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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]].
 
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130480/s1304801.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130480/s1304802.png" /> be smooth vector bundles (cf. also [[Vector bundle|Vector bundle]]) and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130480/s1304803.png" /> be a linear differential operator acting from the module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130480/s1304804.png" /> of smooth sections of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130480/s1304805.png" /> to the module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130480/s1304806.png" />. Denote by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130480/s1304807.png" /> the sheaf of solutions of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130480/s1304808.png" />. To find the cohomology of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130480/s1304809.png" /> with coefficients in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130480/s13048010.png" /> one needs a [[Resolvent|resolvent]] of the sheaf.
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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
 
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 valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130480/s13048011.png" /></td> </tr></table>
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<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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130480/s13048012.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130480/s13048013.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130480/s13048014.png" />. 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130480/s13048015.png" /> a canonical construction ([[#References|[a5]]], [[#References|[a6]]], [[#References|[a1]]]) of a complex, called the second (or sophisticated) Spencer complex. In this complex,
+
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 valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130480/s13048016.png" /></td> </tr></table>
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<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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130480/s13048017.png" /> have the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130480/s13048018.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130480/s13048019.png" /> are prolongations of the differential equation corresponding to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130480/s13048020.png" /> (cf. also [[Prolongation of solutions of differential equations|Prolongation of solutions of differential equations]]) and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130480/s13048021.png" /> are the symbols of these prolongations (cf. also [[Symbol of an operator|Symbol of an operator]]). The differential operators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130480/s13048022.png" /> are first-order partial differential operators whose symbols are induced by the exterior multiplication.
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130480/s13048024.png" />-Poincaré lemma [[#References|[a6]]] shows that the [[Cohomology|cohomology]] of the complex does not depend on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130480/s13048025.png" /> when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130480/s13048026.png" /> is large enough. The stable cohomology <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130480/s13048027.png" /> is called the Spencer cohomology of the differential operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130480/s13048028.png" />.
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130480/s13048029.png" />; however, it does in certain special cases, e.g. when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130480/s13048030.png" /> is analytical operator [[#References|[a6]]].
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130480/s13048031.png" />, and the Dolbeault cohomology corresponds to the Cauchy–Riemann <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130480/s13048033.png" />-operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130480/s13048034.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130480/s13048035.png" /> is a determined operator such that not all covectors are characteristic, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130480/s13048036.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130480/s13048037.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130480/s13048038.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130480/s13048039.png" />. In general, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130480/s13048040.png" /> for each formally integrable operators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130480/s13048041.png" />.
+
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.
 
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130480/s13048042.png" /> is an elliptic partial differential operator (cf. [[Elliptic partial differential equation|Elliptic partial differential equation]]) and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130480/s13048043.png" /> is a compact [[Manifold|manifold]], then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130480/s13048044.png" /> and the [[Euler characteristic|Euler characteristic]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130480/s13048045.png" /> of the Spencer complex is called the index of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130480/s13048046.png" /> (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]]]).
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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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130480/s13048047.png" /> satisfies certain extra conditions.
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130480/s13048048.png" /> is the total space of a smooth bundle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130480/s13048049.png" /> over a simply-connected manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130480/s13048050.png" /> and if the fibres of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130480/s13048051.png" /> are not characteristic for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130480/s13048052.png" />, then there exists a spectral sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130480/s13048053.png" /> converging to the Spencer cohomology <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130480/s13048054.png" />; its second term is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130480/s13048055.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130480/s13048056.png" /> is the fibrewise differential operator corresponding to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130480/s13048057.png" /> [[#References|[a4]]].
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130480/s13048058.png" /> is a homogeneous manifold and the structure group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130480/s13048059.png" /> is a compact connected [[Lie group|Lie group]] of symmetries of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130480/s13048060.png" />, then [[#References|[a4]]] the Spencer cohomology <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130480/s13048061.png" /> coincides with the cohomology of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130480/s13048062.png" />-invariant Spencer complex if the non-trivial characters of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130480/s13048063.png" /> are non-characteristic.
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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====
 
====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>
+
<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>

Revision as of 16:57, 1 July 2020

The de Rham cohomology and Dolbeault cohomology (cf. Duality in complex analysis) can be viewed as cohomologies with coefficients in the 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.

Namely, let $\alpha : E ( \alpha ) \rightarrow M$ and $\beta : E ( \beta ) \rightarrow M$ be smooth vector bundles (cf. also 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 of the sheaf.

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

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 ([a5], [a6], [a1]) of a complex, called the second (or sophisticated) Spencer complex. In this complex,

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) and $g _ { m }$ are the symbols of these prolongations (cf. also 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 [a6] shows that the 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 [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 [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) and $M$ is a compact manifold, then $\operatorname { dim} H _ { S } ^ { i } ( D ) < \infty$ and the 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 theory). For elliptic Lie equations the index can be expressed in terms of characteristic classes corresponding to the geometrical structure ([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) 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 $D$ [a4].

If $M = G / G_0$ is a homogeneous manifold and the structure group $G$ is a compact connected Lie group of symmetries of $D$, then [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

[a1] H. Goldschmidt, "Existence theorems for analytic linear partial differential equations" Ann. Math. , 86 (1967) pp. 246–270
[a2] A. Kumpera, D. Spencer, "Lie equations" Ann. Math. Studies , 73 (1972)
[a3] V. Lychagin, V. Rubtsov, "Topological indices of Spencer complexes that are associated with geometric structures" Math. Notes , 45 (1989) pp. 305–312
[a4] V. Lychagin, L. Zilbergleit, "Spencer cohomologies and symmetry groups" Acta Applic. Math. , 41 (1995) pp. 227–245
[a5] D.G. Quillen, "Formal properties of overdetermined systems of linear partial differential equations" Thesis Harvard Univ. (1964)
[a6] D. Spencer, "Overdetermined systems of linear partial differential operators" Bull. Amer. Math. Soc. , 75 (1969) pp. 179–239
How to Cite This Entry:
Spencer cohomology. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Spencer_cohomology&oldid=50201
This article was adapted from an original article by Valentin Lychagin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article