1. Let be a commutative ring, be a multiplicatively closed subset in the sense that . Then we consider the localization as the quotient , where if for some .

Let . We can construct a m.c.subset , and denote the corresponding local ring by .

Let be a prime ideal of . Then is m.c. We denote the corresponding local ring by .

Let be the set of all prime ideals of . For each ideal , let . The Zariski topology on is defined that the closed subsets are exactly .

A basis for the Zariski topology on can be constructed as follows. For each , let to be the set of prime ideals not containing . Then each is open.

The points corresponding to maximal ideals are closed points in the sense that the singleton .

In the case , we see that each maximal ideal corresponds to a point . So one can interprat this as . A non-max prime ideal (a non-closed point) corresponds an affine variety , which is a closed subset in . Then is called the generic point of the varity .

2. Let be a symplectic manifold, be a Lie group acting on via symplectic diffeomorphisms. Let be the Lie algebra of . Each induces a vector field . Note that , and .

Consider the 1-form induced by the contraction . Clearly this 1-form is closed: since preserves the form .

Then the action is called weakly Hamiltonian, if for every , the one-form is exact: for some smooth function on . Although is only determined up to a constant , the constant can be chosen such that the map becomes linear.

The action is called Hamiltonian, if the map , is a Lie algebra homomorphism with respect to Poisson structure. Then and .

A moment map for a Hamiltonian -action on is a map such that for all . In other words, for each fixed point , the map from to is a linear functional on and is denoted by . Also note that . So .

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