Geometric purity and the frame of smashing ideals
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arXiv
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| Auteurs principaux: | , , , , |
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| Format: | Preprint |
| Publié: |
2026
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| _version_ | 1866918311720124416 |
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| author | Gómez, Juan Omar Medina-Barcenas, Maurcio Stevenson, Greg Villarreal, Bernardo Zaldivar-Corichi, Ángel |
| author_facet | Gómez, Juan Omar Medina-Barcenas, Maurcio Stevenson, Greg Villarreal, Bernardo Zaldivar-Corichi, Ángel |
| contents | We introduce the notion of geometric purity in rigidly-compactly generated tt-categories by considering exact triangles that are pure at each tt-stalk. We develop a systematic study of this concept, including examples and applications. In particular, we show that geometric purity is, in general, strictly stronger than ordinary purity, and that it naturally leads to the notion of geometrically pure-injective objects. We prove that such objects arise as pushforwards of pure-injective objects from suitable tt-stalks. Moreover, we give a detailed analysis of indecomposable geometrically pure-injective objects in the derived category of the projective line. Under mild additional assumptions, we identify the geometric part of the Ziegler spectrum as a closed subset.
As an application, we demonstrate that this new notion of purity can be used to tackle the problem of spatiality of the frame of smashing ideals via the geometric Ziegler spectrum. In particular, we show that our approach rules out the counterexamples of Balchin and Stevenson to existing methods. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_20744 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Geometric purity and the frame of smashing ideals Gómez, Juan Omar Medina-Barcenas, Maurcio Stevenson, Greg Villarreal, Bernardo Zaldivar-Corichi, Ángel Representation Theory Algebraic Geometry Algebraic Topology Category Theory 18G80, 18E35, 18E45, 18F99, 55U35, 03C60 We introduce the notion of geometric purity in rigidly-compactly generated tt-categories by considering exact triangles that are pure at each tt-stalk. We develop a systematic study of this concept, including examples and applications. In particular, we show that geometric purity is, in general, strictly stronger than ordinary purity, and that it naturally leads to the notion of geometrically pure-injective objects. We prove that such objects arise as pushforwards of pure-injective objects from suitable tt-stalks. Moreover, we give a detailed analysis of indecomposable geometrically pure-injective objects in the derived category of the projective line. Under mild additional assumptions, we identify the geometric part of the Ziegler spectrum as a closed subset. As an application, we demonstrate that this new notion of purity can be used to tackle the problem of spatiality of the frame of smashing ideals via the geometric Ziegler spectrum. In particular, we show that our approach rules out the counterexamples of Balchin and Stevenson to existing methods. |
| title | Geometric purity and the frame of smashing ideals |
| topic | Representation Theory Algebraic Geometry Algebraic Topology Category Theory 18G80, 18E35, 18E45, 18F99, 55U35, 03C60 |
| url | https://arxiv.org/abs/2601.20744 |