Differential Geometry in Physics by Lugo G

By Lugo G

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In particular, the homotopy category of f -local R-modules is equivalent to the homotopy category of Lf (R)-modules. 9 Examples. Let R = Z, pick a prime p, and let f be the map Z − → Z. Then Lf is smashing, and Lf (X) ∼ Z[1/p] ⊗Z X. 5], which is the total left derived functor of the p-completion functor. In particular, Lf (Z) ∼ Zp and Lf (Z/p∞ ) ∼ ΣZp . Since ΣZp is not equivalent to Zp ⊗Z Z/p∞ , Lf is not smashing in this case. The main positive result about smashing localizations is due to Miller.

15] Let f : A → B be a map of R-modules. If A and B are small, or more generally if the cofibre C of f is equivalent to a coproduct of small R-modules, then Lf is smashing. 11 Lemma. If the cofibre C of f : A → B is equivalent to a coproduct of small R-modules, then the class of f -local R-modules is closed under arbitrary coproducts. Proof. Write C ∼ α Cα , where each Cα is small. 4), which is the case if and only if Y is Cα -null for each α. The lemma now follows from the fact that HomR (Cα , –) commutes up to equivalence with coproducts.

Finally x ¯r = rσ(x) x ¯ where σ(x) is an automorphism of R for all x ∈ G; see [25, p. 2] for further details. We shall assume that all rings have a 1, subrings have the same 1, and ring homomorphisms preserve the 1. We say that the element s of R is a nonzerodivisor (sometimes called a regular element) if sr = 0 = rs whenever 0 = r ∈ R; otherwise s is called a zerodivisor. Let S denote the set of non-zerodivisors of the ring R. The simplest extension to noncommutative rings is when the ring R satisfies the right Ore condition, that is given r ∈ R and s ∈ S, then there exists r1 ∈ R and s1 ∈ S such that rs1 = sr1 .

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