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I have been trying to teach myself the basics of algebraic geometry. I understand the basic premise, how we define geometry spaces (algebraic sets and schemes) in terms of commutative rings. And I understand the importance of commutative algebra, especially the study of ideals, in the understanding these spaces.

But one question I have asked a few people without a totally satisfactory reply is this: Why do modules matter in algebraic geometry? Given a boring old projective variety, what modules should I keep my eye on? Why should I care about Nakayama's lemma or about localization of modules?

Hopefully someone can give some me perspective here.

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    $\begingroup$ Ideals are modules. $\endgroup$
    – Hoot
    Jul 7 '15 at 16:05
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I am not completely sure that this constitutes a satisfactory reply, but my answer is sheaf theory. Let me motivate this from the theory of manifolds.

If we have a manifold $M$, we can associate to any open subset $U \subseteq M$ the ring $\mathcal C^\infty(U)$ of $\mathcal C^\infty$ functions on $U$. It turns out that this makes $\mathcal C^\infty$ into a sheaf on $M$ (this means what I just said, plus some 'glueing' conditions that allow us to define such functions by defining them locally: instead of writing down a function on $U$, we may write down functions on open sets $U_i$ covering $U$, and if they agree on the intersections $U_i \cap U_j$, then they give a function defined on $U$).

Other important objects in the manifold world include the tangent space $T_p M$, the $i$-forms $\Omega^i(M)$, and maybe things like tensors (e.g. curvature is a $(0,4)$-tensor, meaning that it takes $4$ tangent vectors, and spits out a real number (which we think about as $0$ tangent vectors)). It turns out that all of these carry a natural multiplication structure by $\mathcal C^\infty$ functions. If we associate to each open $U \subseteq M$ the sets $\operatorname{VF}(U)$ (vector fields), $\Omega^i(U)$, etc, then this does not just become a sheaf, but it becomes a sheaf of $\mathcal C^\infty$-modules: over every open, we get a module over $\mathcal C^\infty (U)$, with natural compatibilities.

In algebraic geometry, it is not so clear what the tangent space or the $k$-forms should be: we certainly cannot use any differentiable structure to think about these (e.g. the tangent space is defined by infinitesimal paths through a point). It is here that modules step in: we can define a module of differentials $\Omega_{A/k}$ for any $k$-algebra $A$, which plays the role of $\Omega^1(M)$. Taking alternating powers (as an $A$-module!) gives the $\Omega^i_{A/k}$, and taking duals (again over $A$) gives $T_{A/k}$.

In the algebraic world, we can associate to any module $M$ over $A$ a sheaf $\tilde M$ over $\operatorname{Spec} A$, and this association is functorial in both $A$ and $M$. Moreover, the localisation $M_f$ corresponds to restriction to the open subset where $f$ is nonzero. We can now glue the modules $\Omega_{A/k}$ to define a sheaf of differentials $\Omega_{X/k}$ for any variety $X$ over $k$. Moreover, analogously to the analytic case, this is a sheaf of $\mathcal O_X$-modules.

So one could ask: why not stick to vector bundles? In the algebraic world, these correspond to finite projective modules (or equivalently, finite flat modules). However, we quickly run into problems if we do that: for example, the cokernel of a map of projective modules need not be projective (just think about $\mathbb Z \stackrel{n}{\to} \mathbb Z$ as $\mathbb Z$-modules). So if we want to apply any sort of homological methods (which prove to be very powerful in algebraic geometry), we better have a theory that works at least for all finitely generated modules. This gives the notion of coherent sheaves.

Observe, however, that the notion of a coherent sheaf is not restricted to algebraic geometry; if we have any ringed space $(X, \mathcal O_X)$ (both varieties and manifolds are examples of this), then we can define what it means for a sheaf of $\mathcal O_X$-modules to be coherent. It is a theorem that this notion corresponds to the notion of finitely generated modules when we are working with a variety.

On the other hand, for smooth projective varieties over $\mathbb C$, Serre's GAGA paper proves that the categories of coherent modules in the algebraic and the analytic world are equivalent! Thus, our notion is truly very close to that of (holomorphic) vector bundles over a complex manifold.

As we showed in the example above, there are many natural examples of vector bundles over manifolds, and we can now define all those things on varieties as well. But there are many more uses for modules: for example, line bundles correspond to divisors (codimension $1$ subvarieties) modulo some equivalence. Ample line bundles classify morphisms into $\mathbb P^n$. So there are lots of different examples where a priori algebraic objects (modules/coherent sheaves) turn out to have important geometric interpretations.

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    $\begingroup$ +1 for a thoughtful answer to an old imprecise question. $\endgroup$ Oct 30 '15 at 0:28

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