Homogeneous coordinate ring

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In the algebraic geometry, the homogeneous coordinate rings R of various algebra V are given as a subvariety of the objective space of a certain dimension N with the definition of the quotient ring X ₀ , X X ₁ , X ₂ sub>]/ I

where I is an ideal homogeneous definition V , K is a closed field of algebra where V is defined, and ₀ , X ₁ , < i> X ₂ ,..., X _N ]

is the polynomial ring in N 1 variable X _i . Therefore the polynomial ring is the homogeneous coordinate ring of the projective space itself, and the variable is the homogeneous coordinate, for the given basic choice (in the vector space underlying the projective space). The basic choice means that this definition is not intrinsic, but can be made using symmetric algebra.

Video Homogeneous coordinate ring

Formulation

Since V is considered a variation, and so an irreducible collection of algebra, the ideal I can be chosen to be the primary ideal, and so R is an integral domain. The same definition can be used for general homogeneous ideals, but the resulting coordinate rings can then contain non-zero nilpotent elements and other zero dividers. From a theoretical point of view the schemes of these cases can be handled on the same footing using the Proj construction.

The correspondence between homogeneous idealism and its variation is biological to ideals that do not contain the ideal J generated by all X _i , which corresponds to the empty set because not all homogeneous coordinates can disappear at the point of the projective space. This correspondence is known as Projektif Nullstellensatz.

Maps Homogeneous coordinate ring

Resolutions and syzygies

In the application of algebraic homology techniques to algebraic geometry, it has been traditional since David Hilbert (though different modern terminology) to apply the free resolution R , is considered a graded module over the polynomial ring. This produces information about syzygies, the relationship between an ideal generator I . In a classical perspective, such generators are merely equations written to define V . If V is a hypersurface then there should be only one equation, and for complete intersection the number of equations can be taken as kodimension; but general projection does not have a very transparent set of equations. Detailed studies, for example the canonical curves and equations that define abelian varieties, show geometric interest in systematic techniques for handling these cases. The subject also grew out of the theory of elimination in its classical form, in which modulo reduction I should be an algorithmic process (now handled by the Grøbner base in practice).

There is a general reason for free resolution R as a graded module above K [ X ₀ /i> ₁ X ₂ i> ]. Resolution is defined as minimized if the images in each module modules morphism are free F _i -> F _{i> - 1}

in resolution is at JF _{i - 1} . As a consequence of lemma Nakayama? then take a certain base in the F _i to the minimal generator set in F _{i - 1} . The concept of minimal free resolution is well defined in a strong sense, in such a resolution is unique (up to complex chain isomorphism) and occurs as a direct summary in every free resolution. This property becomes intrinsic for R allowing the definition of graded numbers graded , ie? _{i, j} which is the number of grade- j images coming from F _i (more precisely, by thinking of? as a homogeneous polynomial matrix, the count of entries from a homogeneous level increased by the inductively obtained gradation from the right). In other words, the weights in all the free modules can be inferred from the resolution, and the graded number Betti calculates the number of generators of the weight given in the given resolution module. The discussion of these invariants V in the projective embedding provided is the research area, even in the case of the curve.

There are some instances where minimal free resolutions are known explicitly. For a rational normal curve it is the Eagon-Northcott complex. For elliptic curves in projective spaces, resolutions can be constructed as Eagon-Northcott conical mapping cones.

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Regularity

The Castelnuovo-Mumford crowd can be read from the ideal minimum resolution I defines projective. In the case of "shifts" that are taken into account in the module- to F _i , it is the maximum of i of a _{i , j} - i ; It is therefore small when the shift only increases by 1 increment as we move to the left in resolution (linear syzygies only).

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Projective normality

The variation V in its projective embedding is projectively normal if R is integrally closed. This condition implies that V is a normal variation, but not the other way around: the property of projective normality is not independent of project embedding, as shown by the example of the rational quartic curve in three dimensions. Another equivalent condition is in terms of the linear divisor system on V that is cut by the tautological bundle L in the projective space, and d its power-to- for d = 1, 2, 3,... Ã,; when V is non-singular, proactively normal if and only if each such linear system is a complete linear system. In a more geometric way, one can think of L as a twist twist Serre O (1) in the projective space, and use it to rotate the sheaf structure O _v k times, for any k . Then V is called k -normal if the global part O ( k ) maps retroactively with O _V ( k ), for the given k ; if V is 1-normal is called normal linear , and projective normality is a condition that V is k all k > = 1. Linear normality can be said geometrically: V as projective variation can not be obtained by isomorphic linear projection of the higher dimensional projective space except by trivial lie in the right linear subspace. Projective normality can also be translated, using sufficient Veronese mapping to reduce it to linear normality conditions.

Seeing this problem from a very large bundle viewpoint that produces a projective embedding of V , like a row bundle (sheaf can be reversed) is said normally generated if V because it is proactively embedded is normal. Projective normality is the first condition N ₀ of the order of conditions specified by Green and Lazarsfeld. For this

${\ displaystyle \ bigoplus _ {d = 0} ^ {\ infty} H ^ {0} (V, L ^ {d})}$

is considered a graded module of a homogeneous coordinate ring from the projective space, and the minimal free resolution taken. The N _p condition is applied to the first Betti number number p , which requires it to disappear when j & gt; i 1. For the Green curve it shows that the condition N _p is satisfied when deg ( L )> = 2 g 1 p , which for p = 0 is the classic result of Guido Castelnuovo.

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See also

• Various projections
• Hilbert polynomial

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Note

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References

• Oscar Zariski and Pierre Samuel, Commutative Algebra Vol. II (1960), pp.Ã, 168-172.

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