Study on Noetherian Local Rings in Commutative Algebra

大Cohen-Macaulay模的构造及其应用：1970年前后，H.Bass和M.Auslander等人。提出了Noether局部环上有限生成模的几个问题，称为同调猜想。1973年，C.Peskine-L.Szpiro证明了Noetherina局部环上有限生成自由模的复形的交猜想蕴含了上述问题。随后，M.Hochster指出，大Cohen-Macaulay模的存在给出了所谓的单项猜想、直和猜想和新的交集猜想，由此引出了Peskine-Szpiro的交集猜想。他利用Frobenius技巧和M.Artin的逼近定理证明了等特征Noether局部环有大Cohen-Macaulay模。此后，几乎所有的交换代数学家都试图在不等特征的Noether局部环上构造Big Cohen-Macaulay模。我们正在利用完备局部环的结构定理、Witt表示、Flenner的Bertini定理、Jacobian准则和广义Frobenius映射来研究上述问题。利用特征相同的Noether局部环的单项猜想，证明了特征不等的Noether局部环上存在大Cohen-Macaulay模。

Construction of Big Cohen-Macaulay Modules and its applications :Around 1970, H.Bass and M.Auslander et al. asked several problems on finitely generated modules over Noetherian local rings, known as Homological Conjectures. Because these questions are basic and important, they attracted many researchers in this field.In 1973, C.Peskine-L.Szpiro showed that intersection conjecture on complexes of finitely generated free-modules over Noetherina local rings implies the problems above. And they solved the intersection conjecture for Noetherian local rings which contain fields of positive characteristic.Soon after, M.Hochster remarked that the existence of Big Cohen-Macaulay Modules gives so-called monomial conjecture, direct-summand conjecture and new intersection conjecture, which induces Peskine-Szpiro's intersection conjecture. He showed that Noetherian local rings of equal characteristic have Big Cohen-Macaulay Modules, using Frobenius trick and M.Artin's approximation theorem.Since then, almost all commutative algebraists have tried to construct Big Cohen-Macaulay Modules over Noetherian local rings of unequal characteristics.We are studying the question above by using the structure theorem of completer local rings, Witt expression, Bertini theorem of Flenner, Jacobian criteria and generalized Frobenius map. Thanks to monomial conjecture for Noetherian local rings of equal characteristic, we are showing the existence of Big Cohen-Macaulay Modules over Noetherian local rings of unequal characteristics.