A Functorial Version of Chevalley's Theorem on Constructible Sets
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arXiv
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866917692850569216 |
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| author | Blatter, Andreas |
| author_facet | Blatter, Andreas |
| contents | To determine whether an $n\times n$-matrix has rank at most $r$ it suffices to check that the $(r+1)\times (r+1)$-minors have rank at most $r$. In other words, to describe the set of $n\times n$-matrices with the property of having rank at most $r$, we only need the description of the corresponding subset of $(r+1)\times (r+1)$-matrices. We will generalize this observation to a large class of subsets of tensor spaces. A description of certain subsets of a high-dimensional tensor space can always be pulled back from a description of the corresponding subset in a fixed lower-dimensional tensor space. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_09092 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A Functorial Version of Chevalley's Theorem on Constructible Sets Blatter, Andreas Algebraic Geometry To determine whether an $n\times n$-matrix has rank at most $r$ it suffices to check that the $(r+1)\times (r+1)$-minors have rank at most $r$. In other words, to describe the set of $n\times n$-matrices with the property of having rank at most $r$, we only need the description of the corresponding subset of $(r+1)\times (r+1)$-matrices. We will generalize this observation to a large class of subsets of tensor spaces. A description of certain subsets of a high-dimensional tensor space can always be pulled back from a description of the corresponding subset in a fixed lower-dimensional tensor space. |
| title | A Functorial Version of Chevalley's Theorem on Constructible Sets |
| topic | Algebraic Geometry |
| url | https://arxiv.org/abs/2406.09092 |