Closure properties of $\displaystyle \lim_\longrightarrow \mathcal C$

Abstract: Let $\mathcal C$ be a class of (right $R$-) modules closed under finite direct sums. If $\mathcal C$ consists of finitely presented modules, then the class $\displaystyle \lim_\longrightarrow \mathcal C$ of all direct limits of modules from $\mathcal C$ is well-known to enjoy a number of closure properties. Moreover, if $R \in \mathcal C$, $\mathcal C$ consists of FP$_2$-modules, and $\mathcal C$ is closed under extensions and direct summands, then $\displaystyle \lim_\longrightarrow \mathcal C$ can be described homologically: $\displaystyle \lim_\longrightarrow \mathcal C$ is the double perpendicular class of $\mathcal C$ with respect to the Tor$_1^R$ bifunctor [1].

Things change completely when $\mathcal C$ is allowed to contain infinitely generated modules: $\displaystyle \lim_\longrightarrow \mathcal C$ then need not even be closed under direct limits. After presenting some positive general results (and their constraints), we will concentrate on two particular cases: $\mathcal C = add(M)$ and $\mathcal C = Add(M)$, for an arbitrary module $M$. We will prove that if $S = \End M$ and $\mathcal F$ is the class of all flat right $S$-modules, then $\displaystyle \lim_\longrightarrow add(M) = \{ F \otimes _S M \mid F \in \mathcal F \}$. For $\displaystyle \lim_\longrightarrow Add(M)$, we will have a similar formula, involving the contratensor product $\odot _S$ and direct limits of projective right $S$-contramodules (for $S$ endowed with the finite topology). We will also show that for various classes of modules $\mathcal D$, if $M \in \mathcal D$ then $\displaystyle \lim_\longrightarrow add(M) = \displaystyle \lim_\longrightarrow Add(M)$. However, the equality remains open in general, even for (infinitely generated) projective modules.

The talk is based on my recent joint work with Leonid Positselski [2].

[1] L.Angeleri Hügel, J. Trlifaj: Direct limits of modules of finite projective dimension, in Rings, Modules, Algebras, and Abelian Groups, LNPAM 236, M.Dekker, New York 2004, 27-44.

[2] L.Positelski, J.Trlifaj: Closure properties of $\displaystyle \lim_\longrightarrow \mathcal C$, preprint.

rings and algebrasrepresentation theory

Audience: researchers in the topic

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