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  1. Purity, spectra and localisation.Mike Prest - 2009 - New York: Cambridge University Press.
    The central aim of this book is to understand modules and the categories they form through associated structures and dimensions, which reflect the complexity of these, and similar, categories.
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  • Combinatorics with definable sets: Euler characteristics and Grothendieck rings.Jan Krají Cek & Thomas Scanlon - 2000 - Bulletin of Symbolic Logic 6 (3):311-330.
    We recall the notions of weak and strong Euler characteristics on a first order structure and make explicit the notion of a Grothendieck ring of a structure. We define partially ordered Euler characteristic and Grothendieck ring and give a characterization of structures that have non-trivial partially ordered Grothendieck ring. We give a generalization of counting functions to locally finite structures, and use the construction to show that the Grothendieck ring of the complex numbers contains as a subring the ring of (...)
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  • (1 other version)Grothendieck rings of ℤ-valued fields.Raf Cluckers & Deirdre Haskell - 2001 - Bulletin of Symbolic Logic 7 (2):262-269.
    We prove the triviality of the Grothendieck ring of a Z-valued field K under slight conditions on the logical language and on K. We construct a definable bijection from the plane K 2 to itself minus a point. When we specialized to local fields with finite residue field, we construct a definable bijection from the valuation ring to itself minus a point.
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  • Canonical forms for definable subsets of algebraically closed and real closed valued fields.Jan E. Holly - 1995 - Journal of Symbolic Logic 60 (3):843-860.
    We present a canonical form for definable subsets of algebraically closed valued fields by means of decompositions into sets of a simple form, and do the same for definable subsets of real closed valued fields. Both cases involve discs, forming "Swiss cheeses" in the algebraically closed case, and cuts in the real closed case. As a step in the development, we give a proof for the fact that in "most" valued fields F, if f(x),g(x) ∈ F[ x] and v is (...)
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  • Combinatorics with definable sets: Euler characteristics and grothendieck rings.Jan Krajíček & Thomas Scanlon - 2000 - Bulletin of Symbolic Logic 6 (3):311-330.
    We recall the notions of weak and strong Euler characteristics on a first order structure and make explicit the notion of a Grothendieck ring of a structure. We define partially ordered Euler characteristic and Grothendieck ring and give a characterization of structures that have non-trivial partially ordered Grothendieck ring. We give a generalization of counting functions to locally finite structures, and use the construction to show that the Grothendieck ring of the complex numbers contains as a subring the ring of (...)
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  • (1 other version)Grothendieck Rings of $mathbb{Z}$-Valued Fields.Raf Cluckers & Deirdre Haskell - 2001 - Bulletin of Symbolic Logic 7 (2):262-269.
    We prove the triviality of the Grothendieck ring of a $\mathbb{Z}$-valued field K under slight conditions on the logical language and on K. We construct a definable bijection from the plane K$^2$ to itself minus a point. When we specialized to local fields with finite residue field, we construct a definable bijection from the valuation ring to itself minus a point.
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  • Model Theory and Modules.Mike Prest - 1989 - Journal of Symbolic Logic 54 (3):1115-1118.
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