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Decomposing moduli of representations of finite-dimensional algebras

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Abstract

Consider a finite-dimensional algebra A and any of its moduli spaces \({{\mathcal {M}}}(A,\mathbf {d})^{ss}_{\theta }\) of representations. We prove a decomposition theorem which relates any irreducible component of \({{\mathcal {M}}}(A,{\mathbf {d}})^{ss}_{\theta }\) to a product of simpler moduli spaces via a finite and birational map. Furthermore, this morphism is an isomorphism when the irreducible component is normal. As an application, we show that the irreducible components of all moduli spaces associated to tame (or even Schur-tame) algebras are rational varieties.

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Notes

  1. We point out that the inclusion \(\supseteq \) in (7) does not hold if \(\sum _{x \in Q_0}\theta (x)\) is odd. Indeed, if that is the case, then one can easily find elements \(n \in G'_\theta \) such that \(n\cdot f=-f\) for any \(f \in \bigotimes _{i=1}^r S^{m_i}\left( {\text {SI}}(C_i)_{\theta } \right) \), viewed as a regular function on \(C'\).

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Acknowledgements

We wish to thank Grzegorz Bobiński and Alastair King for discussions that led to improvements of our paper. We are especially thankful to Harm Derksen for clarifying discussions on some of the results in [35]. The first author (C.C.) was supported by the NSA under grant H98230-15-1-0022.

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  1. Mathematics Department, University of Missouri-Columbia, Columbia, MO, USA

    Calin Chindris

  2. Department of Mathematics, University of Iowa, Iowa City, IA, USA

    Ryan Kinser

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Chindris, C., Kinser, R. Decomposing moduli of representations of finite-dimensional algebras. Math. Ann. 372, 555–580 (2018). https://doi.org/10.1007/s00208-018-1687-7

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