A few very elementary (yet very important) lemmas in algebraic geometry

I want to go over a few important and useful lemmas in algebraic geometry. These things are very basic, but I wanted to write them down somewhere.

On the way to proving the equivalence of categories \(\text{AffSch} \simeq \text{CRing}^{\text{op}}\), one may prove a stronger result proving a bijection of the hom-sets \(\text{Hom}(X, \text{Spec}(A))\) and \(\text{Hom}(A, \mathcal{O}_X(X))\) where \(X\) is any scheme, not necessarily and affine one. The result is somewhat obvious in the case \(X = \text{Spec}(B)\), so the main point of interest is the proof for the general case.

Of course, we have the map \(\text{Hom}(X, \text{Spec}(A)) \to \text{Hom}(A, \mathcal{O}_X(X))\) in which \(f\) is taken to \(s = f^{\#}(\text{Spec}(A)) : \mathcal{O}_{\text{Spec}(A)}(\text{Spec}(A)) \to \mathcal{O}_X(X)\). Of course, \(\mathcal{O}_{\text{Spec}(A)}(\text{Spec}(A)) \simeq A\) canonically and the stalk at some \(\mathfrak{p} = f(x)\) for \(x \in X\) is canonically isomorphic to \(A_{\mathfrak{p}}\). This gives us a diagram

\begin{equation} \begin{CD} A @>>> \mathcal{O}_X(X) \\ @VVV @VVV \\ A_{\mathfrak{p}} @>>> \mathcal{O}_{X, x} \end{CD} \end{equation}

Because the bottom map is assumed to be local, the maximal ideal \(\mathfrak{p} A_{\mathfrak{p}}\) is taken into \(\mathfrak{m}_x \subset \mathcal{O}_{X, x}\), the maximal ideal in the stalk over \(x\). Therefore, \(\mathfrak{p} A_{\mathfrak{p}} \subset (f^{\#}_{x})^{-1}(\mathfrak{m}_x)\), but this actually must be an equality because of maximality. If we let \(\ell : A \to A_{\mathfrak{p}}\) be the localization map, also notice that \(\ell^{-1}(\mathfrak{p} A_{\mathfrak{p}}) = \mathfrak{p}\). Therefore, if we let \(\ell' : \mathcal{O}_X(X) \to \mathcal{O}_{X, x}\) be the other localization, then

\begin{equation} f(x) = \mathfrak{p} = \ell^{-1}((f^{\#}_x)^{-1}(\mathfrak{m}_x)) = f^{\#}(\text{Spec}(A))^{-1}((\ell')^{-1}(\mathfrak{m}_x)) \end{equation}

In particular, we have recovered \(f(x)\) using only \(f^{\#}(\text{Spec}(A))\). To see that we also recover the map of sheaves \(f^{\#}\), it is clear that the above diagram determines \(f_x^{\#}\), the map on stalks uniquely in terms of the morphism of global sections. The reason for this is because of the universal property of localization: the morphism \(A \to \mathcal{O}_X(X) \to \mathcal{O}_{X, x}\) takes things in \(A - \mathfrak{p}\) to invertible things in \(\mathcal{O}_{X, x}\), hence it factors uniquely through the localization \(A_{\mathfrak{p}}\). The last thing to notice is that a morphism of sheaves is uniquely determined by its stalk maps. Thus, we have proved injectivity.

To prove surjectivity, take some \(\phi : A \to \mathcal{O}_X(X)\). We define \(f : X \to \text{Spec}(A)\) by taking \(x \in X\) to \(\mathfrak{m}_x \subset \mathcal{O}_{X, x}\) and then take the inverse image under the map \(A \to \mathcal{O}_X(X) \to \mathcal{O}_{X, x}\) obtained from composing \(\phi\) with restriction to the stalk. In pursuit of proving that \(f\) is continuous, we need only prove it is locally continuous, and locally \(X\) looks like \(U \simeq \text{Spec}(B)\). This gives rise to a diagram

\begin{equation} \begin{CD} A @>r \circ \phi >> \mathcal{O}_X(U) @>>> B \\ @VVV @VVV @VVV \\ A_{\mathfrak{p}} @>>> \mathcal{O}_{X, x} @>>> B_{\mathfrak{q}} \end{CD} \end{equation}

where \(r\) restricts sections and \(\mathfrak{p} = \phi^{-1}(\mathfrak{m}_x)\). We have \(g : \text{Spec}(B) \to U\) giving an isomorphism of locally ringed spaces. It is easy to see that \(g^{-1}((f|_U)^{-1}(\mathfrak{p}))\) consists of \(\mathfrak{q} \in \text{Spec}(B)\) such that \(\ell^{-1}(\mathfrak{m}_{g(\mathfrak{q})}) \supset \phi(\mathfrak{a})\), which is equivalent to \(r^{-1}(g^{\#}(U)^{-1}((\ell')^{-1}(\mathfrak{q} B_{\mathfrak{q}}))) \supset \phi(\mathfrak{a})\) from the diagram as \(\mathfrak{m}_{g(\mathfrak{q})} = (g^{\#}_{\mathfrak{q}})^{-1}(\mathfrak{q} B_{\mathfrak{q}})\) from locality. Thus, it is actually the collection of \(\mathfrak{q}\) such that \(\mathfrak{q} \supset (g^{\#}(U) \circ r \circ \phi)(\mathfrak{a})\), which is a closed subset, hence our defined \(f\) is continuous.

The only thing that remains is to define a map of sheaves. Let \(D(a) \subset X\) be the collection of \(x \in X\) such that \(a_x \notin \mathfrak{m}_x \subset \mathcal{O}_{X, x}\), where \(a \in \mathcal{O}_X(X)\) and we are localizing. In particular, \(\mathcal{m}_x\) consists of all non-invertible elements in the stalk, so \(a_x\) admits inverse \(b_x \in \mathcal{O}_{X, x}\). In particular, \(a_x \cdot b_x = 1_x\), so there must be a small neighbourhood \(W\) of \(x\) where \(a|_W \cdot b|_W = 1|_W\). Thus, the same is true at the stalks at all points in \(W\), so \(D(a)\) is open. Notice that if \(a_x \notin \mathfrak{m}_x\) and \(b_x \notin \mathfrak{m}_x\), then \(a_x b_x \notin \mathfrak{m}_x\), as if it were, then either \(a_x\) or \(b_x\) is in \(\mathfrak{m}_x\) (it is a prime ideal). On the other hand, if \(a_x b_x \notin \mathfrak{m}_x\), then certainly neither \(a_x\) nor \(b_x\) can be in \(\mathfrak{m}_x\). Thus, \(D(a) \cap D(b) = D(ab)\). It follows that this collection of open sets forms a basis for \(X\).

In principle, one can use these open sets to prove \(f\) is continuous instead of the technique above in which we chose an affine neighbourhood. Indeed,

\begin{align} f^{-1}(D(h)) &= \{x \ | \ \phi^{-1}(\ell^{-1}(\mathfrak{m}_x)) \not\supset (h) \} \\ &= \{x \ | \ \mathfrak{m}_x \not\supset (\phi(h))\} \\ &= D(\phi(h)) \end{align}

When it comes to defining the sheaf maps, notice that the map \(A \to \mathcal{O}_X(f^{-1}(D(h))) = \mathcal{O}_X(D(\phi(h)))\) obtained from composing \(\phi\) with restriction takes \(h\) to \(\phi(h)\) restricted to \(D(\phi(h))\), which by definition consists of points such that the restriction to each stalk is invertible. In particular, \(\phi(h)_x\) is invertible for each \(x \in D(\phi(h))\), hence \(\phi(h)|_{D(\phi(h))}\) itself is invertible, so the above map factors uniquely through the localization to give us \(A_h \to \mathcal{O}_X(f^{-1}(D(h)))\). In the case that we have \(h' \in (h)\) so that \(A_{h'} \supset A_h\) and \(D(h') \subset D(h)\), we also have a unique \(A_{h'} \to \mathcal{O}_X(f^{-1}(D(h')))\) via factoring the map from \(A\) through the localization. This gives two factorizations of the map \(A \to \mathcal{O}_X(f^{-1}(D(h')))\) through \(A_h\), hence they must be equal by uniqueness, and we have the diagrams

\begin{equation} \begin{CD} A_h @>>> \mathcal{O}_X(f^{-1}(D(h))) \\ @VVV @VVV \\ A_{h'} @>>> \mathcal{O}_X(f^{-1}(D(h'))) \end{CD} \end{equation}

The implication is that we have a collection of compatible maps of sections over a basis for our space, and therefore they will glue to give rise to a valid morphism of sheaves. In order to ensure that the morphism is one of local rings, we consider the induced morphism \(A_{f(x)} \to \mathcal{O}_{X, x}\) of stalks. The image of \(\mathfrak{m}_{f(x)} = f(x) A_{f(x)}\), the maximal ideal, is the image of \(f(x)\) under \(A \to \mathcal{O}_{X, x}\), which is exactly \(\ell \circ \phi\). On the other hand, by definition, \(f(x) = \phi^{-1}(\ell^{-1}(\mathfrak{m}_x))\). Thus, \(\mathfrak{m}_{f(x)}\) is sent into \(\mathfrak{m}_x\), so we do in fact have a local ring morphism in each stalk. Finally, notice that if we take global sections of \(f^{\#}\), the resulting map is obviously just \(\phi\). This completes the surjectivity proof.

Now, notice that we have a bijection \(\text{Hom}(X, \text{Spec}(\mathcal{O}_X(X))) \simeq \text{Hom}(\mathcal{O}_X(X), \mathcal{O}_X(X))\). In particular, the identity on the right yields a unique \(\theta : X \to \text{Spec}(\mathcal{O}_X(X))\) called the affinization. Suppose we have some \(f : X \to \text{Spec}(A)\) which yields \(\phi : A \to \mathcal{O}_X(X)\). This \(\phi\) will correspond to a unique \(h : \text{Spec}(\mathcal{O}_X(X)) \to \text{Spec}(A)\) under the bijection. Consider the map \(h \circ \theta : X \to \text{Spec}(A)\). If we take global sections, this is the same as mapping by global sections of \(h\), then global sections of \(\theta\). The former is \(\phi\), the latter is \(\text{id}\), hence by uniqueness, \(h \circ \theta = f\). In the case that \(X\) is affine, meaning we have an isomorphism \(f : X \to \text{Spec}(B)\), the induced global sections map \(\phi : B \to \mathcal{O}_X(X)\) will be an isomorphism. Once again, \(\phi\) corresponds to some \(\psi : \text{Spec}(\mathcal{O}_X(X)) \to \text{Spec}(B)\) which is also an isomorphism. The map \(\psi \circ \theta\) induced \(\phi\) on rings, and is therefore equal to \(f\). In particular, \(\theta\) must be an isomorphism as both \(f\) and \(\psi\) are.

This immediately gives us a nice way to determine if a scheme is affine. Suppose we are given a morphism \(f : X \to \text{Spec}(A)\) such that the induced \(\phi : A \to \mathcal{O}_X(X)\) is an isomorphism. Then immediately, we obtain an isomorphism of schemes \(h : \text{Spec}(\mathcal{O}_X(X)) \to \text{Spec}(A)\) and \(f = h \circ \theta\). If \(X\) is affine, then \(\theta\) is an isomorphism, hence \(f\) must be an isomorphism. In particular, if we we know that \(f\) is not an isomorphism via some basic argument (for example, maybe it isn't surjective), then we can immediately conclude that \(X\) cannot be affine. This is one way to prove that the "affine line with two origins" scheme constructed via gluing is not affine.

Our next goal is to prove the equivalence of categories \(\text{AffSch} \simeq \text{CRing}^{\text{op}}\). To do so, note that we have the functor \(\text{Spec} : \text{CRing}^{\text{op}} \to \text{AffSch}\) and the global sections functor \(\Gamma : \text{AffSch} \to \text{CRing}^{\text{op}}\). Recall that an object in \(\text{AffSch}\) is any locally ringed space isomorphic to \(\text{Spec}(B)\) for some \(B\). We proved above that \(\Gamma\) is fully faithful. Moreover, when we were handed \(\phi\), the \((f, f^{\#})\) constructed above is easily seen to be the usual \(\text{Spec}(\phi)\) construction. In particular, \(f(\mathfrak{p}) = \phi^{-1}(\ell^{-1}(\mathfrak{m}_p)) = \phi^{-1}(\mathfrak{p})\) as \(\mathfrak{m}_{\mathfrak{p}} = \mathfrak{p} A_{\mathfrak{p}}\). Moreover, the \(D(h) = \{\mathfrak{p} \ | \ \frac{h}{1} \notin \mathfrak{p} A_{\mathfrak{p}} \}\) for some \(h \in A\) are easily seen to be the distinguished opens \(D(h) = \{\mathfrak{p} \ | \ \mathfrak{p} \not\supset (h)\}\) for an affine scheme. Thus, the sheaf maps are also those from the \(\text{Spec}\) construction, and from what we concluded earlier, \(\Gamma \circ \text{Spec} = \text{id}\), the identity functor on \(\text{CRing}^{\text{op}}\).

Now, let us consider \(\text{Spec} \circ \Gamma\). In this case, the composition functor sends \(X\) to \(\text{Spec}(\mathcal{O}_X(X))\). We know that in the case of \(X\) affine, \(\theta_X : X \to \text{Spec}(\mathcal{O}_X(X))\) will be an isomorphism. It remains to show that \(\theta\) behaves naturally. In particular, for some morphism \(f : X \to Y\) of affine schemes, we must verify that we have a diagram

\begin{equation} \begin{CD} X @> \theta_X >> \text{Spec}(\mathcal{O}_X(X)) \\ @V f VV @VV \text{Spec}(\Gamma(f)) V \\ Y @> \theta_Y >> \text{Spec}(\mathcal{O}_Y(Y)) \end{CD} \end{equation}

but this simply follows from the fact that \(\theta_Y \circ f : X \to \text{Spec}(\mathcal{O}_Y(Y))\) necessarily factors as \(h \circ \theta_X\) for some \(h : \text{Spec}(\mathcal{O}_X(X)) \to \text{Spec}(\mathcal{O}_Y(Y))\). If we take global sections, \(\Gamma(h)\) must be the same as \(\Gamma(f)\), hence \(\text{Spec}(\Gamma(f))\) is a valid choice for \(h\), and by uniqueness, this must actually be what \(h\) is. Thus, we have demonstrated that \(\text{Spec} \circ \Gamma\) is naturally isomorphic to the identity functor, and we have an equivalence of categories.

One may recall that the Zariski cotangent space at \(x \in X\) (with \(X\) a scheme) is given by \(\mathfrak{m}_x/\mathfrak{m}_x^2\), with \(\mathfrak{m}_x \subset \mathcal{O}_{X, x}\) the usual maximal ideal. One should interpret this as a \(k(x) = \mathcal{O}_{X, x}/\mathfrak{m}_x\) -vector space where \(\mathfrak{m}_x/\mathfrak{m}_x^2\) can just be thought of as a quotient \(B\) -module before we pass to action of the residue field. Note that if \(\mathfrak{m} \subset A\), then \(\mathfrak{m}/\mathfrak{m}^2 \simeq \mathfrak{m} \otimes_B B/\mathfrak{m}\) when we treat both sides as \(B\) -modules. The way to see this, notice that we may define a map \(\mathfrak{m}/\mathfrak{m}^2 \to \mathfrak{m} \otimes_B B/\mathfrak{m}\) via \([v] \mapsto v \otimes [1]\). Such a map is clearly well-defined. We also have a map in the other direction \(v \otimes [w] \mapsto vw\), which is also clearly well-defined. One may easily verify that these maps are \(B\) -module morphisms and are inverses of each other. Via the usual "localization commutes with quotienting", notice that we also have \(B/\mathfrak{m} \simeq B_{\mathfrak{m}}/\mathfrak{m} B_{\mathfrak{m}}\), as \(B\) -modules. Thus,

\begin{equation} \mathfrak{m}/\mathfrak{m}^2 \simeq \mathfrak{m} \otimes_B B/\mathfrak{m} \simeq \mathfrak{m} \otimes_B B_{\mathfrak{m}}/\mathfrak{m} B_{\mathfrak{m}} \simeq \mathfrak{m} B_{\mathfrak{m}}/\mathfrak{m}^2 B_\mathfrak{m} \end{equation}

where the final isomorphism also follows from writing down an explicit isomorphism, similar to the one constructed above. Finally, one may notice that this isomorphism of \(B\) -modules automatically induces one of \(B/\mathfrak{m}\) -vector spaces. We can leverage this fact in order to compute the Zariski tangent space at \(x\), which is define to be \(\text{Hom}(\mathfrak{m}_x/\mathfrak{m}^2_x, k(x))\), the dual of the Zariski cotangent space, where functionals take values in the residue field.

Suppose \(X\) is a \(k\) -scheme of finite type, so that around some \(x \in X\) we can pick an affine open subscheme of the form \(U \simeq \text{Spec}(R/I)\) where \(R = k[x_1, \dots, x_n]\) and \(I = (f_1, \dots, f_m)\). The Zariski tangent space at \(x\) in \(X\) will clearly be the same as that of the analogous point in \(\text{Spec}(R/I)\) as the definition of the Zariski tangent space is local, so we will focus on the case \(X = \text{Spec}(A) = \text{Spec}(R/I)\). Let us assume that \(x \in \text{Spec}(R/I)\) is a \(k\) -point, which is to say that \(x\) is the inverse image of \((0)\) under a morphism \(R/I \to k\). Such a morphism is uniquely determined by a point \((a_1, \dots, a_n) \in k^n\) such that \(f_j(a_1, \dots, a_n) = 0\) for \(j = 1, \dots, m\) (in particular, \(x_j \mapsto a_j\) under the ring morphism). We denote \(\mathfrak{m}_R = (x_1 - a_1, \dots, x_n - a_n) \subset R\). Its image under the quotient map \(R \to R/I\) is \(x = \mathfrak{m}\). Notice that \(\mathcal{O}_{X, x} \simeq A_{\mathfrak{m}}\) and the maximal ideal is \(\mathfrak{m} A_{\mathfrak{m}}\). Hence, using the above isomorphism, the Zariski cotangent space is isomorphic to \(\mathfrak{m}/\mathfrak{m}^2\), which is much easier to work with, as we have:

\begin{equation} \mathfrak{m}/\mathfrak{m}^2 \simeq \mathfrak{m}_R/(I + \mathfrak{m}_R^2) \simeq (\mathfrak{m}_R/\mathfrak{m}_R^2) / ((I + \mathfrak{m}_R^2)/\mathfrak{m}_R^2) \end{equation}

It is easy to see what the vector space \(\mathfrak{m}_R/\mathfrak{m}_R^2\) is: it has a basis consisting of the linear terms \([x_j - a_j]\). On the other hand, for each \(f_j \in I\), we may always write

\begin{equation} f_j(x_1, \dots, x_n) = f_j(a) + \sum_{i = 1}^{n} \frac{d f_j}{dx_i}(a) (x_i - a_i) + \text{higher order} \end{equation}

where \(f_j(a) = 0\) by our original assumption about \(a\), which means that \((I + \mathfrak{m}_R^2)/\mathfrak{m}_R^2\) is the subspace of \(\mathfrak{m}_R/\mathfrak{m}_R^2\) spanned by the elements \(\sum_i \frac{d f_j}{dx_i}(a) [x_i - a_i]\). It follows immediately that some element of \(\text{Hom}(\mathfrak{m}/\mathfrak{m}^2, k)\), the Zariski tangent space, is given by a functional taking \([x_i - a_i] \mapsto v_i \in k\) such that \(\sum_i \frac{d f_j}{dx_i}(a) v_i = 0\). In other words, it is a vector \((v_1, \dots, v_n) \in k^n\) which is in the kernel of \(f_{*}\), the matrix with component \([f_{*}]_{ij} = \frac{d f_i}{dx_j}(a)\), which matches our intuition from smooth manifold theory.