4.9 Theorem on Formal Functions

Because we wind up using it, let’s quote the theorem from  [ 4 , Theorem III.11.1 ] . We need to set up some notation.

Suppose \(f:X\to Y\) is a projective morphism of noetherian schemes, \(\mathcal{F}\) is a coherent sheaf on \(X\) and \(y\in Y\) is a point. For each \(n\in \mathbb {Z}_+\), set \(X_n := X\times _Y \operatorname{\mathrm{Spec}}\mathcal{O}_{Y,y}/\mathfrak {m}_y^n\). It sits in a commutative diagram

So \(X_1\) is just the fiber \(X_y\), but \(X_n\) is a thickened version of the fiber. Then, in fact, we get a sequence of commuting inclusions \(X_1\to X_2\to \cdots X_n\to X\). So, if we write \(\mathcal{F}_n = v_n^*\mathcal{F}\), we get a sequence \(\operatorname{\mathrm{H}}^i(X,\mathcal{F})\to \operatorname{\mathrm{H}}^i(X_n, \mathcal{F}_n)\to \operatorname{\mathrm{H}}_i(X_{n-1}, \mathcal{F}_{n-1})\cdots \to \operatorname{\mathrm{H}}^i(X,\mathcal{F}_1)\). Taking the inverse limit we get \(\varprojlim \operatorname{\mathrm{H}}^i(X_n,\mathcal{F}_n)\).

On the other hand, for each \(n\), we get a base change morphism

coming from 4.35 as a canonical map \(i_n^*R^if_* \mathcal{F}\to R^if'_*v_n^*\mathcal{F}\). To see this, compute the derived pushforward via the Čech complex \(C^*(\mathfrak {U}, \mathcal{F})\) coming from an an affine cover of \(X\). Then restricting the covering to \(X_n\) gives an affine covering of \(X_n\) along with the desired map.

Now, set

If we now take the limit in 4.36 we get a morphism