Can anyone give an example of a sheaf that is supple, but not flabby?

Consider sheafs $\mathcal{F}$ of Abelian groups over $X$.

  • it is flabby if for any $U$ open subset of $X$, the restriction $\rho_{V,U} : \mathcal{F}(U)\to\mathcal{F}(V)$ is surjective.
  • It is said supple if, for $Z_1,Z_2$ closed in $U$ and $Z=Z_1\cup Z_2$, then any $f \in \Gamma_Z(U,\mathcal{F})$ (a section with support contained in $Z$) can be split as a sum $f_1+f_2$, with $f_i\in\Gamma_{Z_i}(U,\mathcal{F})$.

A flabby sheaf is supple, and the converse is not true, but I don't know any example... Actually I don't even know if supple is a standard notion or nomenclature.

Also, a bonus question. Take $Z_1\ldots Z_m$ closed subsets of $U$, and $Z=Z_1\cup Z_m$. Take a given sequence $f_i \in \Gamma_{Z_i}(U,\mathcal{F})$ such that $\sum f_i=0$. Then if $\mathcal{F}$ is flabby, we can split antissymetrically each $f_i$ as $f_i = \sum f_{ij}$ with $f_{ij} = -f_{ji} \in\Gamma_{Z_i\cap Z_j}(U,\mathcal{F})$. Is the same possible for supple sheafs?


This was an exercise for a course I took. I transcribe my submitted solution in the following:

One example is the sheaf of Colombeau generalized functions, as described in theorem 6.2 of Local properties of Colombeau generalized functions by Oberguggenberger, Pilipovic and Scarpalezos. We shall describe roughly the result here, for the one dimensional case.

For $\Omega$ an open set of $\mathbf{R}$, consider $C^\infty(M)$ the locally convex topological algebra with seminorms $$ \| f \|_n = \sup \{ |f^k(x)| \; | \; k\le n, x\in K_n \}$$ where $\{K_n\}$ is an increasing sequence of compact sets whose union is $\Omega$, and $f^k$ denotes the $k$-th derivative of $f$. Nets of functions $f_\epsilon \in C^\infty(\Omega)$ will be indexed by $\epsilon \in (0,1)$ and in particular, we shall deal with the class of moderate nets $\mathcal{E}(\Omega)$ and the class of null nets $\mathcal{N}(\Omega)$ given by the following.

$ \mathcal{E}(\Omega) = \{ (f_\epsilon)_\epsilon \,|\, \forall n\in\mathbf{N}, \exists a\in\mathbf{R} \text{ s.t. } \|{f_\epsilon}\|_n = O(\epsilon^\alpha) \}, $

$ \mathcal{N}(\Omega) = \{ (f_\epsilon)_\epsilon \,|\, \forall n\in\mathbf{N}, \forall a\in\mathbf{R} \text{ s.t. } \|{f_\epsilon}\|_n = O(\epsilon^\alpha) \}, $

That means, moderate nets are such that the $n$-norms does not diverge wildly, , and null nets are such that the $n$-norms vanish rapidly, both as $\epsilon$ goes to zero. $\mathcal{N}$ is an ideal of $\mathcal{N}$, and the Colombeau generalized functions at $\Omega$ are defined as $\mathcal{G}(\Omega) = \mathcal{E}(\Omega)/\mathcal{N}(\Omega)$.

That $\mathcal{G}$ is a sheaf is obvious.

The easiest way to see that $\mathcal{G}$ is not flabby is by taking a counterexample. For $\Omega = (0,\infty)$, take $f_\epsilon(x) = \epsilon^{-1/x}$. Then $f = [(f_\epsilon)_\epsilon]$ is a non-zero element of $\mathcal{G}(\Omega)$ which has no lift to $\mathbf{R}$ since it diverges at the origin for any $\epsilon >0$.

To prove that it is supple, take $Z = Z_1 \cup Z_2$ with $Z_i$ closed sets in $\Omega$. Also, for $C$ a set, denote $C_\epsilon = \{x\in\Omega, d(x,C) < \epsilon\}$. Suppose that $f \in \Gamma_Z(\Omega,\mathcal{G})$, such that $f = [(f_\epsilon)_\epsilon]$. Now, we can pick a net of positive functions $\eta_\epsilon \in C^\infty(\Omega)$ such that $|\eta_\epsilon^k(x)| \le C_k\epsilon^-{p_k}$ for $x\in\Omega$ and such that

$ \eta_\epsilon(x) = \begin{cases} 1 & \text{if } x \in Z_1, \\ 0 & \text{if } x \in \Omega\setminus (Z_1)_\epsilon. \end{cases}$

Then $g_\epsilon = f_\epsilon\eta_\epsilon$ and $h_\epsilon = f_\epsilon(1-\eta_\epsilon)$ are moderate nets, with $f_\epsilon = g_\epsilon + h_\epsilon$. Now, $g = [(g_\epsilon)_\epsilon]$ has support in $Z_1$ since, for any $K$ compact in $\Omega\setminus Z_1$, there is a $\delta>0$ such that $K \cap (Z_1)_\epsilon = \emptyset$ for $\epsilon<\delta$, and hence $|g^n(x)|\sim |f^i(x)\eta^j(x)| = 0$ for $x\in K$. Likewise, $h = [(h_\epsilon)_\epsilon]$ has support in $Z_2$. Hence the split $f=g+h$ is such that $g$ has support in $Z_1$ and $h$, in $Z_2$, thus proving the suppleness of $\mathcal{G}$.


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