Frobenius and Cartier Algebras of Stanley–Reisner Rings (II)

Abstract

It is known that the Frobenius algebra of the injective hull of the residue field of a complete Stanley–Reisner ring (i.e., a formal power series ring modulo a squarefree monomial ideal) can be only principally generated or infinitely generated as algebra over its degree zero piece, and that this fact can be read off in the corresponding simplicial complex; in the infinite case, we exhibit a 1–1 correspondence between potential new generators appearing on each graded piece and certain pairs of faces of such a simplicial complex, and we use it to provide an alternative proof of the fact that these Frobenius algebras can only be either principally generated or infinitely generated.

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Acknowledgements

The authors would like to thank Eran Nevo, Claudiu Raicu and Kevin Tucker for their comments on an earlier draft of this manuscript. Part of this work was done when the first named author visited Northwestern University funded by the CASB fellowship program.

Funding

The first author is supported by Israel Science Foundation (grant No. 844/14) and Spanish Ministerio de Economía y Competitividad MTM2016-7881-P. The second author is supported by Spanish Ministerio de Economía y Competitividad MTM2016-7881-P.

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Correspondence to Alberto F. Boix.

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To our friend Lê Tuân Hoa for his 60th birthday

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Boix, A.F., Zarzuela, S. Frobenius and Cartier Algebras of Stanley–Reisner Rings (II). Acta Math Vietnam 44, 571–586 (2019). https://doi.org/10.1007/s40306-018-00314-1

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Keywords

  • Frobenius algebras
  • Cartier algebras
  • Stanley–Reisner rings
  • Simplicial complexes
  • Free faces

Mathematics Subject Classification (2010)

  • 13A35
  • MSC 13F55