Coproduct decomposition of the quot scheme functor

Question

I am reading FGA explained and stuck on this argument: The functor $\mathfrak{Q}uot_{E/X/S}$ naturally decomposes as a co-product

$$\mathfrak{Q}uot_{E/X/S}=\coprod_{\Phi\in \mathbb{Q}[\lambda]} \mathfrak{Q}uot_{E/X/S}^{\Phi,L}$$

(Here is Nitsure's article where you can find the definition of quot scheme)

I am familiar with the definition of coproduct of objects in a category, but confused about how is coproduct defined for functors. I would also like to know why the above statement is true.

Thanks!

Answer 1

The "naive" coproduct of functors where you define $(\coprod_i F_i)(T)=\coprod_i F_i(T)$ in not a sheaf in general. If you sheafify, (at least in the case you're interested in) you should get a description like this: an element of $(\overline{\coprod_i F_i})(T)$ (the sheafification) is a decomposition $T_i$ of $T$ as a disjoint union $T=\coprod_i T_i$ (same indices) plus a family of elements $\xi_i\in F_i(T_i)$.

Since the Quot scheme is a sheaf, you should interpret that coproduct as the "sheaf coproduct". In this case, the decomposition as a disjoint union corresponds to the fact that the Hilbert polynomial is locally constant.