Abstract
We show that endomorphism rings of cogenerators in the module category of a finite-dimensional algebra A admit a canonical tilting module, whose tilted algebra B is related to A by a recollement. Let M be a gen-finite A-module, meaning there are only finitely many indecomposable modules generated by M. Using the canonical tilts of endomorphism algebras of suitable cogenerators associated to M, and the resulting recollements, we construct desingularisations of the orbit closure and quiver Grassmannians of M, thus generalising all results from previous work of Crawley-Boevey and the second author in 2017. We provide dual versions of the key results, in order to also treat cogen-finite modules.
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Acknowledgements
We would like to thank Teresa Conde and William Crawley-Boevey for helpful comments. These thanks are extended to the anonymous referee for useful suggestions, such as the addition of Example 4.9. The first author also thanks Henning Krause for an invitation to visit Bielefeld, funded by the Deutsche Forschungsgemeinschaft grant SFB 701, from which this project resulted. Subsequently he was supported by a postdoctoral fellowship from the Max-Planck-Gesellschaft. The second author is supported by the Alexander von Humboldt-Stiftung in the framework of an Alexander von Humboldt Professorship endowed by the Federal Ministry of Education and Research.
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Pressland, M., Sauter, J. On Quiver Grassmannians and Orbit Closures for Gen-Finite Modules. Algebr Represent Theor 25, 413–445 (2022). https://doi.org/10.1007/s10468-021-10028-y
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DOI: https://doi.org/10.1007/s10468-021-10028-y