CT5
Lecture 5
CT Generated by Pure Injectives
Technique: Tor-torsion Theory
A tor-torsion theory induces a cotorsion theory with good proeprty.
Say $(\mathcal{A}, \mathcal{B})$ consists of a tor-torsion theory, provided
$$\begin{equation} \mathcal{A}^⊺ := \ker \mathrm{Tor}(\mathcal{A}, -) = \mathcal{B}; \quad {}^⊺\mathcal{B} := \ker \mathrm{Tor}(-, \mathcal{B}) = \mathcal{A}. \end{equation}$$
Show that both $\mathcal{A}$ and $\mathcal{B}$ are closed under summands, extensions, and filtered colimits.
- In general, the left derives of the left adjoint commutes with filtered colimits.
The following lemma embeds the tor-torsion theory into general cotorsion theory.
For any $X ∈ 𝐌𝐨𝐝 _R$ and $Y ∈ 𝐌𝐨𝐝 _{R^{\mathrm{op}}}$, $\mathrm{Ext}^1(X,Y^+) =0$ iff $\mathrm{Tor}_1(X,Y) = 0$.
Note that $(-)^+$ preserves and reflects exactness and zero-objects.
$$\begin{equation} \mathrm{Ext}^1(X, Y^+) = H^1((P(X), Y^+)) ≃ (H^1(P(X) ⊗ Y))^+ = \mathrm{Tor}_1(X, Y)^+. \end{equation}$$
$(𝐅𝐥𝐚𝐭 , \mathcal{A})$ and $(\mathcal{A}, 𝐅𝐥𝐚𝐭 )$ are tor-torsion theories. Hence $(𝐅𝐥𝐚𝐭 , 𝐅𝐥𝐚𝐭 ^⟂ )$ is a cotorsion theory. $𝐅𝐥𝐚𝐭 ^⟂$ is known as Enoch’s cotorsion module.
$$\begin{equation} \mathcal{A}^+ ⊆ \text{Pure Injective} ⊆ 𝐅𝐥𝐚𝐭 ^⟂. \end{equation}$$
(Torsion-free module). Recall that
$$\begin{equation} 0 → \mathrm{Tor}_1(X, R/(r)) → X ⊗ (r) → X → X ⊗ R/(r) → 0. \end{equation}$$
If $R$ has no zero-divisor, then $\mathrm{Tor}_1(X, R/(r)) = \{xr = 0 ∣ x ∈ X\}$ is a subgroup. If $R$ is commutative domain with factional field $K$, then
$$\begin{equation} \mathrm{Tor}_1(X,K/R) = \varinjlim_{r ∈ R ∖ \{0\}} \! \!\! ^{fil} \ \ \mathrm{Tor}(X,R/(r)) = \text{torsion subgrp of } \ X. \end{equation}$$
A torsion-free tor-torsion theory (for general rings) is cogenerated by cyclic modules, i.e.,
$$\begin{equation} 𝐓𝐅 = {}^⊺ 𝐂𝐲𝐜𝐥𝐢𝐜 = {}^⟂ \{(R/(r))^+ : r ∈ R\}. \end{equation}$$