Contracting endomorphisms and dualizing complexes

Saeed Nasseh; Sean Sather-Wagstaff

Czechoslovak Mathematical Journal (2015)

  • Volume: 65, Issue: 3, page 837-865
  • ISSN: 0011-4642

Abstract

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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 ∼ 𝐑 Hom R ( n R , C ) for some n > 0 ; (iii) G C -dim n R < ∞ and C is derived 𝐑 Hom R ( n R , C ) -reflexive for some n > 0 ; and (iv) G C -dim n R < ∞ for infinitely many n > 0 .

How to cite

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Nasseh, Saeed, and Sather-Wagstaff, Sean. "Contracting endomorphisms and dualizing complexes." Czechoslovak Mathematical Journal 65.3 (2015): 837-865. <http://eudml.org/doc/271823>.

@article{Nasseh2015,
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) $C\sim \{\mathbf \{R\}\}\{\rm Hom\}_R(^nR,C)$ for some $n>0$; (iii) $\{\rm G\}_C\text\{\rm -dim\} ^nR <\infty $ and $C$ is derived $\{\mathbf \{R\}\}\{\rm Hom\}_R(^nR,C)$-reflexive for some $n>0$; and (iv) $\{\rm G\}_C\text\{\rm -dim\} ^nR <\infty $ for infinitely many $n>0$.},
author = {Nasseh, Saeed, Sather-Wagstaff, Sean},
journal = {Czechoslovak Mathematical Journal},
keywords = {Bass classes; contracting endomorphisms; dualizing complex; Frobenius endomorphisms; $\{\rm G\}_\{C\}$-dimension; semidualizing complex},
language = {eng},
number = {3},
pages = {837-865},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Contracting endomorphisms and dualizing complexes},
url = {http://eudml.org/doc/271823},
volume = {65},
year = {2015},
}

TY - JOUR
AU - Nasseh, Saeed
AU - Sather-Wagstaff, Sean
TI - Contracting endomorphisms and dualizing complexes
JO - Czechoslovak Mathematical Journal
PY - 2015
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 65
IS - 3
SP - 837
EP - 865
AB - 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) $C\sim {\mathbf {R}}{\rm Hom}_R(^nR,C)$ for some $n>0$; (iii) ${\rm G}_C\text{\rm -dim} ^nR <\infty $ and $C$ is derived ${\mathbf {R}}{\rm Hom}_R(^nR,C)$-reflexive for some $n>0$; and (iv) ${\rm G}_C\text{\rm -dim} ^nR <\infty $ for infinitely many $n>0$.
LA - eng
KW - Bass classes; contracting endomorphisms; dualizing complex; Frobenius endomorphisms; ${\rm G}_{C}$-dimension; semidualizing complex
UR - http://eudml.org/doc/271823
ER -

References

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