Wild torsion modules over Weyl algebras and general torsion modules over HNPs. (English) Zbl 0834.16013
The central result of this very interesting paper shows that the representation theory of the first Weyl algebra \(A_1\) over a field \(F\) of characteristic 0 is wild, in the following sense: Let \(R\) be any (say) countably generated \(F\)-algebra. Then there is an exact embedding \({\mathcal T}_R\) of the category of right \(R\)-modules into the category of right \(A_1\) modules, such that \({\mathcal T}_R\) preserves endomorphism groups and such that \({\mathcal T}_R(M)\) has code length 2 for all non-zero \(M\). Moreover, if \(R\) is a finitely generated \(F\)-algebra then \({\mathcal T}_R (M)\) has finite length for all finite dimensional \(M\). Various examples exploiting and illustrating this result are given, and the paper concludes with a pretty result completing the structure theory of finitely generated modules over HNP rings, as begun by D. Eisenbud and J. C. Robson [J. Algebra 16, 86-104 (1970; Zbl 0211.05701)].
Reviewer: K.A.Brown (Glasgow)
MSC:
| 16G60 | Representation type (finite, tame, wild, etc.) of associative algebras |
| 16E60 | Semihereditary and hereditary rings, free ideal rings, Sylvester rings, etc. |
| 16P40 | Noetherian rings and modules (associative rings and algebras) |
Keywords:
representation type; hereditary Noetherian prime rings; representation theory; first Weyl algebra; countably generated algebras; category of right modules; endomorphism groups; finite length categories; finitely generated modules; HNP ringsCitations:
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