By Massey

ISBN-10: 038797430X

ISBN-13: 9780387974309

This ebook is meant to function a textbook for a path in algebraic topology before everything graduate point. the most subject matters coated are the category of compact 2-manifolds, the elemental staff, overlaying areas, singular homology thought, and singular cohomology thought. those themes are constructed systematically, averting all unecessary definitions, terminology, and technical equipment. anyplace attainable, the geometric motivation at the back of some of the suggestions is emphasised. The textual content involves fabric from the 1st 5 chapters of the author's previous ebook, ALGEBRAIC TOPOLOGY: AN creation (GTM 56), including just about all of the now out-of- print SINGULAR HOMOLOGY conception (GTM 70). the cloth from the sooner books has been conscientiously revised, corrected, and taken brand new.

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**Example text**

The fundamental result that gets the theory oﬀ the ground is the existence of a “structure sheaf” with respect to the Tate topology. 3. Let A be a k-aﬃnoid algebra. The assignment U → AU of the coordinate ring to every aﬃnoid subdomain of M (A) uniquely extends to a sheaf OA with respect to the Tate topology on M (A). In particular, if {Ui } is a ﬁnite collection of aﬃnoid subdomains with U = ∪Ui also an aﬃnoid subdomain of M (A) then the evident sequence 0 → AU → AUi → AUi ∩Uj is exact. ˇ Tate proved this theorem by heavy use of Cech-theoretic methods to reduce to the special case of a Laurent covering of M (A) by the pair {M (A a ), M (A 1/a )} for a ∈ A.

For this reason, it requires some thought to deﬁne the canonical topology. The motivation for the deﬁnition comes from the following concrete description of M (A). 1. Let A be a k-aﬃnoid algebra, and k an algebraic closure of k. For each x ∈ M (A) if we choose a k-embedding i : k(x) → k then we get a k-algebra map A → k whose image lies in a subextension of ﬁnite degree over k. Let A(k) denote the set of k-algebra maps A → k with image contained in a subﬁeld of ﬁnite degree over k; this set has contravariant functorial dependence on A.

Also show that admissibility for subsets is a “local” property in the following sense: if U ⊆ M (A) is an admissible open and {Ui } is an admissible covering of U by admissible opens then a subset V ⊆ U is admissible open in M (A) if and only if V ∩ Ui is admissible open in M (A) for all i. 24 1. 12. Let A = k x, y . Inside M (A) we will construct a subset U that is open for the canonical topology but is not an admissible open subset. Fix c ∈ k× with 0 < |c| < 1, and for i ≥ 1 let Ui = {|x| ≤ |c|i , |y| ≤ |c|1/i } ⊆ M (A).

### A basic course in algebraic topology by Massey

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