Download 2-D Shapes Are Behind the Drapes! by Tracy Kompelien PDF

By Tracy Kompelien

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Title: 2-D Shapes Are in the back of the Drapes!
Author: Kompelien, Tracy
Publisher: Abdo Group
Publication Date: 2006/09/01
Number of Pages: 24
Binding sort: LIBRARY
Library of Congress: 2006012570

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Example text

3 Exercise Exercise 10. Apply the previous paragraph to the case of a cyclic group Γ = Let σ be a generator of the group. We have the exact sequence 0 −→ IΓ −→ [Γ] −→ /n . −→ 0 and IΓ = [Γ](1 − σ). (a) In the case n = 0 (the infinite group) we have that IΓ is a free module. This gives simple formulae for the cohomology and shows H ν (Γ,M ) = 0 for ν ≥ 2. 4 The Functors Ext and Tor 33 (b) In the case of a finite group the map [Γ] −→ IΓ r −→ r(1 − σ) has the kernel [Γ](1 + . . + σ n−1 ). Construct a “periodic” resolution from this and compute the cohomology.

2 ..... . . ... . . ... ... .......... . . h ϕ . ... ... . 3 ...... .... ... . . . ...... h (J )Γ ............................................ (J 1 )Γ ........................................... (J 2 )Γ ................................................. . ) But now it is clear that ϕ• induces zero in the cohomology. If we have a cycle cν ∈ (I ν )Γ representing a given cohomology class then ϕν (cν ) = d ◦ h(cν ) and hence it represents the trivial class. If we apply this to a module M and the identity Id : M −→ M and two different resolutions of M , then we get a unique isomorphism between the resulting cohomology groups.

HomR (P0 ,M ) .................... HomR (P0 ,I 0 ) ................... HomR (P0 ,I 1 ) ................................................ . . .. ... .... 0 ................................................. ......... ... ... . .......... ... ... .. .. ... ... HomR (N,M ) ...................... HomR (N,I 0 ) ....................... HomR (N,I 1 ) ................................................. . . .. .. ... ......... ... ... . 0 0 .. .. .. ... 0 L Ext•R (P,M ) Now the first vertical Complex computes the and the horizontal complex at the bottom computes R Ext•R (P,M ).

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