Schemes associated to vector spaces - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-19T16:35:03Z http://mathoverflow.net/feeds/question/96760 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/96760/schemes-associated-to-vector-spaces Schemes associated to vector spaces Harry 2012-05-12T10:29:22Z 2012-05-12T17:45:04Z <p>Let $k$ be a field. Let $F$ be a covariant functor on the category of $k$-algebras to the category of sets. Assume that the opposite functor $F^{op}$ on the category of affine $k$-schemes is a sheaf. (This last assumption might not be necessary.)</p> <p>Now, suppose that $F(R)$ is a finitely generated $R$-module for all $k$-algebras $R$ and that </p> <p>$F(R) = F(k)\otimes_k R$.</p> <p>It seems that in this case $F$ is "representable" by the vector space $$F(k).$$ </p> <p>I don't know what this means and I'm probably missing some very standard construction here. It should mean that $F^{op}$ is representable by a finite type $k$-scheme.</p> <p>I know one can associate to a finite-dimensional vector space $E$ over $k$ the affine variety $\mathrm{Spec} \ \mathrm{Sym}(E).$ Is it clear that in this case the functor $F$ is representable by the affine variety $\mathrm{Spec} \ \mathrm{Sym}(E)$?</p> http://mathoverflow.net/questions/96760/schemes-associated-to-vector-spaces/96762#96762 Answer by Martin Brandenburg for Schemes associated to vector spaces Martin Brandenburg 2012-05-12T10:42:42Z 2012-05-12T17:45:04Z <p>Since $F(k) \cong k^n$ for some $n$, we have $F(R) \cong R^n$, naturally in $R$. But $R^n \cong (\mathbb{A}^n)(R)$, so that $F$ is just the scheme $\mathbb{A}^n$. This is also isomorphic to $\mathrm{Spec}(\mathrm{Sym}(F(k))$.</p> <p>EDIT: More generally and coordinate free: When $S$ is some base scheme and $\mathcal{E}$ is some quasi-coherent module over $S$, then $\mathbb{V}(\mathcal{E}^*) := \mathrm{Spec}(\mathrm{Sym}(\mathcal{E}))$ represents the contravariant functor which sends an $S$-scheme $p : T \to S$ the set of $\mathcal{O}_S$-homomorphisms <code>$\mathcal{E} \to p_* \mathcal{O}_T$</code>. When $S=\mathrm{Spec}(k)$ is affine, this is the set of $k$-module homomorphisms $\Gamma(S,\mathcal{E}) \to \Gamma(T,\mathcal{O}_T)$. For example, $\mathbb{V}(\mathcal{O}_S^n) = \mathbb{A}^n_S$.</p> <p>Now if $F$ is as in your question, we see that $F$ represents the same functor as $\mathbb{V}(F(k))$.</p>