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==Geometrical interpretation==
In the preceding formulation, the [[polynomial ring]] <math>R[\mathbf x]=R[x_1, \ldots, x_n]</math> defines a [[morphism of schemes|morphism]] of [[Scheme (mathematics)|schemes]] (which are algebraic varieties if {{mvar|R}} is finitely generated over a field)
:<math>\mathbb{P}^{n-1}_R = \operatorname{Proj}(R[\mathbf x]) \to \operatorname{Spec}(R).</math>
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*{{cite book|last=Milne|first=James S.|title=The Abel Prize 2008–2012|chapter=The Work of John Tate|publisher=Springer|year=2014|isbn=9783642394492|author-link=James Milne (mathematician)}}
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