Research on duality theorems of modules on fans

石田把不一定有理范作为复曲面簇理论的推广进行了研究。他构建了粉丝的Zariski-Riemann空间理论。原定义了与固定理想相关的理想的紧闭包，作为理想紧闭包理论的推广。伊藤从奇异性理论和变形理论的角度研究了椭圆曲线的Mordell-Weil格。鹿原研究了赋值环上定义的非奇异代数簇的l-adic etale上同调的Deligne猜想。

Ishida studied on not necessarily rational fan as a generalization of the theory of toric varieties. He constructed the theory of Zariski-Riemann spaces of fans.Hara defined the tight closure of ideals associated with a fixed ideals as a generalization of the theory of tight closures of ideals.Ito studied on the Mordell-Weil lattices of elliptic curves from the view point of singularity theory and deformation theory.Kajiwara studied on the Deligne conjecture on the l-adic etale cohomology of nonsingular algebraic varieties defined over a valuation ring.