On Theories of Superalgebras of Differentiable Functions
On Theories of Superalgebras of Differentiable Functions
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论可微函数的超代数理论
DOI：
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发表时间：
2012
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通讯作者：
Dmitry Roytenberg
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作者：
D. Carchedi;Dmitry Roytenberg
这是第一次在一系列文件奠定基础的微分分级方法派生微分几何（和其他几何的特征零）。本文研究了无穷可微函数可在元素上求值的超交换代数理论。这种理论被称为超级费马理论。任何超空间和光滑函数的范畴都有这样的理论。这包括了真实的和复的超流形，以及代数超流形。特别是，有一个C-无限超代数的超费马理论。C-无限超代数是C-无限环世界中超交换代数的适当概念，后者对综合微分几何和所有导出光滑流形的现有模型都至关重要。超费马理论是对费马理论概念的自然推广。Dubuc和A.科克。我们表明，任何费马理论承认一个典型的超化，但不是每一个超级费马理论产生这种方式。对于一个固定的超费马理论，我们继续研究一个特殊的代数子类，称为近点确定代数，并推导出它们的许多代数性质。
This is the first in a series of papers laying the foundations for a differential graded approach to derived differential geometry (and other geometries in characteristic zero). In this paper, we study theories of supercommutative algebras for which infinitely differentiable functions can be evaluated on elements. Such a theory is called a super Fermat theory. Any category of superspaces and smooth functions has an associated such theory. This includes both real and complex supermanifolds, as well as algebraic superschemes. In particular, there is a super Fermat theory of C-infinity superalgebras. C-infinity superalgebras are the appropriate notion of supercommutative algebras in the world of C-infinity rings, the latter being of central importance both to synthetic differential geometry and to all existing models of derived smooth manifolds. A super Fermat theory is a natural generalization of the concept of a Fermat theory introduced by E. Dubuc and A. Kock. We show that any Fermat theory admits a canonical superization, however not every super Fermat theory arises in this way. For a fixed super Fermat theory, we go on to study a special subcategory of algebras called near-point determined algebras, and derive many of their algebraic properties.