My advisor showed me a definition of formal schemes as follows (acknowledging that these hypotheses may not be minimal): A formal Noetherian scheme is a sequence
$$Y_1 \hookrightarrow Y_2 \hookrightarrow Y_4 \hookrightarrow \cdots$$
of closed immersions of Noetherian schemes such that for all $i$,

(a) $(Y_i)_{red} \to (Y_{i+1})_{red}$ is an isomorphism,

(b) $\mathcal{I}_m / \mathcal{I}_m^2 \leftarrow \mathcal{I}_{m+1} / \mathcal{I}_{m+1}^2$ is an isomorphism,

where $\mathcal{I}_m$ is the coherent ideal sheaf on $Y_m$ defining the closed subscheme $Y_1$.

Morphisms from $Y = (Y_n)$ to $Z = (Z_l)$ are then specified as $$ \mathrm{Hom}(Y,Z) = \varprojlim_k \varinjlim_l \mathrm{Hom}(Y_k, Z_l). $$ Having specified morphisms, we can determine when two sets of data, even if quite distinct, give isomorphic formal schemes.

According to my advisor, this point of view is quite helpful for understanding certain results and proofs that use formal schemes in a not-entirely-essential way, such as the theorem on formal functions.

Can anyone point out a decent exposition of formal schemes from this point of view? I need to know, at least in a cursory way, about formal schemes (as a category), as well as coherent sheaves, sheaf cohomology, and higher direct images. And completions along a closed subset, of course.

easierto understand and use as (topologically) ringed spaces, not inv. systems, and theproofof thm on formal fns is way more transparent this way. Look at "FGA Explained", Ch. 8. – BCnrd Sep 13 '10 at 19:19