Frobenius函子和投射余可解Gorenstein平坦模

Frobenius Functors and Projectively Coresolved Gorenstein Flat Modules

该文研究投射余可解Gorenstein平坦模及其维数在Frobenius函子作用下的同调不变性.设R和S是环,_(S)M_(R)是Frobenius双模且M_(R)是生成子,则R-模X是投射余可解Gorenstein平坦的当且仅当MRX是投射余可解Gorenstein平坦S-模,并且当F:_(R)M→_(S)M是忠实的Frobenius函子时有PGfd(_(R)X)=PGfd(_(S)F(X)),从而投射余可解Gorenstein平坦模及其维数在环的Frobenius扩张下是保持的.还证明了环的PGF整体维数在环的可裂Frobenius扩张下是不变的.

This paper investigates the homological invariance of projectively coresolved Gorenstein flat modules and their dimensions under a Frobenius functor.Let R and S be rings,_(S)M_(R) be a Frobenius bimodule and M_(R) be a generator.Then an R-module X is projectively coresolved Gorenstein flat if and only if M_(R)X is a projectively coresolved Gorenstein flat S-module;and we have PGfd(_(R)X)=PGfd(_(S)F(X))when F:_(R)M→_(S)M is a faithful Frobenius functor,so projectively coresolved Gorenstein flat modules and their dimensions are preserved under a Frobenius extension of rings.It is also shown that the PGF global dimension of a ring is invariant under a split Frobenius extension of rings.