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Homogeneous coordinate ring
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<p>In <a href="/facts/Algebraic_geometry/rh7RHVp3">algebraic geometry</a>, the homogeneous coordinate ring is a certain <a href="/facts/Commutative_ring/QS1r2ovR">commutative ring</a> assigned to any <a href="/facts/Projective_variety/aB2XdB7B">projective variety</a>. If <i>V</i> is an <a href="/facts/Algebraic_variety/Vk7f97vn">algebraic variety</a> given as a <a href="/facts/Algebraic_variety/Vk7f97vn">subvariety</a> of <a href="/facts/Projective_space/7MSpA9cM">projective space</a> of a given dimension <i>N</i>, its homogeneous coordinate ring is by definition the <a href="/facts/Quotient_ring/X2WkXyEw">quotient ring</a> </p> <i>R</i> = <i>K</i>[<i>X</i>0, <i>X</i>1, <i>X</i>2, ..., <i>X</i><i>N</i>] / <i>I</i> <p>where <i>I</i> is the <a href="/facts/Homogeneous_ideal/AIUjkSaP">homogeneous ideal</a> defining <i>V</i>, <i>K</i> is the <a href="/facts/Algebraically_closed_field/HuVQClok">algebraically closed field</a> over which <i>V</i> is defined, and </p> <i>K</i>[<i>X</i>0, <i>X</i>1, <i>X</i>2, ..., <i>X</i><i>N</i>] <p>is the <a href="/facts/Polynomial_ring/ua3vk8Ih">polynomial ring</a> in <i>N</i> + 1 variables <i>X</i><i>i</i>. The polynomial ring is therefore the homogeneous coordinate ring of the projective space itself, and the variables are the <a href="/facts/Homogeneous_coordinates/TG9ebKN4">homogeneous coordinates</a>, for a given choice of basis (in the <a href="/facts/Vector_space/M1pxMLx2">vector space</a> underlying the projective space). The choice of basis means this definition is not intrinsic, but it can be made so by using the <a href="/facts/Symmetric_algebra/A8FthY0N">symmetric algebra</a>. </p><p>The definition mimics the <a href="/facts/Coordinate_ring/WVynz2Kn">coordinate ring</a> as it is introduced for affine varieties. </p>