Koszulity of algebras with non-pure resolutions

Pure resolutions of unbounded complexes of modules

Let $R$ be a commutative ring. We show that pure injective resolutions and pure projective resolutions can be constructed for unbounded complexes of $R$-modules. We use these to obtain a closed symmetric monoidal structure on the unbounded pure derived category.

Cosupports and minimal pure-injective resolutions of affine rings

We prove that any affine ring $R$ over a field $k$ has full cosupport, i.e., the cosupport of $R$ is equal to ${\rm Spec}\, R$. Using this fact, we give a complete description of all terms in a minimal pure-injective resolution of $R$, provided that $|k|=\aleph_1$ and ${\rm dim}\, R\geq 2$, or $|k|\geq \aleph_1$ and ${\rm dim}\, R=2$. As a corollary, we obtain a partial answer to a conjecture by Gruson.

正则局部环的纯子环、自同态环与Frobenius态射

Pure subrings of regular local rings, endomorphism rings and Frobenius morphisms

本文研究正则局部环纯子环的自同态环，证明其在极大Cohen-Macaulay条件下为非交换crepant消解，并利用Frobenius态射构造Morita等价的另一消解，同时探讨非交换Frobenius态射与整体维数有限性的关系，并应用于wild商奇点的强F-正则性判定。