In this paper, we construct the complex associated to an overdetermined system of PDE of order one with constant coefficients using the theory of exterior differential system of Cartan (see [1]). We show that if the tableau associated to the system of PDE is in involution then the complex contains only operators of order one given by the successive torsions of the system (see section 2 for the details). Moreover in section 3, we give a simple direct proof of “the involution of tableau” by prolongation process in this particular case (theorem of Cartan-Kuranishi in this context). Finally in the last section, we construct the complex associated to the Cauchy-Fueter operator (in fact, more general system of PDE containing Cauchy-Fueter equations) using the previous results. We can note that the system of PDE associated to the Cauchy-Fueter operator is not in involution contrarily of the De Rham or Dolbeault operators (this complex was first constructed by [5] by using computer algebra methods, and by [6] and [7] with the help of theory of Leray's spectral sequence). We can add that the key point to obtain many of the above results is a useful characterization of the involution of a tableau given in section 2.

在本文中，我们使用 Cartan 的外微分系统理论构造了与具有常系数的一阶偏微分方程超定系统相关的复形（参见 [1]）。我们表明，如果与 PDE 系统相关的画面是对合的，那么复合体仅包含由系统的连续扭转给出的阶数为 1 的算子（有关详细信息，请参见第 2 节）。此外，在第 3 节中，我们通过在这种特殊情况下的延展过程（在这种情况下 Cartan-Kuranishi 定理）给出了“画面的对合”的简单直接证明。最后，在最后一节中，我们使用先前的结果构造了与 Cauchy-Fueter 算子（实际上，包含 Cauchy-Fueter 方程的更一般的 PDE 系统）相关联的复形。我们可以注意到，与 Cauchy-Fueter 算子相关的 PDE 系统并不与 De Rham 或 Dolbeault 算子相反地对合（该复合体首先由 [5] 使用计算机代数方法构造，并由 [6] 和 [ 7] 在 Leray 谱序列理论的帮助下）。我们可以补充一点，获得上述许多结果的关键点是对第 2 节中给出的表格对合的有用表征。