Abstract
For K a field, consider a finite subgroup G of \({\text {GL}}_n(K)\) with its natural action on the polynomial ring \(R:= K[x_1,\dots ,x_n]\). Let \(\mathfrak {n}\) denote the homogeneous maximal ideal of the ring of invariants \(R^G\). We study how the local cohomology module \(H^n_{\mathfrak {n}}(R^G)\) compares with \(H^n_{\mathfrak {n}}(R)^G\). Various results on the a-invariant and on the Hilbert series of \(H^n_\mathfrak {n}(R^G)\) are obtained as a consequence.
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Acknowledgements
We have benefitted from several examples computed using the computer algebra systems Macaulay2 [11] and Magma [2]; the use of these is gratefully acknowledged. We are also deeply grateful to Gregor Kemper for sharing the database [15] and for related discussions, and to Mitsuyasu Hashimoto for valuable discussions. We thank the referees for a careful proofreading and for useful comments.
Funding
K.G. was supported by a Fulbright-Nehru Postdoctoral Research Fellowship at the University of Utah, J.J. by NSF CAREER Award DMS 2044833, and A.K.S. by NSF grants DMS 1801285 and DMS 2101671.
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Goel, K., Jeffries, J. & Singh, A.K. Local Cohomology of Modular Invariant Rings. Transformation Groups (2024). https://doi.org/10.1007/s00031-024-09851-6
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DOI: https://doi.org/10.1007/s00031-024-09851-6