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Contracting endomorphisms and dualizing complexes

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Abstract

We investigate how one can detect the dualizing property for a chain complex over a commutative local Noetherian ring R. Our focus is on homological properties of contracting endomorphisms of R, e.g., the Frobenius endomorphism when R contains a field of positive characteristic. For instance, in this case, when R is F-finite and C is a semidualizing R-complex, we prove that the following conditions are equivalent: (i) C is a dualizing R-complex; (ii) C ∼ RHom R (n R,C) for some n > 0; (iii) G C -dimn R < ∞ and C is derived RHom R (n R,C)-reflexive for some n > 0; and (iv) G C -dimn R < ∞ for infinitely many n > 0.

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Authors and Affiliations

  1. Department of Mathematical Sciences, Georgia Southern University, Georgia Avenue 65, P.O. Box 8093, Statesboro, Georgia, 30460, USA

    Saeed Nasseh

  2. Department of Mathematical Sciences Clemson University O-110 Martin Hall, P.O. Box 340975, Clemson, South Carolina, 29634, USA

    Sean Sather-Wagstaff

Authors
  1. Saeed Nasseh
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  2. Sean Sather-Wagstaff
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Correspondence to Saeed Nasseh.

Additional information

This material is based on work supported by North Dakota EPSCoR and NSF Grant EPS-0814442. Sather-Wagstaff was supported in part by a grant from the NSA.

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Nasseh, S., Sather-Wagstaff, S. Contracting endomorphisms and dualizing complexes. Czech Math J 65, 837–865 (2015). https://doi.org/10.1007/s10587-015-0212-3

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