---
tag: BKKAA5JKT0
kind: concept
name: Spencer cohomology
msc: [58A20, 35N10, 58J10]
sources:
  - url: https://en.wikipedia.org/wiki/Spencer_cohomology
---

## Summary

Spencer cohomology is a family of cohomology groups attached to the symbol of a system of linear or nonlinear [[partial differential equations|GC5BDZ6QBC]].
It is computed from the Spencer complex, an algebraic analogue of the de Rham complex built from polynomials and [[exterior forms|0MDMZ965VE]].
It measures the obstructions to solving a system order by order in formal power series.
It also gives an algebraic criterion for involutivity in the sense of Cartan.
The name also refers to a related sheaf cohomology, introduced by D. C. Spencer.
It is the cohomology of a manifold with coefficients in the sheaf of solutions of a linear differential operator.

## Explanations

### L1 human

When one tries to solve a system of partial differential equations by power series, one finds the coefficients degree by degree.
At each degree some equations may contradict each other or may force new hidden equations.
Spencer cohomology is the bookkeeping device that records these problems in the language of linear algebra.
When the relevant groups vanish, the order-by-order construction never gets stuck.