Let \(A\) be a noetherian ring, let \(X=\Spec A\), let \(\mfa \subseteq A\) be an ideal, and let \(U \subseteq X\) be the open set \(X-V(\mfa)\).
For any \(A\)-module \(M\), establish the following formula of Deligne: \[ \Gamma(U, \tilde{M}) \cong \colim_n \Hom_A(\mfa^n, M). \]
Apply this in the case of an injective \(A\)-module \(I\), to give another proof of (3.4).
(A product of generators can vanish on the open set)
For \(A=k[x,y]/(xy)\) and \(\mfa=(x,y)\), the open set \(U=D(x)\cup D(y)\) is nonempty, but \(xy=0\) in \(A\). Multiplication by any positive power of \(xy\) therefore sends the nonzero section \(1\in\Gamma(U,\OO_U)\) to zero. Thus a denominator correction using a product of all generators need not preserve a section on their union of distinguished opens. Step <1>3 instead restricts a homomorphism to a power of the whole ideal, which is the unit ideal everywhere on \(U\).
Solution
Use indices \(n\ge1\) and transition maps \[ \Hom_A(\mfa^n,M)\longrightarrow\Hom_A(\mfa^{n+1},M) \] given by restriction along \(\mfa^{n+1}\subseteq\mfa^n\). Including \(n=0\) would give the same direct limit. No finiteness hypothesis is imposed on \(M\).
1. There is a natural \(A\)-linear map\[ \Phi:\varinjlim_n\Hom_A(\mfa^n,M)\longrightarrow\Gamma(U,\widetilde M). \]
For every \(n\), the inclusion \(\widetilde{\mfa^n}\hookrightarrow\OO_X\) becomes an isomorphism on \(U\). Indeed, a prime outside \(V(\mfa)\) omits some element of \(\mfa\), which becomes a unit after localization, so \(\mfa^nA_{\mathfrak p}=A_{\mathfrak p}\). A homomorphism \(h:\mfa^n\to M\) therefore gives, by sheafification and restriction, a map \(\OO_U\to\widetilde M|_U\). Its image of \(1\) is the corresponding section \(\Phi_n(h)\).
For \(a\in\mfa\), its restriction to \(D(a)\) is explicitly \[ \Phi_n(h)|_{D(a)}=\frac{h(a^n)}{a^n}\in M_a. \] Restriction of \(h\) to a larger ideal power does not change the sheaf map on \(U\), since both source ideals there are identified with \(\OO_U\) by inclusion. Thus the maps \(\Phi_n\) are compatible and induce \(\Phi\). Their construction commutes with homomorphisms of \(M\) and preserves scalar multiplication.
2. The map \(\Phi\) is injective.
Suppose \(h:\mfa^n\to M\) gives the zero section on \(U\). Then the image module \(N=\im h\) has \(\widetilde N|_U=0\), because the restriction of the entire sheaf map is zero. The module \(N\) is finite: it is a quotient of the finite ideal \(\mfa^n\). By the support and ideal-torsion calculation, each of its generators is killed by a power of \(\mfa\). Choosing one exponent for the finitely many generators gives \(\mfa^eN=0\) for some \(e\ge1\). Consequently \[ h(\mfa^{n+e})=h(\mfa^e\mfa^n)=\mfa^e h(\mfa^n)=0. \] Thus the class of \(h\) becomes zero at a later stage of the direct system and is zero in the direct limit. This proves injectivity.
3. Every section \(s\in\Gamma(U,\widetilde M)\) is obtained from a homomorphism \(\mfa^q\to M\) for some \(q\ge1\).
If \(U=\varnothing\), its only section is represented by the zero homomorphism, so assume \(U\ne\varnothing\). The graph of the section is the coherent subsheaf \[ \mathcal H_U=\{(a,as):a\in\OO_U\} \subseteq(\OO_X\oplus\widetilde M)|_U, \] isomorphic to \(\OO_U\) by the first projection. The coherent-subsheaf extension theorem P-AGH2515EXTCOH, part (c), gives a coherent subsheaf \(\mathcal H\subseteq\OO_X\oplus\widetilde M\) restricting to this graph. Since \(X\) is affine, it corresponds to a finite submodule \(F\subseteq A\oplus M\). Write \(p:F\to A\) and \(v:F\to M\) for the two coordinate projections, and put \(K=\ker p\) and \(\mathfrak b=p(F)\).
On \(U\), the first projection is an isomorphism from the graph onto \(\OO_U\). Thus \(\widetilde K|_U=0\) and \(\widetilde{A/\mathfrak b}|_U=0\). Both modules are finite over the noetherian ring \(A\). The same support argument as in step 2 supplies integers \(e,d\ge1\) with \[ \mfa^eK=0,\qquad\mfa^d\subseteq\mathfrak b. \] Apply the Artin–Rees lemma to \(K\subseteq F\) and \(\mfa\). For some \(c\ge0\) and every \(N\ge c\), \[ \mfa^NF\cap K=\mfa^{N-c}(\mfa^cF\cap K) \subseteq\mfa^{N-c}K. \] Choose \(N\ge c+e\). Then \(p\) is injective on \(\mfa^NF\), and its image is \(\mfa^N\mathfrak b\). Its inverse followed by the second projection is an actual module homomorphism \[ h:\mfa^N\mathfrak b\xrightarrow{\ (p|_{\mfa^NF})^{-1}\ }\mfa^NF \xrightarrow{v}M. \] Since \(\mfa^{N+d}\subseteq\mfa^N\mathfrak b\), restrict \(h\) to this ideal power.
On \(U\), the ideal \(\mfa\) is the unit ideal, so \(\widetilde{\mfa^NF}|_U=\mathcal H_U\). Under the first projection, the second projection of this graph is multiplication by the original section \(s\). The homomorphism \(h\) therefore restricts to the map \(a\mapsto as\) on \(U\). The inclusion of \(\mfa^{N+d}\) also becomes the identity of \(\OO_U\), so step 1 gives \[ \Phi_{N+d}(h|_{\mfa^{N+d}})=s. \] This proves surjectivity without assigning potentially inconsistent values to monomial generators. The only module to which Artin–Rees was applied was the finite graph extension \(F\), not the arbitrary module \(M\).
4. The formula in (a) is the natural isomorphism\[ \boxed{\varinjlim_{n\ge1}\Hom_A(\mfa^n,M)\xrightarrow{\cong}\Gamma(U,\widetilde M),} \]with the map described in step
1.
Steps 2 and 3 prove bijectivity, and step 1 supplies linearity and naturality. Thus the inverse has those properties as well.
5. For an injective \(A\)-module \(I\), the associated sheaf \(\widetilde I\) is flasque, proving (b).
Let \(W\subseteq X\) be any open subset. Write its closed complement as \(V(\mathfrak b)\) for an ideal \(\mathfrak b\) of \(A\). By step 4, a section on \(W\) is represented by a homomorphism \(h:\mathfrak b^n\to I\) for some \(n\). Injectivity of \(I\) extends \(h\) along \(\mathfrak b^n\hookrightarrow A\) to \(\widetilde h:A\to I\). The element \(\widetilde h(1)\in I=\Gamma(X,\widetilde I)\) restricts to the given section, because the ideal inclusion becomes an isomorphism on \(W\) and step 1 describes exactly that restriction. Hence every section on every open subset extends to all of \(X\). For opens \(W\subseteq V\subseteq X\), restricting such a global extension to \(V\) gives surjectivity of \(\widetilde I(V)\to\widetilde I(W)\). This is flasqueness and gives the asserted alternative proof of [@Har10a, Proposition III.3.4].
6. Q.E.D.
Steps <1>1–<1>4 establish (a), and step 5 proves (b).