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In algebraic geometry, a branch of mathematics, Serre duality is a duality present on non-singular projective algebraic varieties V of dimension n (and in greater generality for vector bundles and further, for coherent sheaves). It shows that a cohomology group Hⁱ is the dual space of another one, Hⁿ⁻ⁱ. If the variety is defined over the complex numbers, this yields different information from Poincaré duality, which relates Hⁱ to H²ⁿ⁻ⁱ, considering V as a real manifold of dimension 2n.

In the case for holomorphic vector bundle E over a smooth compact complex manifold V, the statement is in the form:

$H^q(V,E)\cong H^{n-q}(V,K\otimes E^{\ast }{)}^{\ast },$

in which V is not necessarily projective.

Algebraic curve

The case of algebraic curves was already implicit in the Riemann-Roch theorem. For a curve C the coherent groups Hⁱ vanish for i > 1; but H¹ does enter implicitly. In fact, the basic relation of the theorem involves l(D) and l(K−D), where D is a divisor and K is a divisor of the canonical class. After Serre we recognise l(K−D) as the dimension of H¹(D), where now D means the line bundle determined by the divisor D. That is, Serre duality in this case relates groups H¹(D) and H⁰(KD*), and we are reading off dimensions (notation: K is the canonical line bundle, D* is the dual line bundle, and juxtaposition is the tensor product of line bundles).

In this formulation the Riemann-Roch theorem can be viewed as a computation of the Euler characteristic of a sheaf

h⁰(D) − h¹(D),

in terms of the genus of the curve, which is

h¹(C,O_C),

and the degree of D. It is this expression that can be generalised to higher dimensions.

Serre duality of curves is therefore something very classical; but it has an interesting light to cast. For example, in Riemann surface theory, the deformation theory of complex structures is studied classically by means of quadratic differentials (namely sections of L(K²)). The deformation theory of Kunihiko Kodaira and D. C. Spencer identifies deformations via H¹(T), where T is the tangent bundle sheaf K*. The duality shows why these approaches coincide.

Origin and generalisations

The origin of the theory lies in Serre's earlier work on several complex variables. In the generalisation of Alexander Grothendieck, Serre duality becomes a part of coherent duality in a much broader setting. While the role of K above in general Serre duality is played by the determinant line bundle of the cotangent bundle, when V is a manifold, in full generality K cannot merely be a single sheaf in the absence of some hypothesis of non-singularity on V. The formulation in full generality uses a derived category and Ext functors, to allow for the fact that K is now represented by a chain complex of sheaves, namely, the dualizing complex. Nevertheless, the statement of the theorem is recognisably Serre's.

References

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• Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.
  
  Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.
  
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  Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region
  
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  15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.
  
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• Serre duality at the weblog Rigorous trivialities
• A link between Poincaré and Serre dualities via Hodge theory on Stack exchange