By Massey

This publication is meant to function a textbook for a direction in algebraic topology in the beginning graduate point. the most subject matters lined are the category of compact 2-manifolds, the basic team, masking areas, singular homology conception, and singular cohomology idea. those themes are constructed systematically, keeping off all unecessary definitions, terminology, and technical equipment. at any place attainable, the geometric motivation in the back of many of the techniques is emphasised. The textual content comprises fabric from the 1st 5 chapters of the author's previous publication, 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 modern.

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That is, the map p defined by the restrictions F (T ) → Fi (T ) is injective, and the image of p is the kernel {(fi )i∈I ∈ F (Ti ): p1 (fi ) = p2 (fi ) for all i ∈ I} i∈I of the projections p1 and p2 induced by the maps F (Ti ) → F (Ti ∩ Tj ), respectively F (Tj ) → F (Ti ∩ Tj ), for all i, j ∈ I. 14) Example. Given a scheme X over S and let Y be a subscheme of X with immersion i: Y → X. We have that hY is a locally closed subfunctor and that Y = XhY ,i . Hence hY is an open or closed subfunctor of hX if and only if Y is an open respectively closed subscheme of X.

Bt respectively. Hence the (s − r)– minors containing the first row can be expanded as a sum of the (s − r)–minors containing rows α1 + 1, . . , αt + 1 multiplied by b1 , . . , bt . We consequently have that J ⊆ Ir . By (transfinite, if necessary) induction, we obtain that the ideal in A obtained from the (s − r)–minors of the matrix obtained by adding to B rows coming from any set of elements of N , is equal to Ir . In particular we obtain that the ideal Ir is independent of the choice of generators nα of N .

2) Definition. We say that F is m–regular if H i (X, F (m − i)) = 0, → → for i > 0. 3) Remark. 2) there is an m0 (F ) such that F is m–regular for m ≥ m0 (F ). 4) Note. For every field extension k ⊆ K, we have: (1) The OX –module F is m–regular if and only if FSpec K is m–regular. 1) for K βm (Spec K): H 0 (XSpec K , FSpec K (m)) ⊗K H 0 (XSpec K , OXK (1)) → H 0 (XSpec K , FSpec K (m + 1)) → is surjective. 5). 5) Lemma. Assume that k = A is an infinite field. Given a non–zero coherent sheaf G on P(E).