The Stacks project

\begin{equation*} \DeclareMathOperator\Coim{Coim} \DeclareMathOperator\Coker{Coker} \DeclareMathOperator\Ext{Ext} \DeclareMathOperator\Hom{Hom} \DeclareMathOperator\Im{Im} \DeclareMathOperator\Ker{Ker} \DeclareMathOperator\Mor{Mor} \DeclareMathOperator\Ob{Ob} \DeclareMathOperator\Sh{Sh} \DeclareMathOperator\SheafExt{\mathcal{E}\mathit{xt}} \DeclareMathOperator\SheafHom{\mathcal{H}\mathit{om}} \DeclareMathOperator\Spec{Spec} \newcommand\colim{\mathop{\mathrm{colim}}\nolimits} \newcommand\lim{\mathop{\mathrm{lim}}\nolimits} \newcommand\Qcoh{\mathit{Qcoh}} \newcommand\Sch{\mathit{Sch}} \newcommand\QCohstack{\mathcal{QC}\!\mathit{oh}} \newcommand\Cohstack{\mathcal{C}\!\mathit{oh}} \newcommand\Spacesstack{\mathcal{S}\!\mathit{paces}} \newcommand\Quotfunctor{\mathrm{Quot}} \newcommand\Hilbfunctor{\mathrm{Hilb}} \newcommand\Curvesstack{\mathcal{C}\!\mathit{urves}} \newcommand\Polarizedstack{\mathcal{P}\!\mathit{olarized}} \newcommand\Complexesstack{\mathcal{C}\!\mathit{omplexes}} \newcommand\Pic{\mathop{\mathrm{Pic}}\nolimits} \newcommand\Picardstack{\mathcal{P}\!\mathit{ic}} \newcommand\Picardfunctor{\mathrm{Pic}} \newcommand\Deformationcategory{\mathcal{D}\!\mathit{ef}} \end{equation*}

97.3 The moduli stack of elliptic curves

Here is what we are going to do:

  1. Start with your favorite category of schemes $\mathit{Sch}$.

  2. Add a new symbol $\mathcal{M}_{1, 1}$.

  3. A morphism $S \to \mathcal{M}_{1, 1}$ is an elliptic curve $(E, f, 0)$ over $S$.

  4. A diagram

    \[ \xymatrix{ S \ar[rr]_ a \ar[rd]_{(E, f, 0)} & & S' \ar[ld]^{(E', F', 0')} \\ & \mathcal{M}_{1, 1} } \]

    is commutative if and only if there exists a morphism $\alpha : E \to E'$ of elliptic curves over $a : S \to S'$. We say $\alpha $ witnesses the commutativity of the diagram.

  5. Note that commutative diagrams glue as follows

    \[ \xymatrix{ S \ar[rrr]_ a \ar[rrrd]_{(E, f, 0)} & & & S' \ar[d]_{(E', F', 0')} \ar[rrr]_{a'} & & & S'' \ar[llld]^{(E'', F'', 0'')} \\ & & & \mathcal{M}_{1, 1} } \]

    because $\alpha ' \circ \alpha $ witnesses the commutativity of the outer triangle if $\alpha $ and $\alpha '$ witness the commutativity of the left and right triangles.

  6. The composition

    \[ S \xrightarrow {a} S' \xrightarrow {(E', f', 0')} \mathcal{M}_{1, 1} \]

    is given by $(E' \times _{S'} S, f' \times _{S'} S, 0' \times _{S'} S)$.

At the end of this procedure we have enlarged the category $\mathit{Sch}$ of schemes with exactly one object...

Except that we haven't defined what a morphism from $\mathcal{M}_{1, 1}$ to a scheme $T$ is. The answer is that it is the weakest possible notion such that compositions make sense. Thus a morphism $F : \mathcal{M}_{1, 1} \to T$ is a rule which to every elliptic curve $(E, f, 0)/S$ associates a morphism $F(E, f, 0) : S \to T$ such that given any commutative diagram

\[ \xymatrix{ S \ar[rr]_ a \ar[rd]_{(E, f, 0)} & & S' \ar[ld]^{(E', F', 0')} \\ & \mathcal{M}_{1, 1} } \]

the diagram

\[ \xymatrix{ S \ar[rr]_ a \ar[rd]_{F(E, f, 0)} & & S' \ar[ld]^{F(E', F', 0')} \\ & T } \]

is commutative also. An example is the $j$-invariant

\[ j : \mathcal{M}_{1, 1} \longrightarrow \mathbf{A}^1_{\mathbf{Z}} \]

which you may have heard of. Aha, so now we're done...

Except, no we're not! We still have to define a notion of morphisms $\mathcal{M}_{1, 1} \to \mathcal{M}_{1, 1}$. This we do in exactly the same way as before, i.e., a morphism $F : \mathcal{M}_{1, 1} \to \mathcal{M}_{1, 1}$ is a rule which to every elliptic curve $(E, f, 0)/S$ associates another elliptic curve $F(E, f, 0)$ preserving commutativity of diagrams as above. However, since I don't know of a nontrivial example of such a functor, I'll just define the set of morphisms from $\mathcal{M}_{1, 1}$ to itself to consist of the identity for now.

I hope you see how to add other objects to this enlarged category. Somehow it seems intuitively clear that given any “well-behaved” moduli problem we can perform the construction above and add an object to our category. In fact, much of modern day algebraic geometry takes place in such a universe where $\mathit{Sch}$ is enlarged with countably many (explicitly constructed) moduli stacks.

You may object that the category we obtain isn't a category because there is a “vagueness” about when diagrams commute and which combinations of diagrams continue to commute as we have to produce a witness to the commutativity. However, it turns out that this, the idea of having witnesses to commutativity, is a valid approach to $2$-categories! Thus we stick with it.

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