Cartier modules: finiteness results
On a locally Noetherian scheme X over a field of positive characteristic p , we study the category of coherent X -modules M equipped with a p e -linear map, i.e. an additive map C : X X satisfying rC ( m ) C ( r p e m ) for all m M , r X . The notion of nilpotence, meaning that some power of the map...
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| Main Authors: | , |
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| Format: | Article (Journal) |
| Language: | English |
| Published: |
1. Dezember 2011
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| In: |
Journal für die reine und angewandte Mathematik
Year: 2011, Issue: 661, Pages: 85-123 |
| ISSN: | 1435-5345 |
| DOI: | 10.1515/CRELLE.2011.087 |
| Online Access: | Verlag, lizenzpflichtig, Volltext: https://doi.org/10.1515/CRELLE.2011.087 Verlag, lizenzpflichtig, Volltext: https://www.degruyterbrill.com/document/doi/10.1515/CRELLE.2011.087/html 
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| Author Notes: | by Manuel Blickle at Mainz and Gebhard Böckle at Heidelberg |
| Summary: | On a locally Noetherian scheme X over a field of positive characteristic p , we study the category of coherent X -modules M equipped with a p e -linear map, i.e. an additive map C : X X satisfying rC ( m ) C ( r p e m ) for all m M , r X . The notion of nilpotence, meaning that some power of the map C is zero, is used to rigidify this category. The resulting quotient category, called Cartier crystals, satisfies some strong finiteness conditions. The main result in this paper states that, if the Frobenius morphism on X is a finite map, i.e. if X is F -finite, then all Cartier crystals have finite length. We further show how this and related results can be used to recover and generalize other finiteness results of HartshorneSpeiser Ann. Math. 105: 4579, 1977, Lyubeznik J. reine angew. Math. 491: 65130, 1997, Sharp Trans. Amer. Math. Soc. 359: 42374258, 2007, EnescuHochster Alg. Num. Th. 2: 721754, 2008, and Hochster Contemp. Math. 448: 119127, 2007 about the structure of modules with a left action of the Frobenius. For example, we show that over any regular F -finite scheme X Lyubeznik's F -finite modules J. reine angew. Math. 491: 65130, 1997 have finite length. |
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| Item Description: | Gesehen am 30.03.2022 |
| Physical Description: | Online Resource |
| ISSN: | 1435-5345 |
| DOI: | 10.1515/CRELLE.2011.087 |