The Stacks project

52.28 Application to Lefschetz theorems

In this section we discuss the relation between coherent sheaves on a projective scheme $P$ and coherent modules on formal completion along an ample divisor $Q$.

Let $k$ be a field. Let $P$ be a proper scheme over $k$. Let $\mathcal{L}$ be an ample invertible $\mathcal{O}_ P$-module. Let $s \in \Gamma (P, \mathcal{L})$ be a section1 and let $Q = Z(s)$ be the zero scheme, see Divisors, Definition 31.14.8. For all $n \geq 1$ we denote $Q_ n = Z(s^ n)$ the $n$th infinitesimal neighbourhood of $Q$. If $\mathcal{F}$ is a coherent $\mathcal{O}_ P$-module, then we denote $\mathcal{F}_ n = \mathcal{F}|_{Q_ n}$ the restriction, i.e., the pullback of $\mathcal{F}$ by the closed immersion $Q_ n \to P$.

Proposition 52.28.1. In the situation above assume for all points $p \in P \setminus Q$ we have

\[ \text{depth}(\mathcal{F}_ p) + \dim (\overline{\{ p\} }) > s \]

Then the map

\[ H^ i(P, \mathcal{F}) \longrightarrow \mathop{\mathrm{lim}}\nolimits H^ i(Q_ n, \mathcal{F}_ n) \]

is an isomorphism for $0 \leq i < s$.

Proof. We will use More on Morphisms, Lemma 37.50.1 and we will use the notation used and results found More on Morphisms, Section 37.50 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

\[ \text{depth}(\mathcal{F}_{U, u}) + \dim (\mathcal{O}_{\overline{\{ u\} }, z}) > s + 1 \]

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

\[ H^ i_{Z'}(M') = H^ i_ Z(M) \otimes _ A A' \]

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

\[ \xymatrix{ & H^ i(U, \mathcal{F}_ U) \ar[ld] \ar[d] \ar[r] & \mathop{\mathrm{lim}}\nolimits H^ i(U, \mathcal{F}_ U/f^ n\mathcal{F}_ U) \ar@{=}[d] \\ H^ i(U, \mathcal{F}_ U) \otimes _ A A' \ar@{=}[r] & H^ i(U', \mathcal{F}') \ar[r] & \mathop{\mathrm{lim}}\nolimits H^ i(U', \mathcal{F}'/f^ n\mathcal{F}') } \]

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 (37.50.0.2) we have compatible direct sum decompositions

\[ \mathop{\mathrm{lim}}\nolimits H^ i(U, \mathcal{F}_ U/f^ n\mathcal{F}_ U) = \mathop{\mathrm{lim}}\nolimits \left( \bigoplus \nolimits _{m \in \mathbf{Z}} H^ i(Q_ n, \mathcal{F}_ n \otimes \mathcal{L}^{\otimes m}) \right) \]

and

\[ H^ i(U, \mathcal{F}_ U) = \bigoplus \nolimits _{m \in \mathbf{Z}} H^ i(P, \mathcal{F} \otimes \mathcal{L}^{\otimes m}) \]

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

Lemma 52.28.2. Let $k$ be a field. Let $X$ be a proper scheme over $k$. Let $\mathcal{L}$ be an ample invertible $\mathcal{O}_ X$-module. Let $s \in \Gamma (X, \mathcal{L})$. Let $Y = Z(s)$ be the zero scheme of $s$ with $n$th infinitesimal neighbourhood $Y_ n = Z(s^ n)$. Let $\mathcal{F}$ be a coherent $\mathcal{O}_ X$-module. Assume that for all $x \in X \setminus Y$ we have

\[ \text{depth}(\mathcal{F}_ x) + \dim (\overline{\{ x\} }) > 1 \]

Then $\Gamma (V, \mathcal{F}) \to \mathop{\mathrm{lim}}\nolimits \Gamma (Y_ n, \mathcal{F}|_{Y_ n})$ is an isomorphism for any open subscheme $V \subset X$ containing $Y$.

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. $\square$

Example 52.28.3. Let $k$ be a field and let $X$ be a proper variety over $k$. Let $Y \subset X$ be an effective Cartier divisor such that $\mathcal{O}_ X(Y)$ is ample and denote $Y_ n$ its $n$th infinitesimal neighbourhood. Let $\mathcal{E}$ be a finite locally free $\mathcal{O}_ X$-module. Here are some special cases of Proposition 52.28.1.

  1. If $X$ is a curve, we don't learn anything.

  2. If $X$ is a Cohen-Macaulay (for example normal) surface, then

    \[ H^0(X, \mathcal{E}) \to \mathop{\mathrm{lim}}\nolimits H^0(Y_ n, \mathcal{E}|_{Y_ n}) \]

    is an isomorphism.

  3. If $X$ is a Cohen-Macaulay threefold, then

    \[ H^0(X, \mathcal{E}) \to \mathop{\mathrm{lim}}\nolimits H^0(Y_ n, \mathcal{E}|_{Y_ n}) \quad \text{and}\quad H^1(X, \mathcal{E}) \to \mathop{\mathrm{lim}}\nolimits H^1(Y_ n, \mathcal{E}|_{Y_ n}) \]

    are isomorphisms.

Presumably the pattern is clear. If $X$ is a normal threefold, then we can conclude the result for $H^0$ but not for $H^1$.

Before we prove the next main result, we need a lemma.

Lemma 52.28.4. In Situation 52.16.1 let $(\mathcal{F}_ n)$ be an object of $\textit{Coh}(U, I\mathcal{O}_ U)$. Assume

  1. $A$ is a graded ring, $\mathfrak a = A_+$, and $I$ is a homogeneous ideal,

  2. $(\mathcal{F}_ n) = (\widetilde{M_ n}|_ U)$ where $(M_ n)$ is an inverse system of graded $A$-modules, and

  3. $(\mathcal{F}_ n)$ extends canonically to $X$.

Then there is a finite graded $A$-module $N$ such that

  1. the inverse systems $(N/I^ nN)$ and $(M_ n)$ are pro-isomorphic in the category of graded $A$-modules modulo $A_+$-power torsion modules, and

  2. $(\mathcal{F}_ n)$ is the completion of of the coherent module associated to $N$.

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

\[ a : \mathbf{G}_ m \times X \longrightarrow X \]

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

\[ \alpha : (a^*\mathcal{F}_ n) \longrightarrow (p^*\mathcal{F}_ n) \]

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

\[ \beta : (a^*\mathcal{G}_ n) \longrightarrow (p^*\mathcal{G}_ n) \]

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. $\square$

Let $k$ be a field. Let $P$ be a proper scheme over $k$. Let $\mathcal{L}$ be an ample invertible $\mathcal{O}_ P$-module. Let $s \in \Gamma (P, \mathcal{L})$ be a section and let $Q = Z(s)$ be the zero scheme, see Divisors, Definition 31.14.8. Let $\mathcal{I} \subset \mathcal{O}_ P$ be the ideal sheaf of $Q$. We will use $\textit{Coh}(P, \mathcal{I})$ to denote the category of coherent formal modules introduced in Cohomology of Schemes, Section 30.23.

Proposition 52.28.5. In the situation above let $(\mathcal{F}_ n)$ be an object of $\textit{Coh}(P, \mathcal{I})$. Assume for all $q \in Q$ and for all primes $\mathfrak p \in \mathcal{O}_{P, q}^\wedge $, $\mathfrak p \not\in V(\mathcal{I}_ q^\wedge )$ we have

\[ \text{depth}((\mathcal{F}_ q^\wedge )_\mathfrak p) + \dim (\mathcal{O}_{P, q}^\wedge /\mathfrak p) + \dim (\overline{\{ q\} }) > 2 \]

Then $(\mathcal{F}_ n)$ is the completion of a coherent $\mathcal{O}_ P$-module.

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 37.50.1 and we will use the notation used and results found More on Morphisms, Section 37.50 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

\[ \text{depth}(\pi |_ U^*\mathcal{F}''_{n, y}) + \delta _ Z^ Y(y) \geq 3 \]

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

\[ \text{depth}(\pi ^*\mathcal{F}''_{n, y}) + \delta _ Z^ Y(y) \geq \text{depth}(\pi ^*\mathcal{F}''_{n, y}) + \dim (\overline{\{ y\} }) = \text{depth}(\mathcal{F}''_{n, q}) + \dim (\overline{\{ q\} }) + 1 \]

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

\[ M_ n = \Gamma (U, \pi |_ U^*\mathcal{F}_ n) = \bigoplus \nolimits _{m \in \mathbf{Z}} \Gamma (P, \mathcal{F}_ n \otimes _{\mathcal{O}_ P} \mathcal{L}^{\otimes m}) \]

is canonically a graded $A$-module, see More on Morphisms, Equation (37.50.0.2). 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. $\square$

Example 52.28.6. Let $k$ be a field and let $X$ be a proper variety over $k$. Let $Y \subset X$ be an effective Cartier divisor such that $\mathcal{O}_ X(Y)$ is ample and denote $\mathcal{I} \subset \mathcal{O}_ X$ the corresponding sheaf of ideals. Let $(\mathcal{E}_ n)$ an object of $\textit{Coh}(X, \mathcal{I})$ with $\mathcal{E}_ n$ finite locally free. Here are some special cases of Proposition 52.28.5.

  1. If $X$ is a curve or a surface, we don't learn anything.

  2. If $X$ is a Cohen-Macaulay threefold, then $(\mathcal{E}_ n)$ is the completion of a coherent $\mathcal{O}_ X$-module $\mathcal{E}$.

  3. More generally, if $\dim (X) \geq 3$ and $X$ is $(S_3)$, then $(\mathcal{E}_ n)$ is the completion of a coherent $\mathcal{O}_ X$-module $\mathcal{E}$.

Of course, if $\mathcal{E}$ exists, then $\mathcal{E}$ is finite locally free in an open neighbourhood of $Y$.

Proposition 52.28.7. Let $k$ be a field. Let $X$ be a proper scheme over $k$. Let $\mathcal{L}$ be an ample invertible $\mathcal{O}_ X$-module and let $s \in \Gamma (X, \mathcal{L})$. Let $Y = Z(s)$ be the zero scheme of $s$ and denote $\mathcal{I} \subset \mathcal{O}_ X$ the corresponding sheaf of ideals. Let $\mathcal{V}$ be the set of open subschemes of $X$ containing $Y$ ordered by reverse inclusion. Assume that for all $x \in X \setminus Y$ we have

\[ \text{depth}(\mathcal{O}_{X, x}) + \dim (\overline{\{ x\} }) > 2 \]

Then the completion functor

\[ \mathop{\mathrm{colim}}\nolimits _\mathcal {V} \textit{Coh}(\mathcal{O}_ V) \longrightarrow \textit{Coh}(X, \mathcal{I}) \]

is an equivalence on the full subcategories of finite locally free objects.

Proof. To prove fully faithfulness it suffices to prove that

\[ \mathop{\mathrm{colim}}\nolimits _\mathcal {V} \Gamma (V, \mathcal{L}^{\otimes m}) \longrightarrow \mathop{\mathrm{lim}}\nolimits \Gamma (Y_ n, \mathcal{L}^{\otimes m}|_{Y_ n}) \]

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

\[ \text{depth}((\mathcal{O}_{X, y}^\wedge )_\mathfrak p) + \dim (\mathcal{O}_{X, y}^\wedge /\mathfrak p) = \text{depth}(\mathcal{O}_{X, x}) + \dim (\overline{\{ x\} }) - \dim (\overline{\{ y\} }) \]

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. $\square$

[1] We do not require $s$ to be a regular section. Correspondingly, $Q$ is only a locally principal closed subscheme of $P$ and not necessarily an effective Cartier divisor.

Comments (0)


Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.




In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 0EL0. Beware of the difference between the letter 'O' and the digit '0'.