Algebraic K-theory: Proceedings of a conference held at by R. Keith Dennis

By R. Keith Dennis

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Extra resources for Algebraic K-theory: Proceedings of a conference held at Oberwolfach, June 1980, Part I

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The definition is formally extended to the elements {−[Pi ]| i ∈ Q0 } by declaring F−[Pi ] (y) := 1, (∀i ∈ Q0 ). 10. For any M, N ∈ A–mod, FM ⊕N = FM FN . Proof. This proof is due to Derksen–Weyman–Zelevinsky [22]. The 1–dimensional torus T = C∗ acts on M ⊕N by λ·(m, n) := (m, λn). This defines an automorphism of M ⊕ N and hence it descends to an action of T on the quiver Grassmannian Gre (M ⊕ N ). The T–fixed points are direct sums of subrepresentations of M and of N: Gre (M ⊕N )T = f +h=e Grf (M )×Grh (N ).

It follows that they are not isomorphic (to see this one can notice that Gr(1,1,1,2) (E) is Fano, while Gr(1,1,1,2) (F ) is not). 1. Positivity. In this section we prove that quiver Grassmannians which are smooth of minimal dimension have positive Euler characteristic. This is based on the following key result. 5. For every indecomposable representation M of a Dynkin quiver Q, and every dimension vector e, the quiver Grassmannian Gre (M ) has zero odd cohomology. In particular, χ(Gre (M )) ≥ 0.

In other words, the map Hom(N, π) : HomQ (N, E) → HomQ (N, M ) induced by π is surjective and its kernel is HomQ (N, τ M ). From this we see that [N, τ M ⊕ M ] = [N, E]. 1 this yields an embedding of N into E, contradicting the emptiness of GrdimM (E). Thus N M . Since [M, τ M ] = 0, the only embedding of M into τ M ⊕ M is the canonical one, proving that GrdimM (τ M ⊕ M ) is just a point. The tangent space at this point is isomorphic to HomQ (M, τ M ) which is zero dimensional, proving that GrdimM (τ M ⊕ M ) is a reduced point.

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