the following holds: any module P over any k-algebra R with the property that
$P \oplus R \cong R^n$
has a free summand of rank
$r-1$
. Using techniques from
$\mathbb {A}^1$
-homotopy theory, we characterize those n for which the map
$V_r(\mathbb {A}^n) \to V_1(\mathbb {A}^n)$
has a section in the cases
$r=3,4$
under some assumptions on the base field. |
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| ISSN: | 2050-5094 |