Singular compactness and definability for $\Sigma$-cotorsion and Gorenstein modules

Abstract

We introduce a general version of the singular compactness theorem which makes it possible to show that being a $\Sigma$-cotorsion module is a property of the complete theory of the module. As an application of the powerful tools developed along the way, we give a new description of Gorenstein flat modules which implies that, regardless of the ring, the class of all Gorenstein flat modules forms the left-hand class of a perfect cotorsion pair. We also prove the dual result for Gorenstein injective modules.

Appendix A: Manipulating with directed systems

In the whole paper, we often use the following two constructions of directed systems of modules (or morphisms).

Construction A.1

Let $f:M \rightarrow N$ be a homomorphism of modules, $\lambda$ a regular uncountable cardinal, $\mathcal {M} = (M_i, f_{ji}:M_i \rightarrow M_j\mid i<j\in I)$ and $\mathcal {N} = (N_i, g_{ji}:N_i \rightarrow N_j\mid i< j\in J)$$\lambda$-continuous directed systems of $<\lambda$-presented modules such that $\varinjlim \mathcal {M} = M$ and $\varinjlim \mathcal {N} = N$. Then there is a $\lambda$-continuous directed system $\mathcal {U} = (u_k:M_{i_k}\rightarrow N_{j_k}, (f_{i_l,i_k},g_{j_l,j_k})\mid k<l\in K)$ consisting of morphisms with domains in $\mathcal {M}$ and codomains in $\mathcal {N}$ such that $\varinjlim \mathcal {U} = f$.

Subsequently, there is a $\lambda$-continuous directed system $\mathcal {K} = ({{\,\mathrm{Coker}\,}}(u_k)\mid k\in K)$ consisting of $<\lambda$-presented modules (with canonically defined maps).

Proof

For each $i\in I$ and $j\in J$, let us denote by $f_i:M_i\rightarrow M$ and $g_j:N_j\rightarrow N$ the colimit maps, and define $f_{ii} = \hbox {id}_{M_i}$ and $g_{jj} = \hbox {id}_{N_j}$.

We define $\mathcal {U}$ as the set of all morphisms $u:M_i \rightarrow N_j$ such that $i\in I, j\in J$, $g_ju = ff_i$. For $u:M_i\rightarrow N_j,v:M_r \rightarrow N_s$ from $\mathcal {U}$, we put $u\le v$ if and only if $i\le r, j\le s$ and $vf_{ri} = g_{sj}u$. We easily check that $(\mathcal {U},\le )$ is a poset. Next, we show that it is directed.

First, fix generating sets $G = \{x_\alpha \mid \alpha <\mu \}$ and $H = \{y_\alpha \mid \alpha < \mu \}$ of $M_i$ and $M_r$, respectively, where $\mu <\lambda$, and let $u,v\in \mathcal {U}$ be as above. We find $a\in I$ such that $i,r<a$. Since $\mathcal {N}$ is $\lambda$-continuous and $M_a$ is $<\lambda$-presented, there is $b\in J, j,s<b$, and a morphism $w_0:M_a\rightarrow N_b$ from $\mathcal {U}$. It need not be the case that $u\le w_0$ and $v\le w_0$, however, for each $\alpha <\mu$, there is a $b_\alpha \ge b$ such that $g_{b_\alpha j}u(x_\alpha ) = g_{b_\alpha b}w_0f_{ai}(x_\alpha )$ and $g_{b_\alpha s}v(y_\alpha ) = g_{b_\alpha b}w_0f_{ar}(y_\alpha )$. Since $\mu <\lambda$ and $(J,\le )$ is $\lambda$-directed, there is $c\in J$ such that $c\ge b_\alpha$ for each $\alpha <\mu$. It follows that $w = g_{cb}w_0$ is in $\mathcal {U}$ and $u,v\le w$. Subsequently, $(\mathcal {U},\le )$ is a directed system of morphisms. Moreover, it is $\lambda$-continuous since $\mathcal {M}$ and $\mathcal {N}$ are such.

To prove that $\varinjlim \mathcal {U} = f$, it is now enough to find, for arbitrary $(i,j)\in I\times J$, a morphism $u:M_i \rightarrow N_s$ in $\mathcal {U}$ with $s\ge j$. This is easy (recall how we found $w_0$). $\square$

The next tool allows us to merge less than $\lambda$ directed systems which are $\lambda$-continuous into one. In its statement, we do not use the notation from Definition 2.6.

Construction A.2

Let $M\in \hbox {{Mod-}}R$, $\lambda$ be a regular uncountable cardinal, $\gamma <\lambda$ and, for each $\alpha \le \gamma$, let $\mathcal {M}^\alpha = (M^\alpha _{ji},f^\alpha _{ji}:M^\alpha _i\rightarrow M^\alpha _j\mid i<j\in I_\alpha )$ be a $\lambda$-continuous directed system consisting of $<\lambda$-presented modules such that $\varinjlim \mathcal {M}^\alpha = M$. Then the systems $\mathcal {M}^\alpha$, $\alpha \le \gamma$, have a common cofinal $\lambda$-continuous subsystem.

More precisely: for each $\alpha \le \gamma$, there exists a $\lambda$-continuous cofinal directed subsystem $\mathcal {N}^\alpha = (M^\alpha _{ji},f^\alpha _{ji}:M^\alpha _i\rightarrow M^\alpha _j\mid i<j\in J_\alpha )$ of $\mathcal {M}^\alpha$; furthermore, for any $\alpha ,\beta \le \gamma$, there is a bijection $\iota :J_\alpha \rightarrow J_\beta$ and a directed system $\mathcal {U}$ with $\varinjlim \mathcal {U} = \hbox {id}_M$ whose objects are isomorphisms $u_i:M^\alpha _i \rightarrow M^\beta _{\iota (i)}$, $i\in J_\alpha$, and for each $i<j\in J_\alpha$, there is only one morphism from $u_i$ to $u_j$ in $\mathcal {U}$, namely $(f^\alpha _{ji},f^\beta _{\iota (j),\iota (i)})$.

Proof

We can assume that $\gamma$ is a cardinal. The proof goes by induction on $\gamma$. For $\gamma = 0$, it is trivial. Let $\gamma = 1$.

We use Construction A.1 with $f = \hbox {id}_M$, $\mathcal {M} = \mathcal {M}^0$ and $\mathcal {N} = \mathcal {M}^1$ to obtain the system $\mathcal {U}$ of morphisms. Using [9, Lemma 2.6], we can w.l.o.g. assume that the objects of $\mathcal {U}$ are isomorphisms. The subsystems $\mathcal {N}^\alpha$, $\alpha = 0, 1$, then consist of domains, codomains, respectively, of the isomorphisms in $\mathcal {U}$.

By induction, we have the proof for any $\gamma$ finite. For $\gamma$ infinite, we use the inductive hypothesis and the following simple fact: for each $\alpha \le \gamma$, if $(\mathcal {N}^\alpha _\beta \mid \beta <\gamma )$ is a family of $\lambda$-continuous cofinal directed subsystems of $\mathcal {M}^\alpha$ such that $\mathcal {N}^\alpha _\beta \supseteq \mathcal {N}^\alpha _\delta$ whenever $\beta \le \delta <\gamma$, then $\bigcap _{\beta <\gamma } \mathcal {N}^\alpha _\beta$ is a $\lambda$-continuous cofinal directed subsystem of $\mathcal {M}^\alpha$ as well. $\square$

We also use freely the following easy.

Observation A.3

Let $\lambda$ be a regular uncountable cardinal and $\mathcal {M}$ a $\lambda$-continuous directed system of modules. Let $\mathcal {K}$ be a directed subsystem of $\mathcal {M}$. Then there is a $\lambda$-continuous directed subsystem $\mathcal {K}^\prime$ of $\mathcal {M}$ with the same direct limit as $\mathcal {K}$.