MEMOIRS

8.1. SLOPE CATEGORIES AND THE RATIONAL HELIX 103for all X, Y ∈Hand all m, n ∈ Z we have Hom D (X[n],Y[m]) = Ext m−nH(X, Y ).Note that Ext i H(−, −) =0fori ∈ Z, i ≠ 0, 1. Here, X[n] denotes the element inthe copy H[n] which corresponds to X ∈H. The automorphism T on D, whichisinduced by the assignment X ↦→ X[1], is called translation functor.8.1.3 (Riemann-Roch formula). Let p be the least common multiple of theweights p 1 ,...,p t . Recall that for any x, y ∈ K 0 (X)∑p−1〈〈x, y〉〉 = 〈τ j x, y〉.j=0For any x, y ∈ K 0 (X) the following formula holds ([66, 70]).〈〈x, y〉〉 = κε∣ rk x rk ydeg x deg y∣ ,which in case rk x ≠0≠rky can also be written as 〈〈x, y〉〉 = κε rk x rk y(µy −µx).As application one gets: If X, Y ∈Hare indecomposable with µ(X)