Now let’s introduce a stepping-stone from sheaves to schemes.

Edit: I was notified that I haven’t explained the direct image of sheaves construction! So briefly, if $f:X\rightarrow Y$ is a continuous map of topological spaces and $F$ is a (pre)sheaf on $X$ we obtain a (pre)sheaf $f_* F$ on $Y$ by defining, for an open set $U\subseteq Y$, $f_* F (U) = F(f^{-1} (U))$. This is a sheaf if $F$ is itself a sheaf. This works well with (pre)sheaf morphisms and gives us a functor $f_* : \text{PSh}(X)\rightarrow \text{PSh}(Y)$, which maps the subcategory of sheaves on $X$ to the subcategory of sheaves on $Y$. When restricting to the categories of sheaves, the direct image functor has a left-adjoint, called the inverse image functor and denoted $f^{-1}$. This functor turns sheaves $G$ on $Y$ to sheaves $f^{-1} G$ on $X$, and is given on open sets of $X$ by

$\left(U\mapsto \text{colim}_{V\supseteq f(U)} G(V)\right)^+$

The inverse image functor won’t be used for a while, but I will note some nice properties:

1. For every $x\in X$, $(f^{-1} G)_x \cong G_{f(x)}$
2. If $i: X\hookrightarrow Y$ is the inclusion of the open subset $X\subseteq Y$ and $G$ is a sheaf on $Y$ then $i^{-1} G = G \vert_X$.

A ringed topological space $(X, \mathcal{O}_X)$ is a topological space $X$ equipped with a sheaf of rings $\mathcal{O}_X$ called its structure sheaf. We say the ringed space is locally ringed in local rings if for each point $x\in X$ the stalk $\mathcal{O}_{X, x}$ is a local ring. Let’s denote the unique maximal ideal of $\mathcal{O}_{X,x}$ by $\mathfrak{m}_x$ and write $k(x)=\mathcal{O}_{X,x}/\mathfrak{m}_x$ for its residue field.

A morphism from a ringed space $(X,\mathcal{O}_X)$ to a ringed space $(Y, \mathcal{O}_Y)$ is a pair $(f,f^\#)$ where $f:X\rightarrow Y$ is a continous map and $f^\# : \mathcal{O}_Y \rightarrow f_* \mathcal{O}_X$ is a morphism of sheaves. The reason for the inclusion of the morphism $f^\#$ into the definition is because it allows us to formalise the notion of “pulling back” functions defined on open subsets of $Y$ to functions defined on open subsets of $X$. This is because a morphism $f^\# : \mathcal{O}_Y \rightarrow f_* \mathcal{O}_X$ corresponds to giving a ring morphism $f^\#_V : \mathcal{O}_Y (V) \rightarrow \mathcal{O}_X (f^{-1} (V))$ for each open subset $V\subseteq Y$, compatible with restrictions.

As an example, if we think of $\mathcal{O}_Y (V) = C (V) = \left\{\text{continuous maps } g: V\rightarrow \mathbb{R}\right\}$ as the sheaf of continuous functions on open subsets of $Y$ (and similarly for $\mathcal{O}_X$) then $f^\#_V$ could be interpreted as the pull-back map, sending a continuous map $g: V\rightarrow\mathbb{R}$ to the composition $g\circ f : f^{-1} (V) \rightarrow \mathbb{R}$. In practice (as with schemes), if we interpret $\mathcal{O}_X$ and $\mathcal{O}_Y$ as abstract sheaves of rings then this pull-back idea needs to be taken with a pinch of salt. But it does allow us to connect the two structure sheaves via the continuous map in a natural way.

If $(X,\mathcal{O}_X)$ and $(Y,\mathcal{O}_Y)$ are locally ringed spaces then we also have an additional requirement on morphisms: namely, for each $x\in X$, the induced ring homomorphism on stalks $f^\#_x : \mathcal{O}_{Y, f(x)} \rightarrow \mathcal{O}_{X,x}$ is a local homomorphism. A local homomorphism means that $f^\#_x$ takes the unique maximal ideal $\mathfrak{m}_{f(x)}$ to $\mathfrak{m}_x$. To get some grounding on this idea, if you interpret $\mathfrak{m}_{f(x)}$ as the set of continuous functions on $Y$ which vanish at $f(x)$ then when you pull back each of these functions by $f$ they all vanish at $x$. When we consider the structure sheaves as abstract rings instead of just rings of continuous functions on our spaces, including this condition allows us to think of our structure sheaves as if they were rings of functions on a space.

In fact, this is why Grothendieck’s conception of schemes is so powerful and was so revolutionary: in classical algebraic geometry, dealing with varieties, one started with an algebraic variety and then looked at its “ring of regular functions” (and those defined on open subsets). The algebraic object here – the ring – was secondary to the geometric object – the variety. In contrast, Grothendieck essentially started his theory of schemes by thinking: what if we can realise any commutative ring $R$ as *something like* the ring of globally-defined functions on a black-box geometric object $X$? He called the geometric object $X$ the spectrum of $R$ and denoted it by $\text{Spec} (R)$. As we will see later when I formally introduce schemes, they are locally ringed spaces locally patched together from these ring spectra, and thus are geometric objects on which their rings act like rings of functions on more conventional geometric spaces.