Zero-stability: Examples « Tom’s Math Weblog

Zero-stability: Examples

By trdunlap2

Example zero

Suppose I consists of a single element. Then a zero stable module is pair of maps $a:W\rightarrow V$ and $b:V\rightarrow W$ such that $a$ is surjective, $b$ is injective and $ab=0$ .

The existence of zero-stable modules is only possible then if the dimension of W is at least twice the dimension of V. In the case of the smallest possible example of this (when W is 2-dimensionl) we are essentially picking two non-zero orthogonal vectors in W.

Example 1

Let $(I,\Omega)$ be a directed graph and V be chosen so that $Hom(V_{s(h)},V_{e(h)})=0$ for all h. Then zero-stable modules are possible only when $\dim W_i \ge 2 \dim V_i$ for all $i\in I$ , and will correspond to a choice of zero-stable module of the type in example zero for each non-trivial $V_i$ .

As a more specific example consider the directed graph consisting of four vertices and four edges formed into a circle. Let $V_1=V_3=\mathbb{C}$ and $V_2=V_0=0$ and $W_i=\mathbb{C}^2$ . Then you can think of a zero-stable module as a pair of bases (each an orthogonal basis) for $\mathbb{C}^2$ .

This entry was posted on May 4, 2009 at 2:46 pm and is filed under Uncategorized. You can follow any responses to this entry through the RSS 2.0 feed. You can leave a response, or trackback from your own site.

One Response to “Zero-stability: Examples”

trdunlap2 Says:
May 4, 2009 at 2:55 pm | Reply
Note: I calculate $\mathfrak{z}^s_0$ in that last example to be $(\mathbb{C}^*)\times (\mathbb{C}^*)$ . With sufficient choice of $\rho$ the defining $\mathbb{C}^*$ -action will have four fixed points in the closure, none in the interior.

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