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 is a continuous map of topological spaces and is a (pre)sheaf on we obtain a (pre)sheaf on by defining, for an open set , . This is a sheaf if is itself a sheaf. This works well with (pre)sheaf morphisms and gives us a functor , which maps the subcategory of sheaves on to the subcategory of sheaves on . When restricting to the categories of sheaves, the direct image functor has a **left-adjoint**, called the inverse image functor and denoted . This functor turns sheaves on to sheaves on , and is given on open sets of by

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

- For every ,
- If is the inclusion of the open subset and is a sheaf on then .

A ringed topological space is a topological space equipped with a sheaf of rings called its structure sheaf. We say the ringed space is locally ringed in local rings if for each point the stalk is a local ring. Let’s denote the unique maximal ideal of by and write for its residue field.

A morphism from a ringed space to a ringed space is a pair where is a continous map and is a morphism of sheaves. The reason for the inclusion of the morphism into the definition is because it allows us to formalise the notion of “pulling back” functions defined on open subsets of to functions defined on open subsets of . This is because a morphism corresponds to giving a ring morphism for each open subset , compatible with restrictions.

As an example, if we think of as the sheaf of continuous functions on open subsets of (and similarly for ) then could be interpreted as the pull-back map, sending a continuous map to the composition . In practice (as with schemes), if we interpret and 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 and are *locally ringed spaces* then we also have an additional requirement on morphisms: namely, for each , the induced ring homomorphism on stalks is a **local homomorphism**. A local homomorphism means that takes the unique maximal ideal to . To get some grounding on this idea, if you interpret as the set of continuous functions on which vanish at then when you pull back each of these functions by they all vanish at . 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 as *something like* the ring of globally-defined functions on a black-box geometric object ? He called the geometric object the **spectrum** of and denoted it by . 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.

What’s up with the notation $f_*$? You didn’t mention it before now. I’ve most recently seen it in the context of pushforwards in differential geometry; were you trying to draw an analogy there?

(I just took the $f^\#_V$ comment to be the definition of $f_*$ and was able to read the rest of the post, but I’m wondering if you were quoting standard notation or were trying to hint at something.)

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Apologies, again something is not defined and in this case I’ll add it in to the post as it’s important and very standard notation. You’re right that it’s analogous to pushforwards in differential geometry – it’s called the pushforward or direct image of sheaves:

If is a continuous map of topological spaces and is a (pre)sheaf on then we get a (pre)sheaf on , called the pushforward of by and given by where is open. This actually gives us a functor from the category of presheaves on to those on , which restricts to a functor on the subcategories of sheaves.

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