28.25 Sections of quasi-coherent sheaves
Here is a computation of sections of a quasi-coherent sheaf on a quasi-compact open of an affine spectrum.
Lemma 28.25.1. Let $A$ be a ring. Let $I \subset A$ be a finitely generated ideal. Let $M$ be an $A$-module. Then there is a canonical map
\[ \mathop{\mathrm{colim}}\nolimits _ n \mathop{\mathrm{Hom}}\nolimits _ A(I^ n, M) \longrightarrow \Gamma (\mathop{\mathrm{Spec}}(A) \setminus V(I), \widetilde{M}). \]
This map is always injective. If for all $x \in M$ we have $Ix = 0 \Rightarrow x = 0$ then this map is an isomorphism. In general, set $M_ n = \{ x \in M \mid I^ nx = 0\} $, then there is an isomorphism
\[ \mathop{\mathrm{colim}}\nolimits _ n \mathop{\mathrm{Hom}}\nolimits _ A(I^ n, M/M_ n) \longrightarrow \Gamma (\mathop{\mathrm{Spec}}(A) \setminus V(I), \widetilde{M}). \]
Proof.
Since $I^{n + 1} \subset I^ n$ and $M_ n \subset M_{n + 1}$ we can use composition via these maps to get canonical maps of $A$-modules
\[ \mathop{\mathrm{Hom}}\nolimits _ A(I^ n, M) \longrightarrow \mathop{\mathrm{Hom}}\nolimits _ A(I^{n + 1}, M) \]
and
\[ \mathop{\mathrm{Hom}}\nolimits _ A(I^ n, M/M_ n) \longrightarrow \mathop{\mathrm{Hom}}\nolimits _ A(I^{n + 1}, M/M_{n + 1}) \]
which we will use as the transition maps in the systems. Given an $A$-module map $\varphi : I^ n \to M$, then we get a map of sheaves $\widetilde{\varphi } : \widetilde{I^ n} \to \widetilde{M}$ which we can restrict to the open $\mathop{\mathrm{Spec}}(A) \setminus V(I)$. Since $\widetilde{I^ n}$ restricted to this open gives the structure sheaf we get an element of $\Gamma (\mathop{\mathrm{Spec}}(A) \setminus V(I), \widetilde{M})$. We omit the verification that this is compatible with the transition maps in the system $\mathop{\mathrm{Hom}}\nolimits _ A(I^ n, M)$. This gives the first arrow. To get the second arrow we note that $\widetilde{M}$ and $\widetilde{M/M_ n}$ agree over the open $\mathop{\mathrm{Spec}}(A) \setminus V(I)$ since the sheaf $\widetilde{M_ n}$ is clearly supported on $V(I)$. Hence we can use the same mechanism as before.
Next, we work out how to define this arrow in terms of algebra. Say $I = (f_1, \ldots , f_ t)$. Then $\mathop{\mathrm{Spec}}(A) \setminus V(I) = \bigcup _{i = 1, \ldots , t} D(f_ i)$. Hence
\[ 0 \to \Gamma (\mathop{\mathrm{Spec}}(A) \setminus V(I), \widetilde{M}) \to \bigoplus \nolimits _ i M_{f_ i} \to \bigoplus \nolimits _{i, j} M_{f_ if_ j} \]
is exact. Suppose that $\varphi : I^ n \to M$ is an $A$-module map. Consider the vector of elements $\varphi (f_ i^ n)/f_ i^ n \in M_{f_ i}$. It is easy to see that this vector maps to zero in the second direct sum of the exact sequence above. Whence an element of $\Gamma (\mathop{\mathrm{Spec}}(A) \setminus V(I), \widetilde{M})$. We omit the verification that this description agrees with the one given above.
Let us show that the first arrow is injective using this description. Namely, if $\varphi $ maps to zero, then for each $i$ the element $\varphi (f_ i^ n)/f_ i^ n$ is zero in $M_{f_ i}$. In other words we see that for each $i$ we have $f_ i^ m\varphi (f_ i^ n) = 0$ for some $m \geq 0$. We may choose a single $m$ which works for all $i$. Then we see that $\varphi (f_ i^{n + m}) = 0$ for all $i$. It is easy to see that this means that $\varphi |_{I^{t(n + m - 1) + 1}} = 0$ in other words that $\varphi $ maps to zero in the $t(n + m - 1) + 1$st term of the colimit. Hence injectivity follows.
Note that each $M_ n = 0$ in case we have $Ix = 0 \Rightarrow x = 0$ for $x \in M$. Thus to finish the proof of the lemma it suffices to show that the second arrow is an isomorphism.
Let us attempt to construct an inverse of the second map of the lemma. Let $s \in \Gamma (\mathop{\mathrm{Spec}}(A) \setminus V(I), \widetilde{M})$. This corresponds to a vector $x_ i/f_ i^ n$ with $x_ i \in M$ of the first direct sum of the exact sequence above. Hence for each $i, j$ there exists $m \geq 0$ such that $f_ i^ m f_ j^ m (f_ j^ n x_ i - f_ i^ n x_ j) = 0$ in $M$. We may choose a single $m$ which works for all pairs $i, j$. After replacing $x_ i$ by $f_ i^ mx_ i$ and $n$ by $n + m$ we see that we get $f_ j^ nx_ i = f_ i^ nx_ j$ in $M$ for all $i, j$. Let us introduce
\[ K_ n = \{ x \in M \mid f_1^ nx = \ldots = f_ t^ nx = 0\} \]
We claim there is an $A$-module map
\[ \varphi : I^{t(n - 1) + 1} \longrightarrow M/K_ n \]
which maps the monomial $f_1^{e_1} \ldots f_ t^{e_ t}$ with $\sum e_ i = t(n - 1) + 1$ to the class modulo $K_ n$ of the expression $f_1^{e_1} \ldots f_ i^{e_ i - n} \ldots f_ t^{e_ t}x_ i$ where $i$ is chosen such that $e_ i \geq n$ (note that there is at least one such $i$). To see that this is indeed the case suppose that
\[ \sum \nolimits _{E = (e_1, \ldots , e_ t), |E| = t(n - 1) + 1} a_ E f_1^{e_1} \ldots f_ t^{e_ t} = 0 \]
is a relation between the monomials with coefficients $a_ E$ in $A$. Then we would map this to
\[ z = \sum \nolimits _{E = (e_1, \ldots , e_ t), |E| = t(n - 1) + 1} a_ E f_1^{e_1} \ldots f_{i(E)}^{e_{i(E)} - n} \ldots f_ t^{e_ t}x_{i(E)} \]
where for each multiindex $E$ we have chosen a particular $i(E)$ such that $e_{i(E)} \geq n$. Note that if we multiply this by $f_ j^ n$ for any $j$, then we get zero, since by the relations $f_ j^ nx_ i = f_ i^ nx_ j$ above we get
\begin{align*} f_ j^ nz & = \sum \nolimits _{E = (e_1, \ldots , e_ t), |E| = t(n - 1) + 1} a_ E f_1^{e_1} \ldots f_ j^{e_ j + n} \ldots f_{i(E)}^{e_{i(E)} - n} \ldots f_ t^{e_ t}x_{i(E)} \\ & = \sum \nolimits _{E = (e_1, \ldots , e_ t), |E| = t(n - 1) + 1} a_ E f_1^{e_1} \ldots f_ t^{e_ t}x_ j = 0. \end{align*}
Hence $z \in K_ n$ and we see that every relation gets mapped to zero in $M/K_ n$. This proves the claim.
Note that $K_ n \subset M_{t(n - 1) + 1}$. Hence the map $\varphi $ in particular gives rise to an $A$-module map $I^{t(n - 1) + 1} \to M/M_{t(n - 1) + 1}$. This proves the second arrow of the lemma is surjective. We omit the proof of injectivity.
$\square$
Example 28.25.2. We will give two examples showing that the first displayed map of Lemma 28.25.1 is not an isomorphism.
Let $k$ be a field. Consider the ring
\[ A = k[x, y, z_1, z_2, \ldots ]/(x^ nz_ n). \]
Set $I = (x)$ and let $M = A$. Then the element $y/x$ defines a section of the structure sheaf of $\mathop{\mathrm{Spec}}(A)$ over $D(x) = \mathop{\mathrm{Spec}}(A) \setminus V(I)$. We claim that $y/x$ is not in the image of the canonical map $\mathop{\mathrm{colim}}\nolimits \mathop{\mathrm{Hom}}\nolimits _ A(I^ n, A) \to A_ x = \mathcal{O}(D(x))$. Namely, if so it would come from a homomorphism $\varphi : I^ n \to A$ for some $n$. Set $a = \varphi (x^ n)$. Then we would have $x^ m(xa - x^ ny) = 0$ for some $m > 0$. This would mean that $x^{m + 1}a = x^{m + n}y$. This would mean that $\varphi (x^{n + m + 1}) = x^{m + n}y$. This leads to a contradiction because it would imply that
\[ 0 = \varphi (0) = \varphi (z_{n + m + 1} x^{n + m + 1}) = x^{m + n}y z_{n + m + 1} \]
which is not true in the ring $A$.
Let $k$ be a field. Consider the ring
\[ A = k[f, g, x, y, \{ a_ n, b_ n\} _{n \geq 1}]/ (fy - gx, \{ a_ nf^ n + b_ ng^ n\} _{n \geq 1}). \]
Set $I = (f, g)$ and let $M = A$. Then $x/f \in A_ f$ and $y/g \in A_ g$ map to the same element of $A_{fg}$. Hence these define a section $s$ of the structure sheaf of $\mathop{\mathrm{Spec}}(A)$ over $D(f) \cup D(g) = \mathop{\mathrm{Spec}}(A) \setminus V(I)$. However, there is no $n \geq 0$ such that $s$ comes from an $A$-module map $\varphi : I^ n \to A$ as in the source of the first displayed arrow of Lemma 28.25.1. Namely, given such a module map set $x_ n = \varphi (f^ n)$ and $y_ n = \varphi (g^ n)$. Then $f^ mx_ n = f^{n + m - 1}x$ and $g^ my_ n = g^{n + m - 1}y$ for some $m \geq 0$ (see proof of the lemma). But then we would have $0 = \varphi (0) = \varphi (a_{n + m}f^{n + m} + b_{n + m}g^{n + m}) = a_{n + m}f^{n + m - 1}x + b_{n + m}g^{n + m - 1}y$ which is not the case in the ring $A$.
We will improve on the following lemma in the Noetherian case, see Cohomology of Schemes, Lemma 30.10.5.
Lemma 28.25.3. Let $X$ be a quasi-compact scheme. Let $\mathcal{I} \subset \mathcal{O}_ X$ be a quasi-coherent sheaf of ideals of finite type. Let $Z \subset X$ be the closed subscheme defined by $\mathcal{I}$ and set $U = X \setminus Z$. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_ X$-module. The canonical map
\[ \mathop{\mathrm{colim}}\nolimits _ n \mathop{\mathrm{Hom}}\nolimits _{\mathcal{O}_ X}(\mathcal{I}^ n, \mathcal{F}) \longrightarrow \Gamma (U, \mathcal{F}) \]
is injective. Assume further that $X$ is quasi-separated. Let $\mathcal{F}_ n \subset \mathcal{F}$ be subsheaf of sections annihilated by $\mathcal{I}^ n$. The canonical map
\[ \mathop{\mathrm{colim}}\nolimits _ n \mathop{\mathrm{Hom}}\nolimits _{\mathcal{O}_ X}(\mathcal{I}^ n, \mathcal{F}/\mathcal{F}_ n) \longrightarrow \Gamma (U, \mathcal{F}) \]
is an isomorphism.
Proof.
Let $\mathop{\mathrm{Spec}}(A) = W \subset X$ be an affine open. Write $\mathcal{F}|_ W = \widetilde{M}$ for some $A$-module $M$ and $\mathcal{I}|_ W = \widetilde{I}$ for some finite type ideal $I \subset A$. Restricting the first displayed map of the lemma to $W$ we obtain the first displayed map of Lemma 28.25.1. Since we can cover $X$ by a finite number of affine opens this proves the first displayed map of the lemma is injective.
We have $\mathcal{F}_ n|_ W = \widetilde{M_ n}$ where $M_ n \subset M$ is defined as in Lemma 28.25.1 (details omitted). The lemma guarantees that we have a bijection
\[ \mathop{\mathrm{colim}}\nolimits _ n \mathop{\mathrm{Hom}}\nolimits _{\mathcal{O}_ W}( \mathcal{I}^ n|_ W, (\mathcal{F}/\mathcal{F}_ n)|_ W) \longrightarrow \Gamma (U \cap W, \mathcal{F}) \]
for any such affine open $W$.
To see the second displayed arrow of the lemma is bijective, we choose a finite affine open covering $X = \bigcup _{j = 1, \ldots , m} W_ j$. The injectivity follows immediately from the above and the finiteness of the covering. If $X$ is quasi-separated, then for each pair $j, j'$ we choose a finite affine open covering
\[ W_ j \cap W_{j'} = \bigcup \nolimits _{k = 1, \ldots , m_{jj'}} W_{jj'k}. \]
Let $s \in \Gamma (U, \mathcal{F})$. As seen above for each $j$ there exists an $n_ j$ and a map $\varphi _ j : \mathcal{I}^{n_ j}|_{W_ j} \to (\mathcal{F}/\mathcal{F}_{n_ j})|_{W_ j}$ which corresponds to $s|_{U \cap W_ j}$. By the same token for each triple $(j, j', k)$ there exists an integer $n_{jj'k}$ such that the restriction of $\varphi _ j$ and $\varphi _{j'}$ as maps $\mathcal{I}^{n_{jj'k}} \to \mathcal{F}/\mathcal{F}_{n_{jj'k}}$ agree over $W_{jj'k}$. Let $n = \max \{ n_ j, n_{jj'k}\} $ and we see that the $\varphi _ j$ glue as maps $\mathcal{I}^ n \to \mathcal{F}/\mathcal{F}_ n$ over $X$. This proves surjectivity of the map.
$\square$
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