By Howard Eves
From the book's preface:
Since writing the preface of the 1st version of this paintings, the gloomy plight there defined of starting collegiate geometry has brightened significantly. The pendulum turns out certainly to be swinging again and a goodly quantity of fine textual fabric is showing.
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Extra resources for A Survey of Geometry (Revised Edition)
Let [M ], [N ] ∈ Iso(A), then we see that G∗ ([M ] · [N ]) = M, N A m R FM,N [G (R)], [R]∈Iso(A) whilst G∗ ([M ]) · G∗ ([N ]) = G (M ), G (N ) B m FGS(M ),G (N ) [S]. [S]∈Iso(B) Therefore these two expressions are equal if G preserves the Euler-form and if it preserves the R sets FM,N for all objects M, N, R of A. Equivalently, this is so if G is an exact functor and ∼ if it is extremely faithful: it induces isomorphisms ExtiA (M, N ) −→ ExtiB (G (M ), G (N )) for all M, N ∈ Ob(A), for all i 0.
20. Let Q be a quiver. The dimension of the space Ext1Q (S(i), S(j)) equals the number cij of arrows in Q from i to j. 1 Generalities on quivers Proof. 14. When i = j, the domain of the map c ≡ cS(i),S(j) : Homk (S(i)s(α) , S(j)t(α) ) ∼ = Homk (S(i)l , S(j)l ) −→ l∈Q0 Homk (k, k) α:i→j α∈Q1 is the zero-dimensional vector space. Therefore, the cokernel of c equals the codomain of c, which in turn equals Ext1Q (S(i), S(j)). The isomorphism in the above equation shows that the dimension of the codomain of c is precisely cij .
Choose a vertex i0 such that Vi0 = 0, and take a non-zero vector v ∈ Vi0 . Since M is nilpotent, there exists a path p = (i1 |αhn | . . |αh1 |i0 ) in Q such that p · v = 0 but (α · p) · v = 0 for any arrow leaving the vertex i1 . But this means that S(i1 ) M is a non-zero subrepresentation. We infer that M = S(i1 ) by simplicity of M . 20. Let Q be a quiver. The dimension of the space Ext1Q (S(i), S(j)) equals the number cij of arrows in Q from i to j. 1 Generalities on quivers Proof. 14. When i = j, the domain of the map c ≡ cS(i),S(j) : Homk (S(i)s(α) , S(j)t(α) ) ∼ = Homk (S(i)l , S(j)l ) −→ l∈Q0 Homk (k, k) α:i→j α∈Q1 is the zero-dimensional vector space.