A method for constructing minimal projective resolutions over idempotent subrings
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- by Carlo Klapproth
- Proc. Amer. Math. Soc. 151 (2023), 4579-4592
- DOI: https://doi.org/10.1090/proc/16470
- Published electronically: August 4, 2023
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Abstract:
We show how to obtain minimal projective resolutions of finitely generated modules over an idempotent subring $\Gamma _e ≔(1-e)R(1-e)$ of a semiperfect noetherian basic ring $R$ by a construction inside $\mathsf {mod}\,R$. This is then applied to investigate homological properties of idempotent subrings $\Gamma _e$ under the assumption of $R/\langle 1-e\rangle$ being a right artinian ring. In particular, we prove the conjecture by Ingalls and Paquette that a simple module $S_e ≔eR /\operatorname {rad}eR$ with $\operatorname {Ext}_R^1(S_e,S_e) = 0$ is self-orthogonal, that is $\operatorname {Ext}^k_R(S_e,S_e)$ vanishes for all $k \geq 1$, whenever $\operatorname {gldim}R$ and $\operatorname {pdim}eR(1-e)_{\Gamma _e}$ are finite. Indeed, a slightly more general result is established, which applies to sandwiched idempotent subrings: Suppose $e \in R$ is an idempotent such that all idempotent subrings $\Gamma$ sandwiched between $\Gamma _e$ and $R$, that is $\Gamma _e \subseteq \Gamma \subseteq R$, have finite global dimension. Then the simple summands of $S_e$ can be numbered $S_1, \dots , S_n$ such that $\operatorname {Ext}_R^k(S_i, S_j) = 0$ for $1 \leq j \leq i \leq n$ and all $k > 0$.References
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Bibliographic Information
- Carlo Klapproth
- Affiliation: Department of Mathematics, Aarhus University, 8000 Aarhus C, Denmark
- ORCID: 0000-0002-1395-7800
- Email: carlo.klapproth@math.au.dk
- Received by editor(s): November 19, 2021
- Received by editor(s) in revised form: November 30, 2021, and February 22, 2023
- Published electronically: August 4, 2023
- Additional Notes: This work was financially supported by the Aarhus University Research Foundation (grant no. AUFF-F-2020-7-16).
- Communicated by: Jerzy Weyman
- © Copyright 2023 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 151 (2023), 4579-4592
- MSC (2020): Primary 16E05, 16E10, 16G10
- DOI: https://doi.org/10.1090/proc/16470
- MathSciNet review: 4634865