Remark on the Serre-Swan theorem for non-compact manifolds

Remark on the Serre-Swan theorem for graded manifolds

Combining the Batchelor theorem and the Serre-Swan theorem, we come to that, given a smooth manifold $X$, a graded commutative $C^\infty(X)$-algebra $\cA$ is isomorphic to the structure ring of a graded manifold with a body $X$ iff it is the exterior algebra of some projective $C^\infty(X)$-module of finite rank. In particular, it follows that odd fields in field theory on a smooth manifold $X$ can be represented by graded functions on some graded manifold with body $X$.

超几何中的塞雷-桑定理

The Serre-Swan Theorem in supergeometry

本文在超几何框架下证明了塞雷-桑定理的类比形式，研究局部自由超sheaf与有限生成超投射模之间的等价性。

标准模型——交换情形：旋量、狄拉克算子与 de Rham 代数

The Standard Model - the Commutative Case: Spinors, Dirac Operator and de Rham Algebra

本文以模论方法综述经典场论的数学基础，涵盖旋量丛、狄拉克算子及 Connes 微分代数，并给出外代数丛与 Connes 微分代数同构的新证明。

K-Theory of topological algebras and second quantization

Applying the classical Serre-Swan theorem, as this is extended to topological (non-normed) algebras, one attains a classification of elementary particles via their spin-structure. In this context, our argument is virtually based on a ``correspondence principle'' of S. A. Selesnick, formulated herewith in a sheaf-theoretic language, presisely speaking, in terms of vector sheaves. This then leads directly to second quantization, as well as, to other applications of geometric (pre)quantization theory.