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By R. L. Goodstein

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We claim there is an exact sequence d dt 0 −−−−→ R(U ) −−−−→ D(U ) −−− −→ D(U ) −−−−→ 0 , where the first map is inclusion, while the second map is differentiation by arclength. To see that this sequence is exact, note that U is the disjoint union of open arcs that are diffeomorphic to the real line. On such an open arc, (i) the constants are the only functions that are annihilated by differentiation, and (ii) every smooth function can be integrated to a smooth function. On the other hand, as we will discuss later, the sequence is not exact on the right if U is S1 .

There are corresponding definitions when F is contravariant, and/or when the functor is right-exact. We leave the formulations of these concepts to the reader. 20. Let F be a left-exact covariant additive functor on the category of R-modules. A module A is acyclic for F if Ri F(A) = 0 for i 1. Now let M be an R-module and ι : M −→ A• a quasi-isomorphism, where A• is a bounded below complex consisting of modules which are acyclic for F. It can then be proved that there are natural isomorphisms Ri F(M ) ∼ = H i (F(A• )) for each i 0.

20. −→ Here is one application of this result. 23. Let A be the category of modules over a commutative ring R, and let M be an R-module. The functor F(−) = M ⊗R − is left adjoint to HomR (M, −). Therefore, by the theorem above, if {Ai }i∈I is an I-diagram in A, then there is a natural identification lim(M ⊗R Ai ) = M ⊗R lim(Ai ) . −→ −→ Thus, direct limits commute with tensor products. 24. Let I be a poset. Let R be a commutative ring, and let {Ai }i∈I and {A′i }i∈I be I-diagrams of R-modules. (1) Prove that there is a natural homomorphism of R-modules lim(Ai ⊗R A′i ) −→ lim(Ai ) ⊗R lim(A′i ) .

### Boolean algebra by R. L. Goodstein

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