# Download Abelian Varieties by D. Mumford PDF By D. Mumford

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A full subcategory C of CΓ is thick if it satisfies the following axiom: If 0 −→ M −→ M −→ M −→ 0 is a short exact sequence in CΓ, then M is in C if and only if M and M are. ) A full subcategory F of FH is thick if it satisfies the following two axioms: (i) If f X −→ Y −→ Cf is a cofibre sequence in which two of the three spaces are in F, then so is the third. (ii) If X ∨ Y is in F then so are X and Y . Thick subcategories were called generic subcategories by Hopkins in [Hop87]. 4. THICK SUBCATEGORIES 37 Using the Landweber filtration theorem, one can classify the thick subcategories of CΓ(p) .

Now suppose C ⊂ CΓ(p) is thick. If C = {0}, choose the largest n so that Cp,n ⊃ C. Then C ⊂ Cp,n+1 , and we want to show that C = Cp,n , so we need to verify that C ⊃ Cp,n . −1 M = 0 Let M be a comodule in C but not in Cp,n+1 . Thus vm −1 for m < n but vn M = 0. Choosing a Landweber filtration of M in CΓ, 0 = F0 M ⊂ F1 M ⊂ · · · ⊂ Fk M = M, all Fs M are in C, hence so are all the subquotients Fs M/Fs−1 M = Σds M U∗ /Ip,ms . Since vn−1 M = 0, we must have vn−1 (M U∗ /Ip,ms ) = 0 for some s, so some ms is no more than n.

1(i) is analogous to the 2. SPANIER-WHITEHEAD DUALITY 53 isomorphism Hom(V, W ) ∼ =V∗⊗W for any vector space W . 1(v) is analogous to the isomorphism (V ⊗ W )∗ ∼ = V ∗ ⊗ W ∗. The geometric idea behind Spanier-Whitehead duality is as follows. A finite spectrum X is the suspension spectrum of a finite CW-complex, which we also denote by X. The latter can always be embedded in some Euclidean space RN and hence in S N . Then DX is a suitable suspension of the suspension spectrum of the complement S N − X.