Finitistic dimension and restricted injective dimension

Dejun Wu

Czechoslovak Mathematical Journal (2015)

  • Volume: 65, Issue: 4, page 1023-1031
  • ISSN: 0011-4642

Abstract

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We study the relations between finitistic dimensions and restricted injective dimensions. Let R be a ring and T a left R -module with A = End R T . If R T is selforthogonal, then we show that rid ( T A ) ≤ findim ( A A ) ≤ findim ( R T ) + rid ( T A ) . Moreover, if R is a left noetherian ring and T is a finitely generated left R -module with finite injective dimension, then rid ( T A ) ≤ findim ( A A ) ≤ fin . inj . dim ( R R ) + rid ( T A ) . Also we show by an example that the restricted injective dimensions of a module may be strictly smaller than the Gorenstein injective dimension.

How to cite

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Wu, Dejun. "Finitistic dimension and restricted injective dimension." Czechoslovak Mathematical Journal 65.4 (2015): 1023-1031. <http://eudml.org/doc/276318>.

@article{Wu2015,
abstract = {We study the relations between finitistic dimensions and restricted injective dimensions. Let $R$ be a ring and $T$ a left $R$-module with $A=\mathop \{\rm End\}_RT$. If $_RT$ is selforthogonal, then we show that $\mathop \{\rm rid\}(T_A)\le \mathop \{\rm findim\}(A_A)\le \mathop \{\rm findim\}(_RT)+\mathop \{\rm rid\}(T_A)$. Moreover, if $R$ is a left noetherian ring and $T$ is a finitely generated left $R$-module with finite injective dimension, then $\mathop \{\rm rid\}(T_A)\le \mathop \{\rm findim\}(A_A)\le \mathop \{\rm fin.inj.dim\}(_RR)+\mathop \{\rm rid\}(T_A)$. Also we show by an example that the restricted injective dimensions of a module may be strictly smaller than the Gorenstein injective dimension.},
author = {Wu, Dejun},
journal = {Czechoslovak Mathematical Journal},
keywords = {finitistic dimension; restricted injective dimension; tilting module},
language = {eng},
number = {4},
pages = {1023-1031},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Finitistic dimension and restricted injective dimension},
url = {http://eudml.org/doc/276318},
volume = {65},
year = {2015},
}

TY - JOUR
AU - Wu, Dejun
TI - Finitistic dimension and restricted injective dimension
JO - Czechoslovak Mathematical Journal
PY - 2015
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 65
IS - 4
SP - 1023
EP - 1031
AB - We study the relations between finitistic dimensions and restricted injective dimensions. Let $R$ be a ring and $T$ a left $R$-module with $A=\mathop {\rm End}_RT$. If $_RT$ is selforthogonal, then we show that $\mathop {\rm rid}(T_A)\le \mathop {\rm findim}(A_A)\le \mathop {\rm findim}(_RT)+\mathop {\rm rid}(T_A)$. Moreover, if $R$ is a left noetherian ring and $T$ is a finitely generated left $R$-module with finite injective dimension, then $\mathop {\rm rid}(T_A)\le \mathop {\rm findim}(A_A)\le \mathop {\rm fin.inj.dim}(_RR)+\mathop {\rm rid}(T_A)$. Also we show by an example that the restricted injective dimensions of a module may be strictly smaller than the Gorenstein injective dimension.
LA - eng
KW - finitistic dimension; restricted injective dimension; tilting module
UR - http://eudml.org/doc/276318
ER -

References

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  7. Smal{ø, S. O., Homological differences between finite and infinite dimensional representations of algebras, Infinite Length Modules. Proceedings of the Conference, Bielefeld, Germany, 1998 H. Krause et al. Trends Math. Birkhäuser, Basel (2000), 425-439. (2000) Zbl0991.16002MR1798916
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