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Classification of Algebraic Complexes

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Part of the book series: Algebra and Applications ((AA,volume 17))

Abstract

By an algebraic n-complex over Z[G] we mean an exact sequence of Z[G]-modules

$$E_* = (0 \rightarrow J \rightarrow E_n \stackrel{\partial_n}{\rightarrow} E_{n-1} \stackrel{\partial _{n-1}}{\rightarrow} \cdots\stackrel{\partial_2}{\rightarrow}E_1 \stackrel{\partial_1}{\rightarrow} E_0\rightarrow\mathbf{ Z} \rightarrow0)$$

in which each E r is finitely generated and stably free over Z[G]. The notion is an abstraction from a cell complex X with π 1(X)=G for which \(\pi_{r}(\tilde{X}) = 0\) for 0<r<n. In this chapter we use the Swan homomorphism of Chap. 7 to classify algebraic n-complexes up to homotopy equivalence.

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Notes

  1. 1.

    An interpretation of Yoneda product as composition in the derived module category is given in the thesis of Gollek [33].

References

  1. Gollek, S.: Computations in the derived module category. Ph.D. Thesis, University College London (2010)

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  2. Johnson, F.E.A.: Stable Modules and the D(2)-Problem. LMS Lecture Notes in Mathematics, vol. 301. Cambridge University Press, Cambridge (2003)

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  3. Johnson, F.E.A.: Rigidity of hyperstable complexes. Arch. Math.. 90, 123–132 (2008)

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  4. MacLane, S.: Homology. Springer, Berlin (1963)

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  5. Whitehead, J.H.C.: Combinatorial homotopy II. Bull. Am. Math. Soc. 55, 453–496 (1949)

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Correspondence to F. E. A. Johnson .

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© 2012 Springer-Verlag London Limited

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Johnson, F.E.A. (2012). Classification of Algebraic Complexes. In: Syzygies and Homotopy Theory. Algebra and Applications, vol 17. Springer, London. https://doi.org/10.1007/978-1-4471-2294-4_8

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