By Edward H. Spanier

ISBN-10: 0070598835

ISBN-13: 9780070598836

Meant to be used either as a textual content and a reference, this booklet is an exposition of the elemental rules of algebraic topology. the 1st 3rd of the e-book covers the elemental staff, its definition and its software within the research of protecting areas. the focal point then turns to homology idea, together with cohomology, cup items, cohomology operations, and topological manifolds. the remainder 3rd of the e-book is dedicated to Homotropy concept, protecting easy evidence approximately homotropy teams, functions to obstruction concept, and computations of homotropy teams of spheres. within the later components, the most emphasis is at the software to geometry of the algebraic instruments constructed previous.

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**Example text**

How is this possible? , it must have a common factor with C ( X, Y )! In other words, f λ0 ( X, Y ) = C ( X, Y ) · ( X, Y ) where ( X, Y ) is a linear function — since C ( X, Y ) is quadratic and f λ0 ( X, Y ) is cubic. The zero-set of f λ0 ( X, Y ) still contains all of the intersections of the two sets of lines extending the sides of the hexagon. Since six of the intersection lie in the conic section defined by C, the other three must lie in the set ( X, Y ) = 0 In particular, these three intersections lie on a line — called a Pascal Line.

5 on page 405 implies that rm is integral over F [y1 , . . , yq ] as well. 3 (Hilbert’s Nullstellensatz (weak form)). The maximal ideals of k [ X1 , . . , Xn ] are precisely the ideals for all points I ( a 1 , . . , a n ) = ( X1 − a 1 , X2 − a 2 , . . , X n − a n ) ( a 1 , . . , a n ) ∈ An Consequently every proper ideal a ⊂ k[ X1 , . . , Xn ] has a 0 in An . R EMARK . 29 on page 340 for a discussion of the properties of maximal ideals. P ROOF. Clearly k[ X1 , . . , Xn ]/I( a1 , . . , an ) = k The projection k [ X1 , .

Xn−1 + λn−1 Xn , Xn ) is f d (λ1 , . . , λn−1 , 1). Since F is infinite, there is a point (λ1 , . . , λn−1 ) ∈ F n−1 for which f d (λ1 , . . , λn−1 , 1) = 0 (a fact that is easily established by induction on the number of variables). The following result is called the Noether Normalization Theorem or Lemma. It was first stated by Emmy Noether in [125] and further developed in [126]. 12 on page 77. 2 (Noether Normalization). Let F be an infinite field and suppose A = F [r1 , . . , rm ] is a finitely generated F-algebra that is an integral domain with 40 2.

### Algebraic topology by Edward H. Spanier

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