Differential group delay

In algebraic geometry, a branch of mathematics, an adequate equivalence relation is an equivalence relation on algebraic cycles of smooth projective varieties used to obtain a well-working theory of such cycles, and in particular, well-defined intersection products. Samuel formalized the concept of an adequate equivalence relation in 1958. Since then it has become central to theory of motives. For every adequate equivalence relation, one may define the category of pure motives with respect to that relation.

Possible (and useful) adequate equivalence relations include rational, algebraic, homological and numerical equivalence. They are called "adequate" because dividing out by the equivalence relation is functorial, i.e. push-forward (with change of co-dimension) and pull-back of cycles is well-defined. Codimension one cycles modulo rational equivalence form the classical group of divisors. All cycles modulo rational equivalence form the Chow ring.

Definition

Let Z^*(X) := Z[X] be the free abelian group on the algebraic cycles of X. Then an adequate equivalence relation is a family of equivalence relations, ∼_X on Z^*(X), one for each smooth projective variety X, satisfying the following three conditions:

1. (Linearity) The equivalence relation is compatible with addition of cycles.
2. (Moving lemma) If $\alpha ,\beta \in Z^{*}(X)$ are cycles on X, then there exists a cycle $\alpha '\in Z^{*}(X)$ such that $\alpha$ ~_X $\alpha '$ and $\alpha '$ intersects $\beta$ properly.
3. (Push-forwards) Let $\alpha \in Z^{*}(X)$ and $\beta \in Z^{*}(X\times Y)$ be cycles such that $\beta$ intersects $\alpha \times Y$ properly. If $\alpha$ ~_X 0, then $(\pi _{Y})_{*}(\beta \cdot (\alpha \times Y))$ ~_Y 0, where $\pi _{Y}:X\times Y\to Y$ is the projection.

The push-forward cycle in the last axiom is often denoted

$\beta (\alpha ):=(\pi _{Y})_{*}(\beta \cdot (\alpha \times Y))$

If $\beta$ is the graph of a function, then this reduces to the push-forward of the function. The generalizations of functions from X to Y to cycles on X × Y are known as correspondences. The last axiom allows us to push forward cycles by a correspondence.

Examples of equivalence relations

The most common equivalence relations, listed from strongest to weakest, are gathered below in a table.

	definition	remarks
rational equivalence	Z ∼rat Z' if there is a cycle V on X × P1 flat over P1, such that [V ∩ X × {0}] - [V ∩ X × {∞}] = [Z] - [Z' ].	the finest adequate equivalence relation. "∩" denotes intersection in the cycle-theoretic sense (i.e. with multiplicities) and [.] denotes the cycle associated to a subscheme. see also Chow ring
algebraic equivalence	Z ∼alg Z' if there is a curve C and a cycle V on X × C flat over C, such that [V ∩ X × {c}] - [V ∩ X × {d}] = [Z] - [Z' ] for two points c and d on the curve.	strictly stronger than homological equivalence, see also Néron–Severi group
smash-nilpotence equivalence	Z ∼sn Z' if Z - Z' is smash-nilpotent on X, that is, if $(Z-Z')^{\otimes n}$ ∼rat 0 on Xn for n >> 0.	introduced by Voevodsky in 1995.
homological equivalence	for a given Weil cohomology H, Z ∼hom Z' if the image of the cycles under the cycle class map agrees	depends a priori of the choice of H, not assuming the standard conjecture D
numerical equivalence	Z ∼num Z' if deg(Z ∩ T) = deg(Z' ∩ T), where T is any cycle such that dim T = codim Z (The intersection is a linear combination of points and we add the intersection multiplicities at each point to get the degree.)	the coarsest equivalence relation

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.
  
  In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.
  
  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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