The Stacks project

Proof. We will use More on Morphisms, Lemma 37.51.1 and we will use the notation used and results found More on Morphisms, Section 37.51 without further mention; this proof will not make sense without at least understanding the statement of the lemma. Observe that in our case $A = \bigoplus _{m \geq 0} \Gamma (P, \mathcal{L}^{\otimes m})$ is a finite type $k$-algebra all of whose graded parts are finite dimensional $k$-vector spaces, see Cohomology of Schemes, Lemma 30.16.1.

We may and do think of $s$ as an element $f \in A_1 \subset A$, i.e., a homogeneous element of degree $1$ of $A$. Denote $Y = V(f) \subset X$ the closed subscheme defined by $f$. Then $U \cap Y = (\pi |_ U)^{-1}(Q)$ scheme theoretically. Recall the notation $\mathcal{F}_ U = \pi ^*\mathcal{F}|_ U = (\pi |_ U)^*\mathcal{F}$. This is a coherent $\mathcal{O}_ U$-module. Choose a finite $A$-module $M$ such that $\mathcal{F}_ U = \widetilde{M}|_ U$ (for existence see Local Cohomology, Lemma 51.8.2). We claim that $H^ i_ Z(M)$ is annihilated by a power of $f$ for $i \leq s + 1$.

To prove the claim we will apply Local Cohomology, Proposition 51.10.1. Translating into geometry we see that it suffices to prove for $u \in U$, $u \not\in Y$ and $z \in \overline{\{ u\} } \cap Z$ that

This requires only a small amount of thought.

Observe that $Z = \mathop{\mathrm{Spec}}(A_0)$ is a finite set of closed points of $X$ because $A_0$ is a finite dimensional $k$-algebra. (The reader who would like $Z$ to be a singleton can replace the finite $k$-algebra $A_0$ by $k$; it won't affect anything else in the proof.)

The morphism $\pi : L \to P$ and its restriction $\pi |_ U : U \to P$ are smooth of relative dimension $1$. Let $u \in U$, $u \not\in Y$ and $z \in \overline{\{ u\} } \cap Z$. Let $p = \pi (u) \in P \setminus Q$ be its image. Then either $u$ is a generic point of the fibre of $\pi $ over $p$ or a closed point of the fibre. If $u$ is a generic point of the fibre, then $\text{depth}(\mathcal{F}_{U, u}) = \text{depth}(\mathcal{F}_ p)$ and $\dim (\overline{\{ u\} }) = \dim (\overline{\{ p\} }) + 1$. If $u$ is a closed point of the fibre, then $\text{depth}(\mathcal{F}_{U, u}) = \text{depth}(\mathcal{F}_ p) + 1$ and $\dim (\overline{\{ u\} }) = \dim (\overline{\{ p\} })$. In both cases we have $\dim (\overline{\{ u\} }) = \dim (\mathcal{O}_{\overline{\{ u\} }, z})$ because every point of $Z$ is closed. Thus the desired inequality follows from the assumption in the statement of the lemma.

Let $A'$ be the $f$-adic completion of $A$. So $A \to A'$ is flat by Algebra, Lemma 10.97.2. Denote $U' \subset X' = \mathop{\mathrm{Spec}}(A')$ the inverse image of $U$ and similarly for $Y'$ and $Z'$. Let $\mathcal{F}'$ on $U'$ be the pullback of $\mathcal{F}_ U$ and let $M' = M \otimes _ A A'$. By flat base change for local cohomology (Local Cohomology, Lemma 51.5.7) we have

and we find that for $i \leq s + 1$ these are annihilated by a power of $f$. Consider the diagram

The lower horizontal arrow is an isomorphism for $i < s$ by Lemma 52.13.2 and the torsion property we just proved. The horizontal equal sign is flat base change (Cohomology of Schemes, Lemma 30.5.2) and the vertical equal sign is because $U \cap Y$ and $U' \cap Y'$ as well as their $n$th infinitesimal neighbourhoods are mapped isomorphically onto each other (as we are completing with respect to $f$).

Applying More on Morphisms, Equation (37.51.0.2) we have compatible direct sum decompositions

and

Thus we conclude by Algebra, Lemma 10.98.4. $\square$

Proof. By Proposition 52.28.1 this is true for $V = X$. Thus it suffices to show that the map $\Gamma (V, \mathcal{F}) \to \mathop{\mathrm{lim}}\nolimits \Gamma (Y_ n, \mathcal{F}|_{Y_ n})$ is injective. If $\sigma \in \Gamma (V, \mathcal{F})$ maps to zero, then its support is disjoint from $Y$ (details omitted; hint: use Krull's intersection theorem). Then the closure $T \subset X$ of $\text{Supp}(\sigma )$ is disjoint from $Y$. Whence $T$ is proper over $k$ (being closed in $X$) and affine (being closed in the affine scheme $X \setminus Y$, see Morphisms, Lemma 29.43.18) and hence finite over $k$ (Morphisms, Lemma 29.44.11). Thus $T$ is a finite set of closed points of $X$. Thus $\text{depth}(\mathcal{F}_ x) \geq 2$ is at least $1$ for $x \in T$ by our assumption. We conclude that $\Gamma (V, \mathcal{F}) \to \Gamma (V \setminus T, \mathcal{F})$ is injective and $\sigma = 0$ as desired. $\square$

Proof. Let $(\mathcal{G}_ n)$ be the canonical extension as in Lemma 52.16.8. The grading on $A$ and $M_ n$ determines an action

of the group scheme $\mathbf{G}_ m$ on $X$ such that $(\widetilde{M_ n})$ becomes an inverse system of $\mathbf{G}_ m$-equivariant quasi-coherent $\mathcal{O}_ X$-modules, see Groupoids, Example 39.12.3. Since $\mathfrak a$ and $I$ are homogeneous ideals the closed subschemes $Z$, $Y$ and the open subscheme $U$ are $\mathbf{G}_ m$-invariant closed and open subschemes. The restriction $(\mathcal{F}_ n)$ of $(\widetilde{M_ n})$ is an inverse system of $\mathbf{G}_ m$-equivariant coherent $\mathcal{O}_ U$-modules. In other words, $(\mathcal{F}_ n)$ is a $\mathbf{G}_ m$-equivariant coherent formal module, in the sense that there is an isomorphism

over $\mathbf{G}_ m \times U$ satisfying a suitable cocycle condition. Since $a$ and $p$ are flat morphisms of affine schemes, by Lemma 52.16.9 we conclude that there exists a unique isomorphism

over $\mathbf{G}_ m \times X$ restricting to $\alpha $ on $\mathbf{G}_ m \times U$. The uniqueness guarantees that $\beta $ satisfies the corresponding cocycle condition. In this way each $\mathcal{G}_ n$ becomes a $\mathbf{G}_ m$-equivariant coherent $\mathcal{O}_ X$-module in a manner compatible with transition maps.

By Groupoids, Lemma 39.12.5 we see that $\mathcal{G}_ n$ with its $\mathbf{G}_ m$-equivariant structure corresponds to a graded $A$-module $N_ n$. The transition maps $N_{n + 1} \to N_ n$ are graded module maps. Note that $N_ n$ is a finite $A$-module and $N_ n = N_{n + 1}/I^ n N_{n + 1}$ because $(\mathcal{G}_ n)$ is an object of $\textit{Coh}(X, I\mathcal{O}_ X)$. Let $N$ be the finite graded $A$-module foud in Algebra, Lemma 10.98.3. Then $N_ n = N/I^ nN$, whence $(\mathcal{G}_ n)$ is the completion of the coherent module associated to $N$, and a fortiori we see that (b) is true.

To see (a) we have to unwind the situation described above a bit more. First, observe that the kernel and cokernel of $M_ n \to H^0(U, \mathcal{F}_ n)$ is $A_+$-power torsion (Local Cohomology, Lemma 51.8.2). Observe that $H^0(U, \mathcal{F}_ n)$ comes with a natural grading such that these maps and the transition maps of the system are graded $A$-module map; for example we can use that $(U \to X)_*\mathcal{F}_ n$ is a $\mathbf{G}_ m$-equivariant module on $X$ and use Groupoids, Lemma 39.12.5. Next, recall that $(N_ n)$ and $(H^0(U, \mathcal{F}_ n))$ are pro-isomorphic by Definition 52.16.7 and Lemma 52.16.8. We omit the verification that the maps defining this pro-isomorphism are graded module maps. Thus $(N_ n)$ and $(M_ n)$ are pro-isomorphic in the category of graded $A$-modules modulo $A_+$-power torsion modules. $\square$

Proof. By Cohomology of Schemes, Lemma 30.23.6 to prove the lemma, we may replace $(\mathcal{F}_ n)$ by an object differing from it by $\mathcal{I}$-torsion (see below for more precision). Let $T' = \{ q \in Q \mid \dim (\overline{\{ q\} }) = 0\} $ and $T = \{ q \in Q \mid \dim (\overline{\{ q\} }) \leq 1\} $. The assumption in the proposition is exactly that $Q \subset P$, $(\mathcal{F}_ n)$, and $T' \subset T \subset Q$ satisfy the conditions of Lemma 52.21.2 with $d = 1$; besides trivial manipulations of inequalities, use that $V(\mathfrak p) \cap V(\mathcal{I}^\wedge _ y) = \{ \mathfrak m^\wedge _ y\} \Leftrightarrow \dim (\mathcal{O}_{P, q}^\wedge /\mathfrak p) = 1$ as $\mathcal{I}_ y^\wedge $ is generated by $1$ element. Combining these two remarks, we may replace $(\mathcal{F}_ n)$ by the object $(\mathcal{H}_ n)$ of $\textit{Coh}(P, \mathcal{I})$ found in Lemma 52.21.2. Thus we may and do assume $(\mathcal{F}_ n)$ is pro-isomorphic to an inverse system $(\mathcal{F}_ n'')$ of coherent $\mathcal{O}_ P$-modules such that $\text{depth}(\mathcal{F}''_{n, q}) + \dim (\overline{\{ q\} }) \geq 2$ for all $q \in Q$.

We will use More on Morphisms, Lemma 37.51.1 and we will use the notation used and results found More on Morphisms, Section 37.51 without further mention; this proof will not make sense without at least understanding the statement of the lemma. Observe that in our case $A = \bigoplus _{m \geq 0} \Gamma (P, \mathcal{L}^{\otimes m})$ is a finite type $k$-algebra all of whose graded parts are finite dimensional $k$-vector spaces, see Cohomology of Schemes, Lemma 30.16.1.

By Cohomology of Schemes, Lemma 30.23.9 the pull back by $\pi |_ U : U \to P$ is an object $(\pi |_ U^*\mathcal{F}_ n)$ of $\textit{Coh}(U, f\mathcal{O}_ U)$ which is pro-isomorphic to the inverse system $(\pi |_ U^*\mathcal{F}_ n'')$ of coherent $\mathcal{O}_ U$-modules. We claim

for all $y \in U \cap Y$. Since all the points of $Z$ are closed, we see that $\delta _ Z^ Y(y) \geq \dim (\overline{\{ y\} })$ for all $y \in U \cap Y$, see Lemma 52.18.1. Let $q \in Q$ be the image of $y$. Since the morphism $\pi : U \to P$ is smooth of relative dimension $1$ we see that either $y$ is a closed point of a fibre of $\pi $ or a generic point. Thus we see that

because either the depth goes up by $1$ or the dimension. This proves the claim.

By Lemma 52.22.1 we conclude that $(\pi |_ U^*\mathcal{F}_ n)$ canonically extends to $X$. Observe that

is canonically a graded $A$-module, see More on Morphisms, Equation (37.51.0.2). By Properties, Lemma 28.18.2 we have $\pi |_ U^*\mathcal{F}_ n = \widetilde{M_ n}|_ U$. Thus we may apply Lemma 52.28.4 to find a finite graded $A$-module $N$ such that $(M_ n)$ and $(N/I^ nN)$ are pro-isomorphic in the category of graded $A$-modules modulo $A_+$-torsion modules. Let $\mathcal{F}$ be the coherent $\mathcal{O}_ P$-module associated to $N$, see Cohomology of Schemes, Proposition 30.15.3. The same proposition tells us that $(\mathcal{F}/\mathcal{I}^ n\mathcal{F})$ is pro-isomorphic to $(\mathcal{F}_ n)$. Since both are objects of $\textit{Coh}(P, \mathcal{I})$ we win by Lemma 52.15.3. $\square$

Proof. To prove fully faithfulness it suffices to prove that

is an isomorphism for all $m$, see Lemma 52.15.2. This follows from Lemma 52.28.2.

Essential surjectivity. Let $(\mathcal{F}_ n)$ be a finite locally free object of $\textit{Coh}(X, \mathcal{I})$. Then for $y \in Y$ we have $\mathcal{F}_ y^\wedge = \mathop{\mathrm{lim}}\nolimits \mathcal{F}_{n, y}$ is is a finite free $\mathcal{O}_{X, y}^\wedge $-module. Let $\mathfrak p \subset \mathcal{O}_{X, y}^\wedge $ be a prime with $\mathfrak p \not\in V(\mathcal{I}_ y^\wedge )$. Then $\mathfrak p$ lies over a prime $\mathfrak p_0 \subset \mathcal{O}_{X, y}$ which corresponds to a specialization $x \leadsto y$ with $x \not\in Y$. By Local Cohomology, Lemma 51.11.3 and some dimension theory (see Varieties, Section 33.20) we have

Thus our assumptions imply the assumptions of Proposition 52.28.5 are satisfied and we find that $(\mathcal{F}_ n)$ is the completion of a coherent $\mathcal{O}_ X$-module $\mathcal{F}$. It then follows that $\mathcal{F}_ y$ is finite free for all $y \in Y$ and hence $\mathcal{F}$ is finite locally free in an open neighbourhood $V$ of $Y$. This finishes the proof. $\square$