论文标题
通过Gorenstein投影维度猜想的有限维度猜想
Finitistic Dimension Conjectures via Gorenstein Projective Dimension
论文作者
论文摘要
这是Auslander和Reiten的众所周知的结果，违反$ \ Mathcal {p}^{\ Mathrm {\ Mathrm {fin}} _ \ Inftty $（有限生成的有限投影维度的模块）在ARTIN ALGEBRA上是足够的条件，是有限层的定位率的有效性。由戈伦斯坦投影维度可以计算的代数的有限维度的动机，在这项工作中，我们研究了auslander-reiten条件的戈伦斯坦（Gorenstein有限的Gorenstein投影维度）及其与有限维度猜想的有效性的关系。事实证明，类$ \ mathcal {gp}^{\ mathrm {fin}} _ \ infty $的违反有限性意味着第二个有利维度对左Artinian Rings的有效性。然而，在更特殊的Artin代数环境中，事实证明，Auslander - 占状态及其Gorenstein对应物实际上是等效的，因为违反类$ \ Mathcal {gp}^gp}^{\ mathrm {fin} _ infliant的违反了finities contrical contracity clastiant $\mathcal{P}^{\mathrm{fin}}_\infty$ over any Artin algebra, and the converse holds for Artin algebras over which the class $\mathcal{GP}^{\mathrm{fin}}_0$ (of finitely generated Gorenstein projective modules) is contravariantly finite.
It is a well-known result of Auslander and Reiten that contravariant finiteness of the class $\mathcal{P}^{\mathrm{fin}}_\infty$ (of finitely generated modules of finite projective dimension) over an Artin algebra is a sufficient condition for validity of finitistic dimension conjectures. Motivated by the fact that finitistic dimensions of an algebra can alternatively be computed by Gorenstein projective dimension, in this work we examine the Gorenstein counterpart of Auslander--Reiten condition, namely contravariant finiteness of the class $\mathcal{GP}^{\mathrm{fin}}_\infty$ (of finitely generated modules of finite Gorenstein projective dimension), and its relation to validity of finitistic dimension conjectures. It is proved that contravariant finiteness of the class $\mathcal{GP}^{\mathrm{fin}}_\infty$ implies validity of the second finitistic dimension conjecture over left artinian rings. In the more special setting of Artin algebras, however, it is proved that the Auslander--Reiten sufficient condition and its Gorenstein counterpart are virtually equivalent in the sense that contravariant finiteness of the class $\mathcal{GP}^{\mathrm{fin}}_\infty$ implies contravariant finiteness of the class $\mathcal{P}^{\mathrm{fin}}_\infty$ over any Artin algebra, and the converse holds for Artin algebras over which the class $\mathcal{GP}^{\mathrm{fin}}_0$ (of finitely generated Gorenstein projective modules) is contravariantly finite.
