The differential Galois theory of strongly normal extensions
The differential Galois theory of strongly normal extensions
复制标题
强正规扩张的微分伽罗瓦理论
DOI：
10.1090/s0002-9947-03-03306-3
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发表时间：
2003
影响因子：
1.3
通讯作者：
Jerald J. Kovacic
中科院分区：
文献类型：
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作者：
Jerald J. Kovacic
微分伽罗瓦理论，强正规扩张的理论，不幸的是已经衰落了。这可能是由于它依赖于Kolchin的优雅，但不被广泛采用，公理化的理论代数群。本文试图恢复理论使用的微分计划，在这些公理。我们也避免使用泛微分场，而是依赖于某个张量积。我们确定一个强正常的扩展与最大微分理想的张量积的自同构，从而确定伽罗瓦群与闭点的仿射微分计划。此外，张量积有一个自然的核结构，它可以转化为伽罗瓦群运算：自同构的合成。这个仿射微分方案分裂，即通过从（非微分，不一定是仿射）群方案的基扩展获得。因此，伽罗瓦群与定义在常数上的群概型的闭点或有理点规范同构。我们得到了微分伽罗瓦理论的基本定理，给出了子群概型与中间微分域之间的一个双射对应。在这一结果的方式，我们研究某些方面的微分代数几何，例如封闭浸入，产品，当地的环形空间的常数，和分裂微分计划。
Differential Galois theory, the theory of strongly normal extensions, has unfortunately languished. This may be due to its reliance on Kolchin's elegant, but not widely adopted, axiomatization of the theory of algebraic groups. This paper attempts to revive the theory using a differential scheme in place of those axioms. We also avoid using a universal differential field, instead relying on a certain tensor product. We identify automorphisms of a strongly normal extension with maximal differential ideals of this tensor product, thus identifying the Galois group with the closed points of an affine differential scheme. Moreover, the tensor product has a natural coring structure which translates into the Galois group operation: composition of automorphisms. This affine differential scheme splits, i.e. is obtained by base extension from a (not differential, not necessarily affine) group scheme. As a consequence, the Galois group is canonically isomorphic to the closed, or rational, points of a group scheme defined over constants. We obtain the fundamental theorem of differential Galois theory, giving a bijective correspondence between subgroup schemes and intermediate differential fields. On the way to this result we study certain aspects of differential algebraic geometry, e.g. closed immersions, products, local ringed space of constants, and split differential schemes.