Differential graded orders, their class groups and idèles
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866916388200775680 |
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| author | Zimmermann, Alexander |
| author_facet | Zimmermann, Alexander |
| contents | For a Dedekind domain $R$ with field of fractions $K$ a classical $R$-order in a semisimple $K$-algebra $A$ is an $R$-projective $R$-subalgebra $Λ$ of $A$ such that $KΛ=A$. We study differential graded $K$-algebras which are semisimple as $K$-algebras and define differential graded $R$-orders as a differential graded $R$-subalgebras, which are in addition classical $R$-orders in $A$. We give a series of examples for such differential graded algebras and orders. We show that any differential graded $R$-order is contained in a maximal differential graded order. We develop parts of the classical ring theory in the differential graded setting, in particular the properties of analogues of the Jacobson radical. We further define class groups of differential graded orders as subgroups of the Grothendieck group of locally free differential graded modules. We define idèles in this setting showing that these idèle groups maps surjectively to the differential graded class group. Finally we give a homomorphism to the class group of the homology of the differential graded order and prove a Mayer-Vietoris like sequence for each central idempotent of $A$, including the analogous one for the kernel groups of these morphisms. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2310_06340 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Differential graded orders, their class groups and idèles Zimmermann, Alexander Rings and Algebras K-Theory and Homology Representation Theory 16E45, 16H10, 11R29 For a Dedekind domain $R$ with field of fractions $K$ a classical $R$-order in a semisimple $K$-algebra $A$ is an $R$-projective $R$-subalgebra $Λ$ of $A$ such that $KΛ=A$. We study differential graded $K$-algebras which are semisimple as $K$-algebras and define differential graded $R$-orders as a differential graded $R$-subalgebras, which are in addition classical $R$-orders in $A$. We give a series of examples for such differential graded algebras and orders. We show that any differential graded $R$-order is contained in a maximal differential graded order. We develop parts of the classical ring theory in the differential graded setting, in particular the properties of analogues of the Jacobson radical. We further define class groups of differential graded orders as subgroups of the Grothendieck group of locally free differential graded modules. We define idèles in this setting showing that these idèle groups maps surjectively to the differential graded class group. Finally we give a homomorphism to the class group of the homology of the differential graded order and prove a Mayer-Vietoris like sequence for each central idempotent of $A$, including the analogous one for the kernel groups of these morphisms. |
| title | Differential graded orders, their class groups and idèles |
| topic | Rings and Algebras K-Theory and Homology Representation Theory 16E45, 16H10, 11R29 |
| url | https://arxiv.org/abs/2310.06340 |