Reading: Class notes. Atiyah-MacDonald: Exercises 2.14-2.19 and the Topology and Completions section from Chapter 10. Eisenbud: Introduction to Chapter 6, Sections 6.1, 6.2, 6.3 (Proposition 6.1, Corollary 6.3 through Corollary 6.5), 7.1, 7.2 (through Theorem 7.2), and 7.5 (Proposition 7.12, Corollary 7.13, and proofs of Theorems 7.1 and 7.2)]
Exercises: Eisenbud: 1.23, 6.1, 7.11.
Exercise G: Show that the functor of direct limit on the category of R-modules is exact.
Exercise H: Let R be a ring, x ∈ R and M an R-module. Show
that the direct limit of the following system
M → M →
M → ...,
where each arrow is multiplication by x, is isomorphic to
localization Mx.
Exercise I: Show that the functor of inverse limit on the
category of R-modules is left-exact in the following, strong
sense. If
0 → M'i → M i →
M''i
is eventually exact, then the sequence of inverse limits
0 → lim M'i → lim M i →
lim M''i
is exact.
Last modified: (2020-02-12 01:36:03 CST)