Spencer cohomology

Definition

Let $V$ and $W$ be finite-dimensional real vector spaces, with $\dim V = n$. Let $S^k = S^k V^* \otimes W$ be the space of $W$-valued homogeneous polynomials of degree $k$ on $V$, with $S^k = 0$ for $k < 0$, and let $\Lambda^j = \Lambda^j V^*$ be the exterior power. Choose a basis $e_1, \dots, e_n$ of $V$ with dual basis $e^1, \dots, e^n$, and let $\partial_i$ be the derivative of polynomials in the direction $e_i$. The Spencer operator is the linear map $$\delta \colon S^k \otimes \Lambda^j \to S^{k-1} \otimes \Lambda^{j+1}, \qquad \delta(P \otimes \omega) = \sum_{i=1}^n \partial_i P \otimes (e^i \wedge \omega).$$ It does not depend on the basis, and $\delta \circ \delta = 0$.

Let $q \ge 1$ and let $g_q \subseteq S^q$ be a linear subspace, the symbol of a system of order $q$. Put $g_k = S^k$ for $k < q$, and for $k \ge q$ define the prolongations inductively by $$g_{k+1} = \set{ P \in S^{k+1} : \partial_v P \in g_k \text{ for all } v \in V }.$$ Then $\delta(g_k \otimes \Lambda^j) \subseteq g_{k-1} \otimes \Lambda^{j+1}$, so the spaces $g_k \otimes \Lambda^j$ form chain complexes, the Spencer complexes of $g_q$. The Spencer cohomology of $g_q$ is $$H^{k,j}(g_q) = \frac{\ker\big(\delta \colon g_k \otimes \Lambda^j \to g_{k-1} \otimes \Lambda^{j+1}\big)}{\delta(g_{k+1} \otimes \Lambda^{j-1})}.$$