可表示的切结构用于仿射方案
Representable tangent structures for affine schemes
AI总结：
本文研究了仿射方案范畴中可表示的切结构，引入了切oids概念，并证明在交换有单位元代数中，切oids等价于交换结合非单位元代数，从而确定了仿射方案上仅存在两种可表示的切结构。
摘要：
仿射方案的范畴是一个切范畴，其切包函子由Kähler微分诱导，为代数几何与切范畴理论提供了直接联系。此外，该切包函子由双数环表示。本文研究了仿射方案范畴中可表示的切结构。为此，引入了切oids的概念，即在单形范畴中诱导切结构的物件。进一步，可指数化的切oids诱导了相反范畴上的切结构。因此，我们首先证明在交换有单位元代数中，切oids等价于交换结合非单位元代数，即乘法为同构的交换结合非单位元代数。从那里，我们解释了仿射方案上的可表示切结构如何对应于有限生成的射影交换结合非单位元代数。特别是，对于主理想整环上的仿射方案，我们证明仅存在两种可表示的切结构：平凡的和由Kähler微分给出的。
英文摘要：
The category of affine schemes is a tangent category whose tangent bundle functor is induced by Kähler differentials, providing a direct link between algebraic geometry and tangent category theory. Moreover, this tangent bundle functor is represented by the ring of dual numbers. How special is this tangent structure? Are there any other (non-trivial) tangent structure on the category of affine schemes? In this paper, we characterize the representable tangent structures on the category of affine schemes. To this end, we introduce a useful tool, the notion of tangentoids, which are precisely the objects in a monoidal category that induce a tangent structure via tensoring. Furthermore, coexponentiable tangentoids induce tangent structures on the opposite category. As such, we first prove that tangentoids in the category of commutative unital algebras are equivalent to commutative associative solid non-unital algebras, that is, commutative associative non-unital algebras whose multiplication is an isomorphism. From there, we explain how representable tangent structures on affine schemes correspond to finitely generated projective commutative associative solid non-unital algebras. In particular, for affine schemes over a principal ideal domain, we show that there are precisely two representable tangent structures: the trivial one and the one given by Kähler differentials.
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