View source for Spencer cohomology

<!--This article has been texified automatically. Since there was no Nroff source code for this article, 
the semi-automatic procedure described at https://encyclopediaofmath.org/wiki/User:Maximilian_Janisch/latexlist
was used.
If the TeX and formula formatting is correct and if all png images have been replaced by TeX code, please remove this message and the {{TEX|semi-auto}} category.

Out of 63 formulas, 59 were replaced by TEX code.-->

{{TEX|semi-auto}}{{TEX|part}}
The [[De Rham cohomology|de Rham cohomology]] and Dolbeault cohomology (cf. [[Duality in complex analysis|Duality in complex analysis]]) can be viewed as cohomologies with coefficients in the [[Sheaf|sheaf]] of locally constant, respectively harmonic, functions. Spencer cohomology is a generalization of these two cohomologies for the case of the solution sheaf of an arbitrary [[Linear differential operator|linear differential operator]].

Namely, let $\alpha : E ( \alpha ) \rightarrow M$ and $\beta : E ( \beta ) \rightarrow M$ be smooth vector bundles (cf. also [[Vector bundle|Vector bundle]]) and let $D : \Gamma ( \alpha ) \rightarrow \Gamma ( \beta )$ be a linear differential operator acting from the module $\Gamma ( \alpha )$ of smooth sections of $\alpha$ to the module $\Gamma ( \beta )$. Denote by ${\frak G} _ { D }$ the sheaf of solutions of $D a = 0$. To find the cohomology of $M$ with coefficients in ${\frak G} _ { D }$ one needs a [[Resolvent|resolvent]] of the sheaf.

Spencer cohomology appears as a result of constructing a resolvent by a locally exact complex of differential operators

<table class="eq" style="width:100%;"> <tr><td style="width:94%;text-align:center;" valign="top"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130480/s13048011.png"/></td> </tr></table>

where $\alpha = \alpha _ { 0 }$, $\alpha _ { 1 } = \beta$, $D = D _ { 0 }$. The condition that the complex be locally exact is too strong, and therefore D. Spencer proposed the weaker condition that the complex should be  "formally exact" . In this setting, there exists for a formally integrable differential operator $D$ a canonical construction ([[#References|[a5]]], [[#References|[a6]]], [[#References|[a1]]]) of a complex, called the second (or sophisticated) Spencer complex. In this complex,

<table class="eq" style="width:100%;"> <tr><td style="width:94%;text-align:center;" valign="top"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130480/s13048016.png"/></td> </tr></table>

the vector bundles $C _ { k }$ have the form $C _ { k } = \Lambda ^ { k } T ^ { * } M \otimes R _ { m } / \delta ( \Lambda ^ { k - 1 } T ^ { * } M \otimes g _ { m + 1 } )$, where $R _ { m } \subset J ^ { m } ( \alpha )$ are prolongations of the differential equation corresponding to $D$ (cf. also [[Prolongation of solutions of differential equations|Prolongation of solutions of differential equations]]) and $g _ { m }$ are the symbols of these prolongations (cf. also [[Symbol of an operator|Symbol of an operator]]). The differential operators $D _ { k }$ are first-order partial differential operators whose symbols are induced by the exterior multiplication.

The $\delta$-Poincaré lemma [[#References|[a6]]] shows that the [[Cohomology|cohomology]] of the complex does not depend on $m$ when $m$ is large enough. The stable cohomology $H _ { S } ^ { * } ( D )$ is called the Spencer cohomology of the differential operator $D$.

In general, the second Spencer complex does not produce a resolvent of ${\frak G} _ { D }$; however, it does in certain special cases, e.g. when $D$ is analytical operator [[#References|[a6]]].

Almost-all cohomologies encountered in applications are of Spencer type. For example, de Rham cohomology corresponds to the differential $D = d : C ^ { \infty } ( M ) \rightarrow \Omega ^ { 1 } ( M )$, and the Dolbeault cohomology corresponds to the Cauchy–Riemann $\overline { \partial }$-operator $\overline { \partial } : \Omega ^ { p , 0 } ( M ) \rightarrow \Omega ^ { p , 1 } ( M )$. If $D$ is a determined operator such that not all covectors are characteristic, then $H _ { S } ^ { 0 } ( D ) =\operatorname{ ker} D$, $H _ { S } ^ { 1 } ( D ) = \text { coker } D$ and $H _ { S } ^ { i } ( D ) = 0$ for $i \geq 2$. In general, $H _ { S } ^ { 0 } ( D ) =\operatorname{ ker} D$ for each formally integrable operators $D$.

In the case of Lie equations and corresponding geometrical structures (see [[#References|[a2]]]), the first Spencer cohomology gives an estimate of the set of deformations of the structure.

If $D$ is an elliptic partial differential operator (cf. [[Elliptic partial differential equation|Elliptic partial differential equation]]) and $M$ is a compact [[Manifold|manifold]], then $\operatorname { dim} H _ { S } ^ { i } ( D ) &lt; \infty$ and the [[Euler characteristic|Euler characteristic]] $\chi ( D ) = \sum ( - 1 ) ^ { i } \operatorname { dim } H _ { S } ^ { i } ( D )$ of the Spencer complex is called the index of $D$ (cf. also [[Index formulas|Index formulas]]; [[Index theory|Index theory]]). For elliptic Lie equations the index can be expressed in terms of characteristic classes corresponding to the geometrical structure ([[#References|[a3]]]).

As is well-known, there are two main methods for calculating the de Rham cohomology: the Leray–Serre spectral sequence (cf. also [[Spectral sequence|Spectral sequence]]) and the theorem on coincidence of de Rham cohomology with invariant cohomology on homogeneous manifolds. These methods also apply to Spencer cohomology, provided the operator $D$ satisfies certain extra conditions.

Thus, if the base manifold $M$ is the total space of a smooth bundle $\pi : M \rightarrow B$ over a simply-connected manifold $B$ and if the fibres of $\pi$ are not characteristic for $D$, then there exists a spectral sequence $( E _ { r } ^ { p q } , d _ { r } ^ { p q } )$ converging to the Spencer cohomology $H _ { S } ^ { * } ( D )$; its second term is $E _ { 2 } ^ { p q } = H ^ { p } ( B ) \otimes H _ { S } ^ { q } ( D _ { \pi } )$, where $D _ { \pi }$ is the fibrewise differential operator corresponding to $D$ [[#References|[a4]]].

If $M = G / G_0$ is a homogeneous manifold and the structure group $G$ is a compact connected [[Lie group|Lie group]] of symmetries of $D$, then [[#References|[a4]]] the Spencer cohomology $H _ { S } ^ { * } ( D )$ coincides with the cohomology of the $G$-invariant Spencer complex if the non-trivial characters of $( G , G _ { 0 } )$ are non-characteristic.

====References====
<table><tr><td valign="top">[a1]</td> <td valign="top">  H. Goldschmidt,   "Existence theorems for analytic linear partial differential equations"  ''Ann. Math.'' , '''86'''  (1967)  pp. 246–270</td></tr><tr><td valign="top">[a2]</td> <td valign="top">  A. Kumpera,   D. Spencer,   "Lie equations"  ''Ann. Math. Studies'' , '''73'''  (1972)</td></tr><tr><td valign="top">[a3]</td> <td valign="top">  V. Lychagin,   V. Rubtsov,   "Topological indices of Spencer complexes that are associated with geometric structures"  ''Math. Notes'' , '''45'''  (1989)  pp. 305–312</td></tr><tr><td valign="top">[a4]</td> <td valign="top">  V. Lychagin,   L. Zilbergleit,   "Spencer cohomologies and symmetry groups"  ''Acta Applic. Math.'' , '''41'''  (1995)  pp. 227–245</td></tr><tr><td valign="top">[a5]</td> <td valign="top">  D.G. Quillen,   "Formal properties of overdetermined systems of linear partial differential equations"  ''Thesis Harvard Univ.''  (1964)</td></tr><tr><td valign="top">[a6]</td> <td valign="top">  D. Spencer,   "Overdetermined systems of linear partial differential operators"  ''Bull. Amer. Math. Soc.'' , '''75'''  (1969)  pp. 179–239</td></tr></table>