By Bruck R.H.

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If F is a sheaf on a digraph, G, we say that F is edge simple if F(e) is of dimension 0 or 1 for each e ∈ EG . 22. Let F be an inﬁnite ﬁeld. Let F be an edge simple sheaf of F-vector spaces on a digraph, G. (F). htwist 1 Proof. Let {e1 , . . , er } ⊂ E be the edges where F(e) = 0. ,r be indeterminates, and let F(V )(ψ) = F(V ) ⊗F F(ψ). For each ei choose a wi ∈ F(ei ) with wi = 0, and let vi = ai + ψi bi ∈ F(V )(ψ), with ai = F(h, ei )(wi ), bi = F(t, ei )(wi ). Say that a vj is critical for v1 , .

Vr is critical, we may assume it is v1 ; in this case, if some element of v2 , . . , vr is critical for that set, we may assume it is v2 ; continuing in this fashion, there is an s such that for all i < s, vi is critical for vi , . . , vr , and no element of vs , . . , vr is critical for that set. Consider the sheaf F which agrees with F everywhere except that F (ei ) = 0 for i < s (and so F and F agree at all vertices and all ei with i s). Then {vs , . . , vr } is of size r − s + 1, but also the span of {vs , .

Iνv v ∈ν −1 (v) We leave it to the reader to verify that the restriction maps of ν! F are just the natural maps induced by F. Now we see that (ν! F)(VG ) F(VG ), (ν! F)(EG ) F(EG ), 24 1. FOUNDATIONS OF SHEAVES ON GRAPHS with dν! F and dF identiﬁed under the isomorphism. Hence they have the same homology groups, same adjacency matrix, etc. The main diﬀerence is that one is a sheaf on G, the other a sheaf on G . 4. ν! and Contagious Vanishing Theorems. In this section, we comment that vanishing of homology groups of a sheaf implies the vanishing of certain homology groups of related sheaves.

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