By C. R. F. Maunder

ISBN-10: 0521231612

ISBN-13: 9780521231619

Thorough, glossy therapy, basically from a homotopy theoretic standpoint. themes comprise homotopy and simplicial complexes, the elemental staff, homology concept, homotopy concept, homotopy teams and CW-Complexes and different themes. each one bankruptcy includes workouts and recommendations for additional interpreting. 1980 corrected version.

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**Example text**

Show that (t, u) → (t2 − 1, t(t2 − 1)) deﬁnes a bijective morphism Z1 → Z2 which is not an isomorphism. (b) Show that the morphism A1 (k) → V (Y 2 − X 3 ) ⊂ A2 (k), t → (t2 , t3 ) is a bijective morphism that is not an isomorphism. 13. Show that for n ≥ 2 the open subprevariety An (k) \ {0} ⊂ An (k) is not an aﬃne variety. Is A1 (k) \ {0} aﬃne? 14. Let X be a prevariety and let Y be an aﬃne variety. Show that the map Hom(X, Y ) → Hom(k-Alg) (Γ(Y ), Γ(X)), f → f ∗ : ϕ → ϕ ◦ f, is bijective. Deduce that Hom(X, An (k)) = Γ(X)n .

Let Y, Z be linear subspaces of Pn (k). Show that Y ∩ Z is again a linear subspace of dimension ≥ dim(Y ) + dim(Z) − n. Deduce that Y ∩ Z is always non-empty if dim(Y ) + dim(Z) ≥ n. Conversely let Y1 , . . , Yr ⊆ Pn (k) be ﬁnitely many linear subspaces and let 0 ≤ d ≤ n be an integer such that maxi dim(Yi ) + d < n. Show that there exists a linear subspace Z of Pn (k) of dimension d such that Yi ∩ Z = ∅ for all i = 1, . . , r. Deduce that for any ﬁnite subset X ⊂ Pn (k) there exists a hyperplane Z of Pn (k) such that X ∩ Z = ∅.

We denote by R[X0 , . . , Xn ]d the R-submodule of all homogeneous polynomials of degree d. As we can decompose uniquely every polynomial into its homogeneous parts, we have R[X0 , . . , Xn ] = R[X0 , . . , Xn ]d . 58. Let i ∈ {0, . . , n} and d ≥ 0. There is a bijective R-linear map (d) ∼ Φi = Φi : R[X0 , . . , Xn ]d → { g ∈ R[T0 , . . , Ti , . . , Tn ] ; deg(g) ≤ d }, f → f (T0 , . . , 1, . . , Tn ). ) Proof. We construct an inverse map. Let g be a polynomial in the right hand side set d and let g = j=0 gj be its decomposition into homogeneous parts (with respect to T for = 0, .

### Algebraic topology by C. R. F. Maunder

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