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Difference between revisions of "De Rham cohomology"

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''of an algebraic variety''
 
''of an algebraic variety''
  
A [[Cohomology|cohomology]] theory of algebraic varieties based on differential forms. To every algebraic variety <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030320/d0303201.png" /> over a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030320/d0303202.png" /> is associated a complex of regular differential forms (see [[Differential form|Differential form]] on an algebraic variety); its cohomology groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030320/d0303203.png" /> are called the de Rham cohomology groups of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030320/d0303204.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030320/d0303205.png" /> is a smooth complete variety and if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030320/d0303206.png" />, then de Rham cohomology is a special case of [[Weil cohomology|Weil cohomology]] (see [[#References|[2]]], [[#References|[3]]]). If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030320/d0303207.png" /> is a smooth affine variety and if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030320/d0303208.png" />, then the following analogue of the [[De Rham theorem|de Rham theorem]] is valid:
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A
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[[Cohomology|cohomology]] theory of algebraic varieties based on differential forms. To every algebraic variety $X$ over a field $k$ is associated a complex of regular differential forms (see
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[[Differential form|Differential form]] on an algebraic variety); its cohomology groups $H_{\text{dR}}^p (X/k)$ are called the de Rham cohomology groups of $X$. If $X$ is a smooth complete variety and if $\text{char } k=0$, then de Rham cohomology is a special case of
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[[Weil cohomology|Weil cohomology]] (see
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[[#References|[2]]],
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[[#References|[3]]]). If $X$ is a smooth affine variety and if $k = \mathbf{C}$, then the following analogue of the
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[[De Rham theorem|de Rham theorem]] is valid:
  
<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/d/d030/d030320/d0303209.png" /></td> </tr></table>
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$$H_{\text{dR}^p(X/k) \cong H^p(X^\text{an}, \mathbf{C}), \qquad p \ge 0,$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030320/d03032010.png" /> is the complex-analytic manifold corresponding to the algebraic variety <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030320/d03032011.png" /> (see [[#References|[1]]]). For example, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030320/d03032012.png" /> is the complement of an algebraic hypersurface in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030320/d03032013.png" />, then the cohomology group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030320/d03032014.png" /> can be calculated using rational differential forms on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030320/d03032015.png" /> with poles on this hypersurface.
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where $X^\text{an}$ is the complex-analytic manifold corresponding to the algebraic variety $X$ (see
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[[#References|[1]]]). For example, if $X$ is the complement of an algebraic hypersurface in $P^n(\mathbf{C})$, then the cohomology group $H^p(X, \mathbf{C})$ can be calculated using rational differential forms on $P^n(\mathbf{C})$ with poles on this hypersurface.
  
For any morphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030320/d03032016.png" /> it is possible to define the relative de Rham complex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030320/d03032017.png" /> (see [[Derivations, module of|Derivations, module of]]), which results in the relative de Rham cohomology groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030320/d03032018.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030320/d03032019.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030320/d03032020.png" /> are affine, the relative de Rham complex coincides with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030320/d03032021.png" />. The cohomology groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030320/d03032022.png" /> of the sheaf complex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030320/d03032023.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030320/d03032024.png" /> are called the relative de Rham cohomology sheaves. These sheaves are coherent on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030320/d03032025.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030320/d03032026.png" /> is a [[Proper morphism|proper morphism]].
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For any morphism $f:X\to S$ it is possible to define the relative de Rham complex $\sum_{p\ge 0} \Gamma(\Omega_{X/S}^p)$ (see
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[[Derivations, module of|Derivations, module of]]), which results in the relative de Rham cohomology groups $H^_{\text{dR}}^p (X/S)$. If $X=\text{Spec } A$ and $S=\text{Spec } B$ are affine, the relative de Rham complex coincides with $\Lambda \Omega_{A/B}^1$. The cohomology groups $\mathcal{H}_{\text{dR}}^p (X/S)$ of the sheaf complex $\sum_{p\ge 0} f \ast \Omega_{X/S}^p$ on $S$ are called the relative de Rham cohomology sheaves. These sheaves are coherent on $S$ if $f$ is a
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[[Proper morphism|proper morphism]].
  
 
====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A. Grothendieck,  "On the de Rham cohomology of algebraic varieties"  ''Publ. Math. IHES'' , '''29'''  (1966)  pp. 351–359</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  R. Hartshorne,  "Ample subvarieties of algebraic varieties" , Springer  (1970)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  R. Hartshorne,  "On the de Rham cohomology of algebraic varieties"  ''Publ. Math. IHES'' , '''45'''  (1975)  pp. 5–99</TD></TR></table>
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<table><TR><TD valign="top">[1]</TD>
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<TD valign="top">  A. Grothendieck,  "On the de Rham cohomology of algebraic varieties"  ''Publ. Math. IHES'' , '''29'''  (1966)  pp. 351–359</TD>
 +
</TR><TR><TD valign="top">[2]</TD>
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<TD valign="top">  R. Hartshorne,  "Ample subvarieties of algebraic varieties" , Springer  (1970)</TD>
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</TR><TR><TD valign="top">[3]</TD>
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<TD valign="top">  R. Hartshorne,  "On the de Rham cohomology of algebraic varieties"  ''Publ. Math. IHES'' , '''45'''  (1975)  pp. 5–99</TD>
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</TR></table>

Revision as of 01:15, 23 July 2018

of an algebraic variety

A cohomology theory of algebraic varieties based on differential forms. To every algebraic variety $X$ over a field $k$ is associated a complex of regular differential forms (see Differential form on an algebraic variety); its cohomology groups $H_{\text{dR}}^p (X/k)$ are called the de Rham cohomology groups of $X$. If $X$ is a smooth complete variety and if $\text{char } k=0$, then de Rham cohomology is a special case of Weil cohomology (see [2], [3]). If $X$ is a smooth affine variety and if $k = \mathbf{C}$, then the following analogue of the de Rham theorem is valid:

$$H_{\text{dR}^p(X/k) \cong H^p(X^\text{an}, \mathbf{C}), \qquad p \ge 0,$$

where $X^\text{an}$ is the complex-analytic manifold corresponding to the algebraic variety $X$ (see [1]). For example, if $X$ is the complement of an algebraic hypersurface in $P^n(\mathbf{C})$, then the cohomology group $H^p(X, \mathbf{C})$ can be calculated using rational differential forms on $P^n(\mathbf{C})$ with poles on this hypersurface.

For any morphism $f:X\to S$ it is possible to define the relative de Rham complex $\sum_{p\ge 0} \Gamma(\Omega_{X/S}^p)$ (see Derivations, module of), which results in the relative de Rham cohomology groups $H^_{\text{dR}}^p (X/S)$. If $X=\text{Spec } A$ and $S=\text{Spec } B$ are affine, the relative de Rham complex coincides with $\Lambda \Omega_{A/B}^1$. The cohomology groups $\mathcal{H}_{\text{dR}}^p (X/S)$ of the sheaf complex $\sum_{p\ge 0} f \ast \Omega_{X/S}^p$ on $S$ are called the relative de Rham cohomology sheaves. These sheaves are coherent on $S$ if $f$ is a proper morphism.

References

[1] A. Grothendieck, "On the de Rham cohomology of algebraic varieties" Publ. Math. IHES , 29 (1966) pp. 351–359
[2] R. Hartshorne, "Ample subvarieties of algebraic varieties" , Springer (1970)
[3] R. Hartshorne, "On the de Rham cohomology of algebraic varieties" Publ. Math. IHES , 45 (1975) pp. 5–99
How to Cite This Entry:
De Rham cohomology. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=De_Rham_cohomology&oldid=43366
This article was adapted from an original article by A.L. Onishchik (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article