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  "body": "\n## Definition\n\nLet $V$ and $W$ be finite-dimensional real [[vector spaces|05EGHEQZAW]], with $\\dim V = n$.\nLet $S^k = S^k V^* \\otimes W$ be the space of $W$-valued [[homogeneous polynomials|2C3PAQ7H50]] of degree $k$ on $V$, with $S^k = 0$ for $k \u003c 0$, and let $\\Lambda^j = \\Lambda^j V^*$ be the [[exterior power|0MDMZ965VE]].\nChoose 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 $e_i$.\nThe *Spencer operator* is the linear map\n$$\\delta \\colon S^k \\otimes \\Lambda^j \\to S^{k-1} \\otimes \\Lambda^{j+1}, \\qquad \\delta(P \\otimes \\omega) = \\sum_{i=1}^n \\partial_i P \\otimes (e^i \\wedge \\omega).$$\nIt does not depend on the basis, and $\\delta \\circ \\delta = 0$.\n\nLet $q \\ge 1$ and let $g_q \\subseteq S^q$ be a linear subspace, the symbol of a system of order $q$.\nPut $g_k = S^k$ for $k \u003c q$, and for $k \\ge q$ define the prolongations inductively by\n$$g_{k+1} = \\set{ P \\in S^{k+1} : \\partial_v P \\in g_k \\text{ for all } v \\in V }.$$\nThen $\\delta(g_k \\otimes \\Lambda^j) \\subseteq g_{k-1} \\otimes \\Lambda^{j+1}$, so the spaces $g_k \\otimes \\Lambda^j$ form [[chain complexes|WQ82C0M3QB]], the *Spencer complexes* of $g_q$.\nThe *Spencer cohomology* of $g_q$ is\n$$H^{k,j}(g_q) = \\frac{\\ker\\big(\\delta \\colon g_k \\otimes \\Lambda^j \\to g_{k-1} \\otimes \\Lambda^{j+1}\\big)}{\\delta(g_{k+1} \\otimes \\Lambda^{j-1})}.$$\n\n## Remarks\n\n$\\delta \\circ \\delta = 0$ because partial derivatives commute while $e^i \\wedge e^l = -e^l \\wedge e^i$.\nIndexing conventions differ between authors, for instance by shifting $k$.\n\nExample: take $g_q = S^q$, the symbol of the empty system.\nThen $g_k = S^k$ for all $k$, and for each $m \\ge 1$ the sequence $0 \\to S^m \\to S^{m-1} \\otimes \\Lambda^1 \\to \\dots \\to S^0 \\otimes \\Lambda^m \\to 0$ is exact.\nIt is the degree $m$ part of the de Rham complex of polynomial forms, and it is dual to the [[Koszul complex|YZY11W08SD]].\nSo $H^{k,j}(S^q) = 0$ whenever $k + j \\ge 1$.\n\nExample: for the Laplace equation on $\\R^2$, $q = 2$, $W = \\R$ and $g_2$ is the space of quadratic forms $a x^2 + b xy + c y^2$ with $a + c = 0$.\nIts prolongation $g_k$ is the space of harmonic homogeneous polynomials of degree $k$, which is $2$-dimensional for each $k \\ge 1$.\n\nConsider a system of [[partial differential equations|GC5BDZ6QBC]] whose symbol has nonzero Spencer cohomology in the relevant degrees.\nSuch a system may have hidden integrability conditions, and its power series solutions cannot always be built degree by degree.\n",
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