Projection lemma

Definition 15.16. LetlA = Xj jg/ be a locally finite open covering of X. A partition of unity for U is a collection of continuous maps y j X - (0,1) such that 

If ll is not assumed to be locally finite, then we define a partition of unity subordinate to U to be a partition of unity for some locally finite refinement ofU. 

Note that the condition (2) in Definition 15.16 can always be weakened in the following way if we found functions Pi iei, ipi X [0,1], such that supp i C Xi and the sum function YhieiTii ) is nowhere zero, then dividing all these functions by we achieve the normalization. 

Definition 15.17. A topological space X is called paracompact if it is Hausdorff and every open covering Xj j / of X has a lomlly finite refinement. 

If X is paracompact, and lA is an open covering of X, then there exists a partition of unity that is subordinate to lA. Moreover, iflA is locally finite, then there exists a partition of unity for lA. 

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Furthermore, by the projection lemma, the map pj hocolimP —> colimP is a homotopy equivalence. Since coIimP = X and hocolimP = N IA), the nerve lemma follows. ... 

Let Si colimPi —> hocolimbe the Segal map defined in the proof of the projection lemma, where it was also shown that it is a homotopy equivalence. Finally, the projection mappj hocolim X>2 —cohmP2 is a homotopy equivalence by the projection lemma, and therefore we conclude that the composition... 

Remark 9.7. It follows from the proof of the lemma that the schemes S X S and S x S x 5 S satisfy the conditions of 9.1. Further we get various commutative diagrams involving lifts of Frobenii and projection maps. This will allow us to use descent arguments. 

Proof. By Proposition 3.5, since V2 is finite dimensional we know that there is an orthogonal projection 112 with range V2. Because p is unitary, the linear transformation 112 is a homomorphism of representations by Proposition 5.4. Thus by Exercise 5.15 the restriction of 112 to Vi is a homomorphism of representations. By hypothesis, this homomorphism cannot be injective. Hence Schur s lemma (Proposition 6.2) implies that since Vi is irreducible, fl2[Vi] is the trivial subspace. In other words, Vi is perpendicidar to V2. ... 

The thin (8 nm) outer cell membrane or "plasma-lemma" (Fig. 1-7) controls the flow of materials into and out of cells, conducts impulses in nerve cells and along muscle fibrils, and participates in chemical communication with other cells. Deep infoldings of the outer membrane sometimes nm into the cytoplasm. An example, is the "T system" of tubules which functions in excitation of muscle contraction (Figs. 19-7, 19-21). Surfaces of cells designated to secrete materials or to absorb substances from the surrounding fluid, such as the cells lining kidney tubules and pancreatic secretory cells, are often covered with very fine projections or microvilli which greatly increase the surface area. 

Then, by substituting v(y) of the above Lemma to the projection problem (6.17) we have... 

Proof. By the lemma it is enough to construct all the finite-dimensional V in Am. Such a V is a subcomodule of the direct sum of its coordinate projections to A, so we may deal just with V in A. The original representation gives us a Hopf algebra surjection of B = k[X11, Xm, 1/det] onto A, and V is contained in the image of some subspace (l/detj j/(Xy) deg(/) < s. These subspaces are B-subcomodules of B, and hence also are /I-subcomodules it will be enough to construct them. 

LEMMA (2, ) If E is finitely generated and projective over A, the same holds for eP,... 

Step 1. For all i G Z/2 Z we take an element 7 Mi which projects onto a (free) generator of the /2/p72-module It follows from Lemma 3.5 that 7 j,... 

Pal ate.—A convex projection on the base of the lower lip of a personate corolla. Pa lea Pal et).—An inner bract of a Grass inflorescence which with the lemma incloses the flower. 

Proof. The construction is a collision-intractable family of hiding functions according to Theorem 8.59 and Lemma 8.65. (The replacement of B by B can be handled as in the proof of Theorem 8.67.) The functions Hr are homomorphisms between groups Gr and Hj according to Theorem 8.16, and is obviously an Abelian group, too. It remains to be shown that Kr is a homomorphism. This is not completely trivial, although TCg is simply a projection, because Gg as a group is not the direct product of Z2T and RQR , but one can immediately see it from the definition of the operation ... 

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