Let $\mathcal{F}, \mathcal{G}$ be quasi-coherent sheaves on a base scheme $S$. Then the $S$-schemes $\mathbb{P}(\mathcal{F}), \mathbb{P}(\mathcal{G})$ are defined by a universal property: to map from an $S$-scheme $T$ (with structure map $g: T \to S$) into $\mathbb{P}(\mathcal{F})$ is the same thing as giving a line bundle $\mathcal{L}$ on $T$ and a surjection $g^\*(\mathcal{F}) \to \mathcal{L}$. The Segre embedding $\mathbb{P}(\mathcal{F}) \times\_S \mathbb{P}(\mathcal{G}) \to \mathbb{P}(\mathcal{F} \otimes \mathcal{G})$ is defined in a canonical manner using this universal property. Similarly, the Plucker embedding of the Grassmannian into a projective scheme is done by giving a morphism of the functors they represent.


The Segre and Plucker embeddings are closed immersions, as is checked locally using the definitions. However, the properness of projective space $\mathbb{P}^n\_{\mathbb{Z}}$ can be easily proved using only the valuative criterion, which seems easier and more elegant (though it can also be done directly). Is there a functorial way to see that the Segre and Plucker embeddings are closed immersions?



Functorial way of showing that the Segre (or Plucker) morphism is a closed embedding?
I think this is a very interesting question. It gives another proof of the Segre embedding, makes it somehow more natural and puts it into a more general context, which I explain below.


I will just talk about the Segre embedding $\mathbb{P}^n \times \mathbb{P}^m \to \mathbb{P}^{n+m+nm}$. The general case of $\mathbb{P}(F) \times \mathbb{P}(G) \to \mathbb{P}(F \otimes G)$ is similar.


Let $x\_0,...,x\_n$ be the homogeneous coordinates of $\mathbb{P}^n$, $y\_0,...,y\_m$ the ones of $\mathbb{P}^m$ and $z\_{00},...,z\_{nm}$ the ones of $\mathbb{P}^{n+m+nm}$. The idea of the Segre embedding is $z\_{ab} = x\_a y\_b$. Observe that with this equation we can interchange $a,c$ and $b,d$ in $z\_{ab} z\_{cd}$ without changing the result. Thus, consider the homogeneous ideal generated by the $z\_{ab} z\_{cd} - z\_{cb} z\_{ad}, z\_{ab} z\_{cd} - z\_{ad} z\_{cb}$ and let $I$ denote the corresponding ideal of the structure sheaf of $\mathbb{P}^{n+m+nm}$. Now I claim that $V(I)$ and $\mathbb{P}^n \times \mathbb{P}^m$ satisfy the same universal property. So these schemes are isomorphic, and we get a closed immersion $\mathbb{P}^n \times \mathbb{P}^m \to \mathbb{P}^{n+m+nm}$.


Let $X$ be a test scheme. Then $\hom(X,\mathbb{P}^n \times \mathbb{P}^m)$ is the set of tuples $(\mathcal{L}\_1,\rho,\mathcal{L}\_2,\tau)$, where $\mathcal{L}\_1$ (resp. $\mathcal{L}\_2$) is a line bundle on $X$ and $\rho\_0,...,\rho\_n$ (resp. $\tau\_0,...,\tau\_m$) are global generators of $\mathcal{L}\_1$ (resp. $\mathcal{L}\_2$).


On the other hand, $\hom(X,V(I))$ is the set of all tuples $(\mathcal{L},\sigma)$, where $\mathcal{L}$ is a line bundles on $X$ and $\sigma\_{00},...,\sigma\_{nm}$ are global generators of $\mathcal{L}$, which satisfy the relations $\sigma\_{ab} \sigma\_{cd} = \sigma\_{cb} \sigma\_{ad}$ and $\sigma\_{ab} \sigma\_{cd} = \sigma\_{ad} \sigma\_{cb}$.


With these descriptions, there is a very natural map $\hom(X,\mathbb{P}^n \times \mathbb{P}^m) \to \hom(X,V(I))$ given by $\mathcal{L} = \mathcal{L}\_1 \otimes \mathcal{L}\_2$ and $\sigma\_{ab} = \rho\_a \otimes \tau\_b$. Actually this map makes sense in every abelian tensor category, but its bijectivity seems to be a special property of the abelian tensor category of quasi-coherent sheaves on a scheme (more generally, sheaves of modules on a ringed space). The usual proof uses the local explicit description of the classifying projective schemes, but you can do it directly by writing down an inverse map:


Let $\mathcal{L},\sigma$ be given as above. Construct a line bundle $\mathcal{L}\_1$ as follows: For every $i$, put $X\_{\rho\_i} = \cup\_j X\_{\sigma\_{ij}}$. On this open set, let $\mathcal{L}\_1$ be generated freely by a symbol $\rho\_i$. The cocycles $\rho\_a / \rho\_i$ are defined as $\sigma\_{aj} / \sigma\_{ij}$ for some $j$; the choice of this $j$ does not matter because of the relations for $\sigma$. Similarily we may define $\mathcal{L}\_2$ with global generators $\tau\_j$. It's easy to show that $\mathcal{L} \cong \mathcal{L}\_1 \otimes \mathcal{L}\_2$ with $\rho\_i \otimes \tau\_j$ corresponding to $\sigma\_{ij}$.


