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We build an infinite dimensional scheme parametrizing isomorphism classes ofcoherent quotients of a quasi-coherent sheaf on a projective scheme. The maintool to achieve the construction is a version of Grothendieck's Grassmannianembedding combined with a result of Deligne, realizing quasi-coherent sheavesas ind-objects in the category of quasi-coherent sheaves of finitepresentation. We end our treatment with the discussion of a special case inwhich we can retain an analog of the Grassmannian embedding.

The quot functor of a quasi-coherent sheaf