Cotorsion pairs and the Mittag-Leffler condition

Dolors Herbera¹

Departament de Matemàtiques,
Universitat Autònoma de Barcelona,
08193 Bellaterra (Barcelona), Spain
dolors@mat.uab.es

Let $\mathcal{C}$ be an abelian category with arbitrary direct sums. Let $(C_n)_{n\in \mathbb{N}}$ be a sequence of compact objects in $\mathcal{C}$, and let $(f_n\colon C_n\to C_{n+1})_{n\in \mathbb{N}}$ be a sequence of morphisms. Then we have the exact sequence

$$0\to \oplus_{n\in \mathbb{N}} C_n\stackrel{\phi}\to \oplus_{n... ...athbb{N}} C_n\to \displaystyle \lim_{\longrightarrow} C_n=A\to0$$

with $\phi \epsilon _n=\epsilon _n- \epsilon _{n+1}f_n$ for any $n\in \mathbb{N}$ and $\epsilon _n\colon C_n \to \oplus_{n\in \mathbb{N}} C_n$ denoting the canonical morphism.

Let $\mathcal{B}$ be a subcategory of $\mathcal{C}$ closed under direct sums. We show that the inverse system

$$(\mathrm{Hom}_{\mathcal{C}}(C_n,B); \mathrm{Hom}_{\mathcal{C}}(f_n,B))_{n\in \mathbb{N}}$$

is Mittag-Leffler for any object $B\in \mathcal B$ if and only if the morphism $\mathrm{Hom}_{\mathcal{C}}(\phi,B)$ is onto for any object $B\in \mathcal B$.

This result has some interesting consequences when applied to cotorsion pairs in module categories. For example, it is the key tool in showing that $1$-dimensional tilting modules are of finite type. That is, if $T_R$ is a tilting module over a ring $R$ then there exists a set $\mathcal{S}$, consisting of finitely presented right $R$-modules of projective dimension at most one, such that

$$\mathrm{Ker}\, \mathrm{Ext}_R^1(T_R,-)=\bigcap _{S\in \mathcal{S}}\mathrm{Ker}\, \mathrm{Ext}_R^1(S,-).$$

Footnotes

... Herbera¹

joint work with Silvana Bazzoni