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Duality of sheaves with actions

滑轮与动作的二元性

基本信息

批准号：
15540018
负责人：
HASHIMOTO Mitsuyasu
金额：
$ 2.3万
依托单位：
Nagoya University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
2003
资助国家：
日本
起止时间：
2003 至 2005
项目状态：
已结题

项目摘要

As an analogue of the duality of quasi-coherent sheaves with actions over schemes, we studied the duality with actions over formal schemes. As a result, we have proved the following : Let S be a universally catenary Nagata Noetherian scheme, f : X→Y a surjective universally open S-morphism of S-schemes. If X is of finite type over S and Y is reduced, then Y is of finite type over S. As an appliatlon, we obtained a new proof of the finiteness of geometric quotients due to Fogarty. As another application, we proved that if S is a Noetherian scheme, f : X→Y a faithfully flat S-morphism of S-schemes, and X is of finite type over S, then Y is of finite type over S. Later, as a generalization, we proved that we may replace the faithful flatness by purity. Moreover, we obtained a new proof of the global F-regularity of Schubert varieties due to Lauritzen, Raben-Pedersen, and Thomsen. Moreover, we obtained a new geometric proof of the computation of invariant subrings first provedin 70's by De Concini and Procesi. Moreover, we extended the class of rings of which we take the invariant subrings, from the class of polynomial rings to that of determinantal rings. Lastly, we proved the following : Let R be a Dedekind domain, G an affine flat R-group scheme, B a flat R-algebra on which G acts. If a Noetherian R-algebra and an R-algebra map A→B^G is given, and if for any R-algebra which is an algebraically closed field, the induced map Kotimes A→(Kotimes B)^{Kotimes G} is an isomorphism, then for any R-algebra S, the induced map S otimes A→(Sotimes B)^{Sotimes G} is an isomorphism.

作为拟凝聚层作用于概型上的对偶的一个类比，我们研究了拟凝聚层作用于形式概型上的对偶。证明了：设S是泛悬链线NagataNoether概型，f：X→Y是S-概型的满射泛开S-态射.如果X是S上的有限型，Y是约化的，则Y是S上的有限型。作为应用，我们得到了Fogarty几何恒等式有限性的一个新证明.作为另一个应用，我们证明了如果S是Noether概型，f：X→Y是S-概型的忠实平坦S-态射，且X是S上的有限型，则Y是S上的有限型.后来，作为推广，我们证明了我们可以用纯度代替忠实的平坦性。此外，我们还得到了Lauritzen，Raben-Pedersen，and Schubert簇的整体F-正则性的一个新的证明.此外，我们还得到了De Concini和Procesi在70年代首次证明的不变子环计算的一个新的几何证明。此外，我们还将取不变子环的环类从多项式环类推广到行列式环类。最后，证明了：设R是Dedekind整环，G是仿射平坦R-群概型，B是G作用于其上的平坦R-代数。若给定一个Noether R-代数和一个R-代数映射A→B^G，且对任一代数闭域R-代数，诱导映射Kotimes A→（Kotimes B）^{Kotimes G}是同构，则对任一R-代数S，诱导映射Sotimes A→（Sotimes B）^{Sotimes G}是同构.