Totally acyclic complexes over noetherian schemes

The stable category of Gorenstein flat sheaves on a noetherian scheme

For a semi-separated noetherian scheme, we show that the category of cotorsion Gorenstein flat quasi-coherent sheaves is Frobenius and a natural non-affine analogue of the category of Gorenstein projective modules over a noetherian ring. We show that this coheres perfectly with the work of Murfet and Salarian that identifies the pure derived category of F-totally acyclic complexes of flat quasi-coherent sheaves as the natural non-affine analogue of the homotopy category of totally acyclic complexes of projective modules.

Bruhat区间上的无环匹配及其在全正性中的应用

Acyclic matchings on Bruhat intervals and applications to total positivity

本文在任意Bruhat区间上基于反射序显式构造无环完全匹配，并借此证明全非负Springer纤维及全非负投影纤维可缩，从而验证Lusztig猜想并简化全非负部分旗簇正则性的证明。

关于Gorenstein内射覆盖的存在性

On the existence of Gorenstein injective covers

本文刻画了Gorenstein内射覆盖性质，构造反例否定Iacob的问题，证明Noetherian概型的覆盖性质是Zariski局部的，还给出了茎局部不成立的反例。

On continuity of local epsilon factors of $\ell$-adic sheaves

Let $S$ be a noetherian scheme and $f\colon X\to S$ be a smooth morphism of relative dimension 1. For a locally constant sheaf on the complement of a divisor in $X$ at over $S$, Deligne and Laumon proved that the universal local acyclicity is equivalent to the local constancy of Swan conductors. In this article, assuming the universal local acyclicity, we show an analogous result of the continuity of local epsilon factors. We also give a generalization of this result to a family of isolated singularities.

完全可分闭概形上的几何

Geometry on totally separably closed schemes

本文证明基域上拟紧分离概形关于Nisnevich拓扑的Cech上同调与层上同调一致，推广了Artin结果，关键证明分离代数概形绝对整闭包上不可约闭子集交集的不可约性。

Prüfer 环上光滑代数的迷向挠子

Isotropic Torsors on Smooth Algebras over Prüfer Rings

本文针对秩一赋值环上光滑概形的半局部局部化，证明完全迷向约化群概形下的 Grothendieck–Serre 型非 Noether 结果，并借助新的 Gabber 表示引理推广已有结论。