«
»

0-stable Modules Part 2

By trdunlap2

Fix a directed graph (I,\Omega) and its double (I,H).  Fix I-graded vector spaces V,W.

Let h_\bullet=(h_n,... h_1) be a path in (I,H).  That is, h_i\in H and s(h_{k+1})=e(h_k) for 1\le k\le n-1.  The length of h_\bullet is n.  Define s(h_\bullet)=s(h_1) and e(h_\bullet)=e(h_n).

P(H) will denote the set of all paths,  P(i,H) will denote the set of paths begining at i, and P(H,j) will denote the paths ending at j.  In practice we will actually these with finite sets depending on the dimension of V — I’ll address this in a moment.

Let (B,a,b) be a module as defined in the previous post.  B_h:V_{s(h)}\rightarrow V_{e(h)} refers to the component of B associated to a particular edge h and B_{h_\bullet}:V_{s(h_\bullet)}\rightarrow V_{e(h_\bullet)} refers a composition of such maps.

Then we may alternatively define zero-stable as follows: (B,a,b) is zero-stable if for all i\in I and every y \in V_i

  1. there exists a path h_\bullet\in P(i,H) such that b_{e(h_\bullet)}B_{h_\bullet}(y)\neq 0 and
  2. there exists a vector x\in\bigoplus_{h_\bullet\in P(H,i)} W_{s(h_\bullet)} such that \sum B_{h_\bullet}a_{s(h_\bullet)}(x_{s(h_\bullet)})=y.

The proof that this is equivelant to the previous definition is straightforward and also demonstrates that we need only consider paths that visit each vertex i no more than \dim(V_i) times.

This entry was posted on April 25, 2009 at 12:07 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.

Leave a Reply