拓扑可积导出和仿射诱导方案上的加法群作用
Topologically integrable derivations and additive group actions on affine ind-schemes
AI总结：
本文提出拓扑可积导出的概念，建立了仿射诱导方案上的加法群作用与导出之间的一一对应关系，并证明了相关切片定理的类比。
摘要：
我们通过纯粹代数和拓扑框架发展了仿射诱导方案上的加法群作用的理论。仿射诱导方案通过完全、第二可数、线性拓扑化的环来描述，而加法群的作用则通过受限指数同态来编码。我们引入了拓扑可积导出的概念，即一个连续导出，其形式指数在受限幂级数的意义下收敛，并证明这一概念提供了局部-nilpotent导出到无限维设定的正确扩展。我们的第一个主要结果建立了拓扑可积导出与仿射诱导方案上的加法群作用之间的一一对应关系，扩展了经典情况下的对应关系。然后我们研究了允许切片的此类作用的结构。在此背景下，我们证明了一个诱导方案的切片定理的类比：如果加法群作用允许切片，则底层的仿射诱导方案与带有仿射线的乘积是等变同构的，并且作用由第二因子上的平移给出。几个例子说明了拓扑假设的必要性，并突显了有限型情况中不存在的现象。
英文摘要：
We develop a theory of additive group actions on affine ind-schemes through a purely algebraic and topological framework. Affine ind-schemes are described via complete, second-countable, linearly topologized rings, and actions of the additive group are encoded by restricted exponential homomorphisms. We introduce the notion of a topologically integrable derivation, a continuous derivation whose formal exponential converges in the sense of restricted power series, and show that this notion provides the correct extension of locally nilpotent derivations to the infinite-dimensional setting. Our first main result establishes a one-to-one correspondence between topologically integrable derivations and additive group actions on affine ind-schemes, extending the classical correspondence for affine varieties. We then investigate the structure of such actions admitting a slice. In this context, we prove an ind-scheme analog of the classical slice theorem: if an additive group action admits a slice, then the underlying affine ind-scheme is equivariantly isomorphic to a product with the affine line, and the action is given by translation on the second factor. Several examples illustrate the necessity of the topological hypotheses and highlight phenomena absent in the finite-type case.
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