Section 15.1: The quasicoherent sheaf corresponding to a graded module
Exercise 15.1.A:
Give an isomorphism between the stalk of \(\widetilde{M_\bullet}\) at a point corresponding to homogeneous prime \(\pp \subset S_\bullet\) and \(((M_\bullet)_{\pp})_0\). (Remark: You can use this exercise to give an alternate definition of \(\widetilde{M}_\bullet\) in terms of "compatible stalks", c.f. Exercise 4.5.M.)
Proof:
Since stalks are defined as the colimit across all open subsets containing \( \pp \), by the preceding paragraph in the text the natural maps \( \widetilde{M}_bullet(D(f)) = ((M_\bullet)_f)_0 \to (\widetilde{M}_\bullet)_\pp \) together with the natural inclusions \( ((M_\bullet)_f)_0 \hookrightarrow ((M_\bullet)\pp)_0 \) canonically induce a map \( (\widetilde{M}_\bullet)_\pp \to ((M_\bullet)_\pp)_0 \). In the reverse direction, for any \( \frac{m}{g^r} \in ((M_\bullet)_\pp)_0 \) (for \( \deg m = r \cdot \deg g \) with \( g \notin \pp \)) we have an obvious map \( ((M_\bullet)_\pp)_0 \to (\widetilde{M}_\bullet)(D(g))\) – since taking stalks is independent of restrictions to some open neighborhood, this induces a map in the reverse direction. Some care must be taken to show that this is independent of choice of \( D(g) \), though this should effectively follow from our definition of the quasicoherent sheafs \( \widetilde{M_\bullet(f)} \) on the distinguished affine open sets.
$$\tag*{$\blacksquare$}$$
Exercise 15.1.B:
Show that \( \sim \) is an exact functor. (Hint: everything in the construction is exact.)
Proof:
Since exactness may be checked at the level of stalks by Exercise 2.6.C (or as a consequence thereof) and \( \sim \) is effectively localization by the previous exercise — which is exact by Exercise 1.6.F — we have that \( \sim \) itself is exact.
$$\tag*{$\blacksquare$}$$
Exercise 15.1.C:
Show that if \(M_\bullet\) and \( M^\prime_\bullet \) agree in high enough degrees, then \(\widetilde{M} \cong \widetilde{M^\prime}\) . Then show that the map from graded \(S_\bullet\)-modules (up to isomorphism) to quasicoherent sheaves on \(\proj S_\bullet\) (up to isomorphism) is not a bijection. (Really: show this isn’t an equivalence of categories.)
Proof:
The proof should ultimately be quite similar to Exercise 6.4.D. If we let \( N \geq 0 \) be the smallest integer such that \( M_n = M_n^\prime \) for all \( n \geq N \) then we need only consider the Veronese map of \( S_\bullet \)-modules \( M_{N\bullet} \hookrightarrow M_{\bullet} \). This clearly induces a map \( ((M_{N\bullet})_f)_0 \hookrightarrow ((M_\bullet)_f)_0 \) giving us a morphism of sheaves on our distinguished affine base and thus a morphism of sheaves in one direction.
In the reverse direction, we have shown in Exercise 6.4.D that we may cover \( \proj S_\bullet \) by distinguished open sets \( D(f) \) with the degree of \( f \) divisible by \( N \) so that if \( \frac{g}{f^r} \in ((M_\bullet)_f)_0 \) we know that the degree of \( g \) must be divisible by \( N \) (say \( \deg g = Nk \)) so that the inverse map is simply induced by the " identity " taking \( g \mapsto g\) taking the \( nk^{th} \)-graded piece of \( M_\bullet \)to the \( k^{th} \) graded piece of \( M_{n\bullet} \). In other words, we get that \( \widetilde{M_{N\bullet}} \cong \widetilde{M_\bullet} \) as sheaves on \( \proj S_\bullet \).
Describe a map of \(S_\bullet\)-modules \(M_\bullet \to \Gamma(\proj S_\bullet , \widetilde{M_\bullet})\). (This foreshadows the "saturation map" of §15.4.5 that takes a graded module to its saturation, see Exercise 15.4.C.)
Proof:
For any \( f \in S_\bullet \) of positive degree, there is an obvious map \( M_0 \to ((M_\bullet)_f)_0 \) defined by sending \( x \mapsto \frac{x}{1} \). Since the sections \( \frac{x}{1} \) clearly glue on overlaps \( D(f) \cap D(g) \), the sheaf axioms tell us we obtain a global section which can be locally given by the \( \frac{x}{1} \). This in turn yields our map \( M_0 \to \Gamma(\proj S_\bullet, \widetilde{M_\bullet}) \).
$$\tag*{$\blacksquare$}$$
Exercise 15.1.E:
Show that if the functor \( \sim \) is applied to the exact sequence of graded \( S_\bullet \)-modules
$$
\begin{CD}
0 @>>> I_\bullet @>>> S_\bullet @>>> S_\bullet / I_\bullet @>>> 0
\end{CD}
$$
we obtain the closed subscheme exact sequence (13.5.4.1) for \(\proj S_\bullet / I_\bullet \hookrightarrow \proj S_\bullet\).
Proof:
This should ultimately follow from unwinding definitions and the fact that localization commutes with quotients. In particular, exactness of localization tells us that we also have the exact sequence
on the distinguished open set \( D(f)\). Now we know from construction that by evaluating the map \( \OO_{\proj S_\bullet} \to \OO_{\proj S_\bullet / I_\bullet} \) on \( D(f) \) we obtain the map \( ((S_\bullet)_f)_0 \to ((S_\bullet / I_\bullet)_f)_0 \). Upon taking kernels it is easy to see that \( \mathscr{I}(D(f)) = ((I_\bullet)_f)_0 = \widetilde{I_\bullet}(D(f)) \).
$$\tag*{$\blacksquare$}$$
Section 15.2: Invertible sheaves (line bundles) on projective A-schemes
Exercise 15.2.A:
If \(S_\bullet = A[x_0,...,x_m]\), so \(\proj S_\bullet = \P^m_A\) , show \(\widetilde{S(n)_\bullet} = \OO(n)\)
using transition functions (cf. §14.1). (Recall from §14.1.2 that the global sections of \(\OO(n)\) should be identified with the homogeneous degree \(n\) polynomials in \(x_0, \dots, x_m\). Can you see that in the context of this exercise?)
Proof:
As \( S_\bullet \) is generated in degree 1 by \( x_0, \dots, x_m \) it suffices to check that our transition functions are the same on the affine open cover \( D(x_0), \dots, D(x_m) \). By definition we have \( \widetilde{S_\bullet(n)} (D(x_i)) \) is the degree \( n \) graded part of \( A[x_0, \dots, x_m]_{x_i}\) which is the same as a degree \( n \) polynomial in \( A[x_{0/i}, \dots, x_{m/i}] / (x_{i/i} - 1) \). Thus, the well-defined transition map from \( A[x_{0/i}, \dots, x_{m/i}] / (x_{i/i} - 1) \) to \( A[x_{0/j}, \dots, x_{m/j}] / (x_{j/j} - 1) \) satisfying the cocycle condition are precisely multiplication by \( x_{i/j}^n \) so that \( \widetilde{S(n)_\bullet}\OO(n) \)
$$\tag*{$\blacksquare$}$$
Exercise 15.2.B:
If \(S_\bullet\) is generated in degree 1, show that \(\OO_{\proj S_\bullet} (n)\) is an invertible sheaf.
Proof:
We need only show \( \OO_{\proj S_\bullet} (n) \) is locally free rank 1. If \( f \in S_1 \) is any generator we know that \( \OO_{\proj S_\bullet} (n)(D(f)) = ((S_\bullet)_f)_n \) so that we may locally define our isomorphism \( ((S_\bullet)_f)_0 \to ((S_\bullet)_f)_n \) by \( s \mapsto f^n s \). While it is difficult to directly show that this agrees on overlaps since \( fg \) is no longer degree 1, by the previous exercise we know that this at the very least commutes with the transition functions which are simply multiplication by \( \frac{f^n}{g^n} \).
$$\tag*{$\blacksquare$}$$
Exercise 15.2.C:
If \(S_\bullet\) is generated in degree 1, show that \(\widetilde{M_\bullet}(n) = \widetilde{M(n)_\bullet}.
(Hereafter, we can be cavalier with the placement of the "dot" in such situations.)
Proof:
Let \( f \in S_1 \) be any generator, and consider the sections of \( \widetilde{M_\bullet} \) on \( D(f) \). As localization commutes with tensor products, we have \( \widetilde{M_\bullet}(n)(D(f)) = (\widetilde{M_\bullet} \otimes \OO(n))(D(f)) = \Big( ( M_\bullet )_f \otimes (S_\bullet(n))_f \Big)_0 \). Since \( S_\bullet \) is generated in degree 1 we have that the degree zero part of \( ( M_\bullet )_f \otimes (S_\bullet(n))_f \) can only come from the degree zero parts of \( (M_\bullet)_f \) and \( (S_\bullet(n))_f \), i.e. the \(n^{th} \) graded part of \( (S_\bullet)_f \); as \( M_\bullet \) is a graded \( S_\bullet \)-module we have \( S_n \cdot M_0 \subset M_n \) so that multiplication gives us a map \( ((S_\bullet)_f)_n \otimes ((M_\bullet)_f)_0 \to ((M_\bullet)_f)_n \). In the reverse direction, we can take any \( m \in (M_\bullet)_f \) of degree \( n \) and send it to \( \frac{m}{f^n} \otimes f^n \). Since the \( D(f) \) cover \( \proj S_\bullet \), this proves the claim.
$$\tag*{$\blacksquare$}$$
Exercise 15.2.D:
If \(S_\bullet\) is generated in degree 1, show that \(\OO(m + n) \cong \OO(m) \otimes \OO (n) \).
Proof:
Using the previous exercise, we may interpret the right hand side as \( \OO(m)(n) \) which on any distinguished open affine \( D(f) \) corresponds to the \( n^{th} \) graded piece of \( (S(m)_\bullet)_f \), which itself corresponds to the \( (n+m)^{th} \) graded piece of \( (S_\bullet)_f \) thus proving the claim.
$$\tag*{$\blacksquare$}$$
Section 15.3: Globally generated and base-point-free line bundles
Exercise 15.3.A:
Show that every quasicoherent sheaf on every affine scheme is globally generated. Show that every finite type quasi-coherent sheaf on every affine scheme is generated by a finite number of global sections. (Hint for both: for any \(A\)-module \(M\), there is a surjection onto \(M\) from a free \(A\)-module.)
Proof:
As indicated in the hint, if \( X \) is an affine scheme then up to isomorphism we may assume without loss of generality \( X = \spec A \). Then for any quasi-coherent sheaf \( \FF \) over \( X \) we have \( \Gamma(\FF) = M \) for some \( A \)-module \( M \), so that \( \FF \cong \widetilde{M} \) (by definition of quasi-coherent). Since any \( A \)-module \( M \) has a natural map \( c : M \to A^{|M|}\), one may use the universal property of free \( A \)-modules to construct a splitting \( p : A^{|M|} \to M \). Since \( \sim \) is functorial by §15.1, we obtain a morphism \( \widetilde{p} : \widetilde{A^{|M|}} \to \FF \) which is a surjection on stalks and thus a surjection in general.
In the finite-type case, we have \( M = \Gamma(\FF) \) is a finitely generated \(A\)-module so that by definition there exists a surjection \( A^{\oplus n} \to M \) for some \( n \geq 1 \). Then the \( \sim \) functor induces a map \( \OO^{\oplus n} \to \FF \).
$$\tag*{$\blacksquare$}$$
Exercise 15.3.B:
Show that if quasicoherent sheaves \(\FF\) and \(\GG\) are globally generated at a point \(p\),then so is \(\FF \otimes \GG\).
Proof:
Suppose \( \phi : \OO^{|I|} \to \FF \) and \( \psi : \OO^{|J|} \to \GG \) induce surjections on the level of stalks at a point \( p \). There is an obvious induced map \( \phi \otimes \psi : \OO^{|I \times J|} \to \FF \otimes \GG \) which must also be surjective on the stalk at \( p \) since the tensor product is right-exact.
$$\tag*{$\blacksquare$}$$
Exercise 15.3.C:
Suppose \(\FF\) is a finite type quasicoherent sheaf on \(X\).
Show that \(\FF\) is globally generated at \(p\) if and only if "the fiber of \(\FF\) is generated by global sections at \(p\)", i.e., the map from global sections to the fiber \(\FF_p/ \mm \FF_p\) is surjective, where \(\mm\) is the maximal ideal of \(\OO_{X,p}\). (Hint: Geometric Nakayama, Exercise 13.7.E.)
Show that if \(\FF\) is globally generated at \(p\), then "\( \FF \) is globally generated near \(p\)": there is an open neighborhood \(U\) of \(p\) such that \(\FF\) is globally generated at every point of \(U\).
Suppose further that \(X\) is a quasicompact scheme. Show that if \(\FF\) is globally generated at all closed points of \(X\), then \(\FF\) is globally generated at all points of \(X\). (Note that nonempty quasicompact schemes have closed points, Exercise 5.1.E.)
Proof:
We will refer to this StackExchange answer by Keenan Kidwell which shows that the question is not quite worded correctly (as the map \( \Gamma(\FF) \to \FF\vert_p\) need not be surjective).
Since \( \FF \) is assumed to be finite-type and globally generated at \( p \), there exists a map \( \phi : \OO^{\oplus n} \to \FF \) such that the induced map of stalks \( \phi : \OO^{\oplus n}_{X, p} \to \FF_p \) is a surjection. Letting \( a_1, \dots, a_n \) be the images of \(1\) in the various summands, the previous part implies that \( a_1\vert_p, \dots, a_n\vert_p \) generate the fibre \( \FF\vert_p \) so that by Geometric Nakayama, Exercise 13.7.E, there exists an affine open neighborhood \( U \) such that \( a_1\vert_U, \dots, a_n\vert_U \) generate \( \FF_q \) at all \( q \in U \).
By Exercise 5.1.E and part (b) we can cover \( X \) by the neighborhoods at which \( \FF \) is globally generated (this is explained in the paragraph following Exercise 5.1.E since we showed globally generated is an open property).
$$\tag*{$\blacksquare$}$$
Exercise 15.3.D:
If \(\FF\) is a finite type quasicoherent sheaf on \(X\), and \(X\) is quasicompact, show that \(\FF\) is globally generated if and only if it is generated by a finite number of global sections
Proof:
The forward direction is trivial. In the reverse direction, assume \( \FF \) is globally generated so that there exists a surjection \( \OO_X^{\oplus I} \twoheadrightarrow \FF \). Since \( \FF \) is finite type, we may remove all but finitely many of these indices to obtain a surjection \( \OO^{\oplus n} \twoheadrightarrow \FF \).
$$\tag*{$\blacksquare$}$$
Exercise 15.3.E:
An invertible sheaf \(\LL\) on \(X\) is globally generated if and only if for any point \(p \in X\), there is a global section of \( \LL \) not vanishing at \(p\). (See Theorem 16.4.1 for why we care.)
Proof:
For the forward direction, recall by Exercise 2.4.E that we can check whether \( \FF \) is globally generated at the level of stalks. In other words we need only check that if there is a global section of \( \LL \) not vanishing at \( p \) then \( \LL \) is globally generated at \( p \). However, this is obvious given the fact that invertible sheaves are rank 1, so this nonvanishing section is our generator.
In the reverse direction, since surjections can be checked at the level of stalks a surjection \( \OO^{\oplus I} \twoheadrightarrow \FF \) gives us a surjection \( \OO_p^{\oplus I} \twoheadrightarrow \FF_p \) at all points \( p \in X \). Since \( \LL_p \) is a one dimensional vector space over \( \kappa(p) \) we must have one of the global sections is nonvanishing in order for \( \OO_p^{\oplus I} \twoheadrightarrow \FF_p \) to be a surjection.
$$\tag*{$\blacksquare$}$$
Exercise 15.3.F:
Suppose \(s_0, \dots, s_n\) are \(n + 1\) global sections of an invertible sheaf \(\LL\) on a scheme \(X\), with no common zero. Define a corresponding map to \(\P^n\):
$$
\begin{CD}
X @>{[s_0, \dots, s_n]}>> \P^n
\end{CD}
$$
Hint: If \(U\) is an open subset on which \(\LL\) is trivial, choose a trivialization, then translate the \(s_i\) into functions using this trivialization, and use Exercise 6.3.M(a) to obtain a morphism \(U \to \P^n\). Then show that all of these maps (for different \(U\) and different trivializations) "agree", using Exercise 6.3.M(b).
Proof:
As indicated in the hint, choose a trivializing open subset \( U = \spec A \) such that there is an isomorphism \( \phi_U : \LL\vert_U \to \OO\vert_U \). If we let \( \widetilde{s_0}, \dots, \widetilde{s_n} \) denote the images of the \( n+1 \) global sections (considered as functions in \(A\)), we know that the images must also not have any common zeros. Then by Exercise 6.3.M(a) we obtain a morphism \( U \to \P^n \).
If we suppose two trivializing open subsets intersect on some distinguished open affine \( D_A(f) = \spec A_f = \spec B_g = D_B(g) \), then the images of our global sections as functions in \( A \) and \( A_f \) differ by a power of \( f \) on \( D(f) \) — as \( f \) is nowhere vanishing on this set, by Exercise 6.3.M(b) we get that our morphism \( D(f) \to \P^n \) is unchanged. By an identical argument on \( D_B(g) \), we have that the map \( U \cap V \to \P^n \) is independent of taking restrictions and agrees on overlaps.
$$\tag*{$\blacksquare$}$$
Section 15.4: Quasicoherent sheaves and graded modules
Exercise 15.4.A:
Describe a morphism of \(S_0\)-modules \(M_n \to \Gamma(\proj S_\bullet, \widetilde{M(n)}_\bullet)\), extending the \(n = 0\) case of Exercise 15.1.D.
Proof:
By Exercise 15.2.C we can be cavalier about how we interpret \( \widetilde{M}(n)_\bullet \), i.e. the order of sheafification and shifting degrees. Thus, we know that for any \( f \in M_n \), \( f \) is degree 0 in the graded \( S_0 \)-module \( M(n) \) so by the argument of Exercise 15.1.D we get a natural map \( M_n \to (( M_\bullet(n) )_g)_0 \) for any \( g \in S_+ \) by sending \( x \mapsto \frac{x}{1} \).
$$\tag*{$\blacksquare$}$$
Exercise 15.4.B:
Show that \(\Gamma_\bullet(\FF)\) is a graded \(S_\bullet\)-module. (Hint: consider \(S_n \to \Gamma (\proj S_\bullet , \OO (n))\) .)
Proof:
We already defined \( \Gamma_n(\FF) := \Gamma(\proj S_\bullet , \FF(n)) \) so that \( \Gamma_\bullet(\FF) \) has a direct sum decomposition as abelian groups. It remains to show that \( S_n \cdot \Gamma_m(\FF) \subseteq \Gamma_{m + n}(\FF) \). On this note, we follow the hint by considering the natural map \( S_n \to \Gamma(\proj S_\bullet, \OO(n)) \). Since Exercise 15.1.D tells us that \( \FF(n + m) \cong \FF(n) \otimes \OO(m) \), we consider the \( S_\bullet \) action obtained by tensoring a global section of \( \FF(m) \) with the global section of \( \OO(n) \) obtained from the natural map \( S_n \to \Gamma(\proj S_\bullet, \OO(n)) \).
$$\tag*{$\blacksquare$}$$
Exercise 15.4.C:
Show that the map \(M_\bullet \to Γ_\bullet ( \widetilde{M_\bullet}) \) arising from the previous two exercises is a map of \(S_\bullet\)-modules. We call this the saturation map.
Proof:
By Exercise 15.4.A we know that for every \( n \) the map \( M_n \to \Gamma( \proj S_\bullet, \widetilde{M_\bullet (n)} ) \) is in fact a morphism of \( S_0 \)-modules. Letting \( g \in S_+ \) be an arbitrary homogeneous element, we obtain a commuting diagram
It is fairly easy to show that the bottom morphism \( (f, \frac{m}{1}) \mapsto \frac{fm}{1} \) is well-defined, especially under the assumption that \( S_\bullet \) is finitely-generated in degree 1. In particular, this induces an \( S_\bullet \)-action on the global sections of \( S_\bullet \)
$$\tag*{$\blacksquare$}$$
Exercise 15.4.D:
Show that the saturation map need not be injective, nor need it be surjective. (Hint: \(S_\bullet = k[x], M_\bullet = k[x]/(x^2)\) or \(M_\bullet = xk[x]\).)
On the other hand, show that if \( S_\bullet \) is a finitely generated graded ring over a field \(S_0 = k\), and \(M_\bullet\) is finitely generated, then the saturation map is an isomorphism in large degree. In other words, show that there exists an \(n_0\) such that \(M_n \to \Gamma ( \proj S_\bullet, \widetilde{M(n)_\bullet} ) \), is an isomorphism for \( n \geq n_0 \).
Proof:
To see that the map need not be injective, recall by 1.3.3.1 that if a multiplicatively closed set \( S \subset A \) contains zerodivisors then \( S^{-1} A \) is in fact the zero ring. Thus, by choosing \( M_\bullet = k[x] / (x^2) \) (as in the hint) we have that \( (k[x] / (x^2))_x \) is in fact the zero ring so that \( \Gamma( \proj S_\bullet, \widetilde{M_\bullet } ) = 0 \).
To see that the map need not be surjective, we follow the hint and consider \( M_\bullet = x k[x] \). First, notice that \( (xk[x])_x = (k[x])_x \) so that in degree 0 we have \( \Gamma_0( \widetilde{M}_\bullet ) = k \) as usual. However, \( M_\bullet \) is trivial in degree 0 so that \( M_0 \to \Gamma_0( \widetilde{M}_\bullet ) \) is the trivial map which is not surjective.
This is actually a quite difficult theorem proved in EGA relying on filtrations of modules over Noetherian rings: we will refer to Takumi Murayama's adaptation for those interested.
$$\tag*{$\blacksquare$}$$
Exercise 15.4.E:
Show that \(\Gamma_\bullet\) is a functor from \(\mathscr{Q}Coh_{\proj S_\bullet}\) to the category of graded \(S_\bullet\)-modules. In other words, if \(\FF \to \GG\) is a morphism of quasicoherent sheaves on \(\proj S_\bullet\) , describe the natural map \(\Gamma_\bullet \FF \to \Gamma_\bullet \GG\) , and show that such maps respect the identity and composition. Thus the saturation map can be better called the saturation functor.
Proof:
This is trivial since \( \sim \) is shown to be a functor prior to Exercise 15.1.B and \( \Gamma \) is a left-exact functor.
$$\tag*{$\blacksquare$}$$
Exercise 15.4.F:
Define the natural map \(\Gamma_\bullet \FF \to \FF\) as follows. First describe the map on sections over \(D(f)\). Note that sections of the left side are of the form \(m / f^n\) where \(m \in \Gamma_n \deg f(\FF)\), and \(m/f^n = m^\prime/f^{n\, prime}\) if there is some \(N\) with \(f^N(f^{n\, \prime} m − f^n m^\prime) = 0\). Sections on the right are implicitly described in Exercise 13.3.H. Show that your map behaves well on overlaps \(D(f) \cap D(g) = D(fg)\).
Proof:
By taking \( \mathscr{L} = \OO(1) \) and \( s = f \), this should in fact follow from Exercise 13.3.H by construction since \( \FF \otimes \mathscr{L}^{\otimes n} = \FF(n) \) — showing that this agrees on overlaps seems redundant.
$$\tag*{$\blacksquare$}$$
Exercise 15.4.G:
Show that the natural map \( \Gamma_\bullet \FF \to \FF\) is an isomorphism, by showing that it is an isomorphism of sections over \(D(f)\) for any \(f\). First show surjectivity, using Exercise 13.3.H to show that any section of \(\FF\) over \(D(f)\) is of the form \(m/f^n\) where \(m \in \Gamma_n \deg f (\FF )\). Then verify that it is injective.
Proof:
Injectivity should also follow from Exercise 13.3.H considering the map is shown to be an isomorphism.
$$\tag*{$\blacksquare$}$$
Exercise 15.4.H:
Show that each closed subscheme of \(\proj S_\bullet\) arises from a homogeneous ideal \(I_\bullet \subset S_\bullet\) . (Hint: Suppose \(Z\) is a closed subscheme of \(\proj S_\bullet\) . Consider the exact sequence \(0 \to \mathscr{I}_Z \to \OO_{\proj S_\bullet} \to \OO_Z \to 0\). Apply \(\Gamma_\bullet\), and then \( \sim \). Be careful: \(\Gamma_\bullet\) is left-exact, but not necessarily exact.) This fulfills promises made in Exercises 8.2.B and 15.1.E.
Proof:
Although \( \Gamma_\bullet \) is only left-exact in practice, we may use the local construction of our ideal exact sequence to see that on open affines \( D(f) \) it restricts to one of the form \( 0 \to (I_\bullet)_f \to (S_\bullet)_f \to (S_\bullet / I_\bullet)_f \to 0 \) —pon taking the degree 0 part, we obtain an exact sequence via the \( \Gamma_\bullet \) functor.