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| {{for|Grothendieck's algebraic de Rham cohomology|Crystalline cohomology}}
| | A lagging computer is absolutely annoying and is very a headache. Almost every individual that utilizes a computer faces this problem some time or the additional. If a computer additionally suffers within the same issue, there are it difficult to continue functioning as usual. In such a condition, the thought, "what must I do to create my PC run faster?" is repeated and infuriating. There's a solution, nevertheless!<br><br>Before actually ordering the software it is best to check found on the firms that create the software. If you will discover details on the kind of standing every company has, perhaps the risk of malicious programs is reduced. Software from reputed businesses have helped me, and other consumers, to create my PC run faster.. If the product description does not look wise to we, refuses to include details regarding the software, refuses to include the scan functions, you need to go for another 1 that ensures you're paying for what we need.<br><br>It doesn't matter whether we are not especially obvious regarding what rundll32.exe is. However remember that it plays an important character in preserving the stability of the computers plus the integrity of the system. Whenever some software or hardware couldn't reply usually to your program operation, comes the rundll32 exe error, that may be caused by corrupted files or lost data inside registry. Usually, error content might shows up at booting or the beginning of running a system.<br><br>Windows errors is caused by any amount of reasons, however, there's virtually constantly 1 cause. There's a hidden piece of the system that is responsible for making 90% of all Windows errors, and it's called the 'registry'. This really is the central database for your system plus is where a computer stores all its system files and settings. It's a truly important piece of Windows, that is should be capable to function. However, it's additionally one of the largest causes of difficulties on your PC.<br><br>There are several [http://bestregistrycleanerfix.com/registry-reviver registry reviver] software pieces in the internet and the only thing that we should do is to download them. Unfortunately, you can not expect that all of these are as efficient because they claim to be. And due to this, it's important for you to check when a vendor is certainly reliable plus credible.<br><br>The many probable cause of your trouble is the program issue - Registry Errors! That is the reason why individuals who already have over 2 G RAM on their computers are nonetheless consistently bothered by the issue.<br><br>We need a choice to automatically delete unwelcome registry keys. This might protect you hours of laborious checking by your registry keys. Automatic deletion is a key element whenever we compare registry products.<br><br>Often the greatest technique is to read reviews on them plus if many users remark regarding its efficiency, it is likely to be function. The best piece is that there are many top registry products which work; you really have to take a pick. |
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| In [[mathematics]], '''de Rham cohomology''' (after [[Georges de Rham]]) is a tool belonging both to [[algebraic topology]] and to [[differential topology]], capable of expressing basic topological information about [[smooth manifold]]s in a form particularly adapted to computation and the concrete representation of [[cohomology class]]es. It is a [[cohomology theory]] based on the existence of [[differential form]]s with prescribed properties.
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| ==Definition==
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| The '''de Rham complex''' is the [[cochain complex]] of [[exterior differential form]]s on some [[smooth manifold]] ''M'', with the [[exterior derivative]] as the differential.
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| :<math>0 \to \Omega^0(M)\ \stackrel{d}{\to}\ \Omega^1(M)\ \stackrel{d}{\to}\ \Omega^2(M)\ \stackrel{d}{\to}\ \Omega^3(M) \to \cdots</math>
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| where Ω<sup>0</sup>(''M'') is the space of smooth functions on ''M'', Ω<sup>1</sup>(''M'') is the space of 1-forms, and so forth. Forms which are the image of other forms under the [[exterior derivative]], plus the constant 0 function in <math>\Omega^0(M)</math> are called '''exact''' and forms
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| whose exterior derivative is 0 are called '''closed''' (see [[closed and exact differential forms]]); the relationship <math> d^{2}= 0 </math> then says that exact forms are closed.
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| The converse, however, is not in general true; closed forms need not be exact. A simple but significant case is the 1-form of angle measure on the [[unit circle]], written conventionally as dθ (described at [[closed and exact differential forms]]). There is no actual function θ defined on the whole circle of which dθ is the derivative; the increment of 2π in going once round the circle in the positive direction means that we can't take a single-valued θ. We can, however, change the topology by removing just one point.
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| The idea of de Rham cohomology is to classify the different types of closed forms on a manifold. One performs this classification by saying that two closed forms α and β in <math>\Omega^k(M)</math> are '''cohomologous''' if they differ by an exact form, that is, if <math>\alpha-\beta</math> is exact. This classification induces an equivalence relation on the space of closed forms in <math>\Omega^k(M)</math>. One then defines the <math>k</math>-th '''de Rham cohomology group''' <math>H^{k}_{\mathrm{dR}}(M)</math> to be the set of equivalence classes, that is, the set of closed forms in <math>\Omega^k(M)</math> modulo the exact forms.
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| Note that, for any manifold ''M'' with ''n'' [[Connected space|connected components]]
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| :<math>H^{0}_{\mathrm{dR}}(M) \cong \mathbf{R}^n. </math>
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| This follows from the fact that any smooth function on M with zero derivative (i.e. locally constant) is constant on each of the connected components of ''M''.
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| ==De Rham cohomology computed==
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| One may often find the general de Rham cohomologies of a manifold using the above fact about the zero cohomology and a [[Mayer–Vietoris sequence]]. Another useful fact is that the de Rham cohomology is a [[homotopy]] invariant. While the computation is not given, the following are the computed de Rham cohomologies for some common [[topological]] objects:
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| '''The ''n''-sphere:'''
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| For the [[n-sphere|''n''-sphere]], and also when taken together with a product of open intervals, we have the following. Let ''n'' > 0, ''m'' ≥ 0, and ''I'' an open real interval. Then
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| :<math>H_{\mathrm{dR}}^{k}(S^n \times I^m) \simeq \begin{cases} \mathbf{R} & \mbox{if } k = 0,n, \\ 0 & \mbox{if } k \ne 0,n. \end{cases}</math>
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| '''The ''n''-torus:''' | |
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| Similarly, allowing ''n'' > 0 here, we obtain
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| :<math>H_{\mathrm{dR}}^{k}(T^n) \simeq \mathbf{R}^{n \choose k}.</math>
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| '''Punctured Euclidean space:'''
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| Punctured Euclidean space is simply [[Euclidean space]] with the origin removed. For ''n'' > 0, we have:
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| :{|
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| |<math>H_{\mathrm{dR}}^{k}(\mathbf{R}^n \setminus \{0\})</math>
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| |<math>\simeq \begin{cases} \mathbf{R} & \mbox{if } k = 0,n-1 \\ 0 & \mbox{if } k \ne 0,n-1 \end{cases}</math>
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| |-
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| |<math>\simeq H_{\mathrm{dR}}^{k}(S^{n-1}).</math>
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| |}
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| '''The Möbius strip, M:'''
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| This follows from the fact that the [[Möbius strip]] can be [[deformation retract]]ed to the 1-sphere:
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| :<math>H_{\mathrm{dR}}^{k}(M) \simeq H_{\mathrm{dR}}^{k}(S^1).</math>
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| ==De Rham's theorem==
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| [[Stokes' theorem]] is an expression of [[duality (mathematics)|duality]] between de Rham cohomology and the [[homology (mathematics)|homology]] of [[Chain (algebraic topology)|chains]]. It says that the pairing of differential forms and chains, via integration, gives a [[homomorphism]] from de Rham cohomology <math>H^{k}_{\mathrm{dR}}(M)</math> to [[singular cohomology|singular cohomology group]]s ''H<sup>k</sup>(M; '''R''')''. '''De Rham's theorem''', proved by [[Georges de Rham]] in 1931, states that for a smooth manifold ''M'', this map is in fact an isomorphism.
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| More precisely, consider the map <math>I; H_{dR}^p(M) \rightarrow H^p(M; \mathbb{R})</math> defined as follows.
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| For any <math>[\omega] \in H_{dR}^p(M)</math>, let <math>I(\omega)</math> be the element of <math>Hom(H_p(M; \mathbb{R}), \mathbb{R}) \simeq H^p(M; \mathbb{R})</math> that takes <math>[c] \in H_p(M)</math> to <math>\int_c \omega</math>. The theorem of de Rham asserts that this is an isomorphism between de Rham cohomology and singular cohomology.
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| The [[wedge product]] endows the [[Direct sum of groups|direct sum]] of these groups with a [[ring (mathematics)|ring]] structure. A further result of the theorem is that the two [[cohomology ring]]s are isomorphic (as [[graded ring]]s), where the analogous product on singular cohomology is the [[cup product]].
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| ==Sheaf-theoretic de Rham isomorphism==
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| The de Rham cohomology is [[isomorphic]] to the [[Čech cohomology]] ''H''<sup>*</sup>('''U''',''F''), where ''F'' is the [[sheaf (mathematics)|sheaf]] of [[abelian group]]s determined by ''F''(''U'') = '''R''' for all connected open sets ''U'' in ''M'', and for open sets ''U'' and ''V'' such that ''U'' ⊂ ''V'', the group morphism res<sub>V,U</sub> : ''F(V)'' → ''F(U)'' is given by the identity map on '''R''', and where '''U''' is a good [[open cover]] of ''M'' (''i.e.'' all the open sets in the open cover '''U''' are [[Contractible space|contractible]] to a point, and all finite intersections of sets in '''U''' are either empty or contractible to a point).
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| Stated another way, if ''M'' is a compact [[differentiability class|''C''<sup>''m+1''</sup>]] manifold of dimension ''m'', then for each ''k''≤''m'', there is an isomorphism
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| :<math>H^k_{\mathrm{dR}}(M)\cong \check{H}^k(M,\mathbf{R})</math>
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| where the left-hand side is the ''k''-th de Rham cohomology group and the right-hand side is the Čech cohomology for the [[constant sheaf]] with fibre '''R'''.
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| ===Proof===
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| Let Ω<sup>''k''</sup> denote the [[sheaf (mathematics)|sheaf of germs]] of ''k''-forms on ''M'' (with Ω<sup>0</sup> the sheaf of ''C''<sup>''m'' + 1</sup> functions on ''M''). By the [[Poincaré lemma]], the following sequence of sheaves is exact (in the [[category (mathematics)|category]] of sheaves):
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| :<math>0 \to \mathbf{R} \to \Omega^0 \,\xrightarrow{d}\, \Omega^1 \,\xrightarrow{d}\, \Omega^2\,\xrightarrow{d} \dots \xrightarrow{d}\, \Omega^m \to 0.</math>
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| This sequence now breaks up into [[short exact sequence]]s
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| :<math>0 \to d\Omega^{k-1} \,\xrightarrow{\mathrm{incl}}\, \Omega^k \,\xrightarrow{d}\, d\Omega^k\to 0.</math>
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| Each of these induces a [[long exact sequence]] in cohomology.
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| Since the sheaf of ''C''<sup>''m'' + 1</sup> functions on a manifold admits [[partition of unity|partitions of unity]], the sheaf-cohomology ''H''<sup>''i''</sup>(Ω<sup>''k''</sup>) vanishes for ''i''>0. So the long exact cohomology sequences themselves ultimately separate into a chain of isomorphisms. At one end of the chain is the Čech cohomology and at the other lies the de Rham cohomology.
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| ==Related ideas==
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| The de Rham cohomology has inspired many mathematical ideas, including [[Dolbeault cohomology]], [[Hodge theory]], and the [[Atiyah-Singer index theorem]]. However, even in more classical contexts, the theorem has inspired a number of developments. Firstly, the [[Hodge theory]] proves that there is an isomorphism between the cohomology consisting of harmonic forms and the de Rham cohomology consisting of closed forms modulo exact forms. This relies on an appropriate definition of '''harmonic forms''' and of '''the Hodge theorem'''. For further details see [[Hodge theory]].
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| ===Harmonic forms===
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| {{see also|Harmonic differential}}
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| If <math>M</math> is a [[Compact space|compact]] [[Riemannian manifold]], then each equivalence class in <math> H^{k}_{\mathrm{dR}}(M) </math> contains exactly one [[harmonic form]]. That is, every member ω of a given equivalence class of closed forms can be written as
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| :<math>\omega = d\alpha+\gamma \,</math> | |
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| where <math>\alpha</math> is some form, and γ is harmonic: Δγ=0.
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| Any [[harmonic function]] on a compact connected Riemannian manifold is a constant. Thus, this particular representative element can be understood to be an extremum (a minimum) of all cohomologously equivalent forms on the manifold. For example, on a 2-[[torus]], one may envision a constant 1-form as one where all of the "hair" is combed neatly in the same direction (and all of the "hair" having the same length). In this case, there are two cohomologically distinct combings; all of the others are linear combinations. In particular, this implies that the 1st [[Betti number]] of a two-torus is two. More generally, on an ''n''-dimensional torus ''T''<sup>n</sup>, one can consider the various combings of ''k''-forms on the torus. There are ''n'' choose ''k'' such combings that can be used to form the basis vectors for <math>H^k_{\text{dR}}(T^n)</math>; the ''k''-th Betti number for the de Rham cohomology group for the ''n''-torus is thus ''n'' choose ''k''.
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| More precisely, for a [[differential manifold]] ''M'', one may equip it with some auxiliary [[Riemannian metric]]. Then the [[Laplacian]] Δ is defined by
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| :<math>\Delta=d\delta+\delta d \,</math>
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|
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| with ''d'' the [[exterior derivative]] and δ the [[codifferential]]. The Laplacian is a homogeneous (in [[graded algebra|grading]]) [[linear]] [[differential operator]] acting upon the [[exterior algebra]] of [[differential form]]s: we can look at its action on each component of degree ''k'' separately.
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| If ''M'' is [[Compact space|compact]] and [[oriented]], the [[dimension]] of the [[kernel (algebra)|kernel]] of the Laplacian acting upon the space of [[differential form|k-form]]s is then equal (by [[Hodge theory]]) to that of the de Rham cohomology group in degree ''k'': the Laplacian picks out a unique ''harmonic'' form in each cohomology class of [[closed form (calculus)|closed form]]s. In particular, the space of all harmonic ''k''-forms on ''M'' is isomorphic to ''H<sup>k</sup>''(''M'';'''R'''). The dimension of each such space is finite, and is given by the ''k''-th [[Betti number]].
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| ===Hodge decomposition===
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| Letting <math>\delta</math> be the [[codifferential]], one says that a form <math>\omega</math> is '''co-closed''' if <math>\delta\omega=0</math> and '''co-exact''' if <math>\omega=\delta\alpha</math> for some form <math>\alpha</math>. The '''Hodge decomposition''' states that any ''k''-form can be split into three [[lp space|L<sup>2</sup>]] components:
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| :<math>\omega = d\alpha +\delta \beta + \gamma \,</math>
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| where <math>\gamma</math> is harmonic: <math>\Delta\gamma=0</math>. This follows by noting that exact and co-exact forms are orthogonal; the orthogonal complement then consists of forms that are both closed and co-closed: that is, of harmonic forms. Here, orthogonality is defined with respect to the [[lp space|L<sup>2</sup>]] inner product on <math>\Omega^k(M)</math>:
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| :<math>(\alpha,\beta)=\int_M \alpha \wedge *\beta.</math>
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| A precise definition and proof of the decomposition requires the problem to be formulated on [[Sobolev space]]s. The idea here is that a Sobolev space provides the natural setting for both the idea of [[Square-integrable function|square-integrability]] and the idea of differentiation. This language helps overcome some of the limitations of requiring compact support.
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| ==See also==
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| * [[Hodge theory]]
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| ==References==
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| * {{Citation | last1=Bott | first1=Raoul | author1-link=Raoul Bott | last2=Tu | first2=Loring W. | title=Differential Forms in Algebraic Topology | publisher=[[Springer-Verlag]] | location=Berlin, New York | isbn=978-0-387-90613-3 | year=1982}}
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| * {{Citation | last1=Griffiths | first1=Phillip | author1-link=Phillip Griffiths | last2=Harris | first2=Joseph | author2-link=Joe Harris (mathematician) | title=Principles of algebraic geometry | publisher=[[John Wiley & Sons]] | location=New York | series=Wiley Classics Library | isbn=978-0-471-05059-9 | mr=1288523 | year=1994}}
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| * {{Citation | last1=Warner | first1=Frank | title=Foundations of Differentiable Manifolds and Lie Groups | publisher=[[Springer-Verlag]] | location=Berlin, New York | isbn=978-0-387-90894-6 | year=1983}}
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| ==External links==
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| * {{springer|title=De Rham cohomology|id=p/d030320}}
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| {{DEFAULTSORT:De Rham Cohomology}}
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| [[Category:Cohomology theories]]
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| [[Category:Differential forms]]
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| [[Category:Homology theory]]
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