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  "body": "\n## Summary\n\nSpencer cohomology is a family of cohomology groups attached to the symbol of a system of linear or nonlinear [[partial differential equations|GC5BDZ6QBC]].\nIt is computed from the Spencer complex, an algebraic analogue of the de Rham complex built from polynomials and [[exterior forms|0MDMZ965VE]].\nIt measures the obstructions to solving a system order by order in formal power series.\nIt also gives an algebraic criterion for involutivity in the sense of Cartan.\nThe name also refers to a related sheaf cohomology, introduced by D. C. Spencer.\nIt is the cohomology of a manifold with coefficients in the sheaf of solutions of a linear differential operator.\n\n## Explanations\n\n### L1 human\n\nWhen one tries to solve a system of partial differential equations by power series, one finds the coefficients degree by degree.\nAt each degree some equations may contradict each other or may force new hidden equations.\nSpencer cohomology is the bookkeeping device that records these problems in the language of linear algebra.\nWhen the relevant groups vanish, the order-by-order construction never gets stuck.\n",
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