# Get A basic course in algebraic topology PDF

By Massey

ISBN-10: 038797430X

ISBN-13: 9780387974309

This publication is meant to function a textbook for a direction in algebraic topology in the beginning graduate point. the most issues lined are the type of compact 2-manifolds, the basic workforce, masking areas, singular homology concept, and singular cohomology concept. those issues are constructed systematically, heading off all unecessary definitions, terminology, and technical equipment. at any place attainable, the geometric motivation in the back of a number of the options is emphasised. The textual content contains fabric from the 1st 5 chapters of the author's past e-book, ALGEBRAIC TOPOLOGY: AN creation (GTM 56), including just about all of the now out-of- print SINGULAR HOMOLOGY thought (GTM 70). the cloth from the sooner books has been conscientiously revised, corrected, and taken brand new.

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Let R be a ring and I a non-zero ideal of R . Prove that I is a free R -module if and only if it is a principal ideal generated by a non-zerodivisor. Exercise 7. Let R be a ring. Show that the following conditions are equivalent. a) The ring R is a ﬁeld. b) Every ﬁnitely generated R -module is free. c) Every cyclic R -module is free. Exercise 8. Let K be a ﬁeld, P = K[x1 , x2 ] , and I be the ideal in P generated by {x1 , x2 } . Show that I is not a free P -module. Tutorial 1: Polynomial Representation I In what follows we work over the ring K[x, y] , where K is one of the ﬁelds deﬁned in CoCoA.

12. In other words, suppose that T is another R -algebra together with elements t1 , . . , tn ∈ T , such that whenever you have an R -algebra S together with elements s1 , . . , sn ∈ S , then there exists a unique R -algebra homomorphism ψ : T → S satisfying ψ(ti ) = si for i = 1, . . , n . Then show that there is a unique R -algebra isomorphism R[x1 , . . , xn ] → T such that xi → ti for i = 1, . . , n . Exercise 3. Show that the map log : Tn −→ Nn is an isomorphism of monoids. Exercise 4.

If these conditions are satisﬁed, the monoid Γ is called Noetherian. 3 Monomial Ideals and Monomial Modules 43 Proof. First we show a) ⇒ b). Suppose we have a chain ∆1 ⊆ ∆2 ⊆ · · · of monoideals in Γ and a sequence n1 < n2 < · · · such that there exist elements γi ∈ ∆ni+1 \∆ni for all i ≥ 1. Then we claim that the monoideal generated by {γ1 , γ2 , . } is not ﬁnitely generated. It is contained in the union ∪i≥1 ∆i , but not in one of the monoideals ∆i . Now assume that it is generated by a ﬁnite set.

### A basic course in algebraic topology by Massey

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