Frobenius functors, stable equivalences and $K$-theory of Gorenstein projective modules

Gorenstein projective modules and Frobenius extensions

We prove that for a Frobenius extension, if a module over the extension ring is Gorenstein projective, then its underlying module over the the base ring is Gorenstein projective; the converse holds if the Frobenius extension is either left-Gorenstein or separable (e.g. the integral group ring extension $\mathbb{Z}\subset \mathbb{Z}G$). Moreover, for the Frobenius extension $R\subset A=R[x]/(x^2)$, we show that: a graded $A$-module is Gorenstein projective in $\mathrm{GrMod}(A)$, if and only if its ungraded $A$-module is Gorenstein projective, if and only if its underlying $R$-module is Gorenst…

Frobenius functors and Gorenstein projective precovers

We establish relations between Gorenstein projective precovers linked by Frobenius functors. This is motivated by an open problem that how to find general classes of rings for which modules have Gorenstein projective precovers. It is shown that if $F:\C\rightarrow\D$ is a separable Frobenius functor between abelian categories with enough projective objects, then every object in $\C$ has a Gorenstein projective precover provided that every object in $\D$ has a Gorenstein projective precover. This result is applied to separable Frobenius extensions and excellent extensions.

Gorenstein projective and injective dimensions over Frobenius extensions

Let $R\subset A$ be a Frobenius extension of rings. We prove that: (1) for any left $A$-module $M$, $_{A}M$ is Gorenstein projective (injective) if and only if the underlying left $R$-module $_{R}M$ is Gorenstein projective (injective). (2) if $\mathrm{G}\text{-}\mathrm{proj.dim}_{A}M<\infty$, then $\mathrm{G}\text{-}\mathrm{proj.dim}_{A}M = \mathrm{G}\text{-}\mathrm{proj.dim}_{R}M$, the dual for Gorenstein injective dimension also holds. (3) if the extension is split, then $\mathrm{G}\text{-}\mathrm{gldim}(A)= \mathrm{G}\text{-}\mathrm{gldim}(R)$.

阿贝尔范畴中的Gorenstein上同调

Gorenstein cohomology in abelian categories

通过Auslander-Buchweitz近似和严格归结，研究了阿贝尔范畴子范畴上的相对上同调函子.我们验证了这些函子之间的某些比较映射是同构的，并在此背景下引入了完美的概念。我们的主要定理是一个平衡的结果，相对上同调，同时恢复定理的Holm和目前的作者作为特殊情况。

When the Schur functor induces a triangle-equivalence between Gorenstein defect categories

Let $R$ be an Artin algebra and $e$ an idempotent of $R$. Assume that ${\rm Tor}_i^{eRe}(Re,G)=0$ for any $G\in{\rm GProj} eRe$ and $i$ sufficiently large. Necessary and sufficient conditions are given for the Schur functor $S_e$ to induce a triangle-equivalence $\mathbb{D}_{def}(R)\simeq\mathbb{D}_{def}(eRe)$. Combine this with a result of Psaroudakis-Skartsaterhagen-Solberg [29], we provide necessary and sufficient conditions for the singular equivalence $\mathbb{D}_{sg}(R)\simeq\mathbb{D}_{sg}(eRe)$ to restrict to a triangle-equivalence $\underline{{\rm GProj} R}\simeq\underline{{\rm GProj}…

Superized Troesch complexes and cohomology for strict polynomial superfunctors

We adapt a construction due to Troesch to the category of strict polynomial superfunctors in order to construct complexes of injective objects whose cohomology is isomorphic to Frobenius twists of the (super)symmetric power functors. We apply these complexes to construct injective resolutions of the even and odd Frobenius twist functors, to investigate the structure of the Yoneda algebra of the Frobenius twist functor, and to compute other extension groups between strict polynomial superfunctors.