Homogeneous coordinate ring

In algebraic geometry, the homogeneous coordinate ring R of an algebraic variety V given as a subvariety of projective space of a given dimension N is by definition the quotient ring R = K[X0, X1, X2, ..., XN] / I where I is the homogeneous ideal defining V, K is the algebraically closed field over which V is defined, and K[X0, X1, X2, ..., XN] is the polynomial ring in N + 1 variables Xi. The polynomial ring is therefore the homogeneous coordinate ring of the projective space itself, and the variables are the homogeneous coordinates, for a given choice of basis (in the vector space underlying the projective space). The choice of basis means this definition is not intrinsic, but it can be made so by using the symmetric algebra. Since V is assumed to be a variety, and so an irreducible algebraic set, the ideal I can be chosen to be a prime ideal, and so R is an integral domain. The same definition can be used for general homogeneous ideals, but the resulting coordinate rings may then contain non-zero nilpotent elements and other divisors of zero. From the point of view of scheme theory these cases may be dealt with on the same footing by means of the Proj construction. The irrelevant ideal J generated by all the Xi corresponds to the empty set, since not all homogeneous coordinates can vanish at a point of projective space. The projective Nullstellensatz gives a bijective correspondence between projective varieties and homogeneous ideals I not containing J. In application of homological algebra techniques to algebraic geometry, it has been traditional since David Hilbert (though modern terminology is different) to apply free resolutions of R, considered as a graded module over the polynomial ring. This yields information about syzygies, namely relations between generators of the ideal I. In a classical perspective, such generators are simply the equations one writes down to define V. If V is a hypersurface there need only be one equation, and for complete intersections the number of equations can be taken as the codimension; but the general projective variety has no defining set of equations that is so transparent.

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