If $x$ is an object of a category $C$, one usually says that $x$ is if finite presentation (or compact) if for any direct filtered system ($y_i$) in $C$, the canonical map

$$f : \text{colim}_i \, \text{Hom}(x,y_i) \to \text{Hom}(x, \text{colim} y_i)$$

is bijective. One usually says $x$ is of finite type if this holds only for direct filtered systems of monomorphisms.

However, I am interested in objects $x$ for which $f$ is injective (without additional condition on the direct filtered system). It seems to me that, in the category of (right) modules over a ring, such objects are exactly finitely generated modules. My question is, has this been considered, and is there a name for this property of $x$? I considered "finitely generated"... but does anyone have a better idea?

  • 1
    $\begingroup$ I agree with Peter that a more descriptive name is better. Words like "finite", "finitely presented", "small", "compact", and even "finitely generated" are already overused. Just as an example, for model categories, finitely generated means that it's cofibrantly generated with domains of the generating cofibrations "finite", meaning what you call "finitely presented." $\endgroup$ – David White Apr 27 '17 at 15:14
  • $\begingroup$ I agree too, and that's precisely why I'm looking for a better term, but what would be a more descriptive term? $\endgroup$ – Alain Bruguieres Apr 28 '17 at 16:07

(more of a comment, but given in answer-form since the comment format is too small to accomodate it)

It reminds me of what Stenström called fp-injectiveness. This is in the context of left-modules though, but it should be straightforward to transpose it to the category of right-modules. It might help to consult

B. Stenström. Coherent Rings and FP-injective modules.

Journal of the London Mathematical Society 2, p. 323--329 (1970)

Briefly, a left R-module M being fp-injective means that each short exact sequence of left-$R$-modules

$0\rightarrow M\rightarrow A\rightarrow B\rightarrow 0$

is pure-exact. The latter means that for every finitely-presented left-$R$-module $E$ the induced sequence

$0\rightarrow \mathrm{Hom}(E,M)\rightarrow \mathrm{Hom}(E,A)\rightarrow \mathrm{Hom}(E,B)\rightarrow 0$

is exact.


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