In this article we construct a cochain complex of a complex Clifford algebra with coefficients in itself in a combinatorial fashion and we call the corresponding cohomology by Clifford cohomology. We show that Clifford cohomology controls the deformation of a complex Clifford algebra and can classify them up to Morita equivalence. We also study Hochschild cohomology groups and formal deformations of the algebra of smooth sections of a complex Clifford algebra bundle over an even dimensional orientable Riemannian manifold M which admits a \(Spin^{c}\) structure.

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关于克利福德代数上同调的注解

在本文中，我们以组合方式构造了复数 Clifford 代数的上链复形，其本身具有系数，并且我们将相应的上同调称为Clifford 上同调。我们证明Clifford 上同调控制复 Clifford 代数的变形，并且可以将它们分类为 Morita 等价。我们还研究了 Hochschild 上同调群以及复 Clifford 代数束在偶维可定向黎曼流形M上光滑截面代数的形式变形，该流形允许\(Spin^{c}\)结构。