Let $C$ be a small linear category over a commutative ring $R$. A *representation* of $C$ is an $R$-linear functor $C \to \mathrm{Mod}(R)$. For example, for each $c\in C$, there is a representation $[c] = \hom\_C(c,-)$. These representations are called *generators*, because every representation can be written as a quotient $\bigoplus\_{i\in I} [c\_i] / \bigoplus\_{j\in J} [c'\_j]$ for some indexing sets $I,J$ and objects $c\_i,c'\_j \in C$.


A representation $M$ is *finitely generated* if it can be written in such a way with $|I|<\infty$. Equivalently, $M$ is finitely generated if there is a surjection $\bigoplus\_{i=1}^n [c\_i] \to M$ for some finite list $\{c\_i\}$ of objects in $C$. A map of representations is a surjection iff it evaluates to a surjection in $\mathrm{Mod}(R)$ when evaluated at each object in $C$.


The category $C$ is *Noetherian* if every subrepresentation of every generator is finitely generated. This is the **a** natural generalization of the case of algebras (which correspond to categories with only one object, hence subrepresentations are ideals).



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> What strategies are there to prove that some given category is Noetherian?
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For example, is there a notion of "Ore extension" of categories for which Noetherian-ness is preserved?


In my case, I have a few categories that I would like to be Noetherian. They are constructed from smaller, known-to-be-Noetherian categories by some process that looks a bit like Ore extension, and then a bit not.


I'd rather not reinvent the wheel (or the stethoscope or the transistor or...) if I can help it.



Strategies for proving a category is Noetherian?
One way to prove a ring is Noetherian is to use Groebner theory. Recent work of Sam and Snowden generalizes Groebner theory to certain small categories enriched in finite sets (so-called hom-finite categories). Any representation of such a category factors through the universal R-linearization category, and so their work handles any case where $C$ is a quotient of the universal R-linearization of some hom-finite category admitting Groebner bases.


The paper is excellent. <http://arxiv.org/abs/1409.1670>


Unfortunately, a practical criterion for checking if a category is Noetherian seems well beyond current technology, even assuming that the category is the linearization of a hom-finite category.


If you get those Ore extensions working, I'd love to hear about it :)


---Edit---


I have just posted a paper <http://arxiv.org/abs/1508.04107> explaining when the representation theory of a category is simultaneously Noetherian and Artinian. In Corollary 1.8, I give a new way to show that a category is Noetherian: if it has what I call a "homological modulus." A homological modulus is an explicit gadget that you can look for computationally. For example, my REU student Andrew Gitlin has recently shown that the category of finite sets with relations <https://en.wikipedia.org/wiki/Category_of_relations> admits a homological modulus over $\mathbb{Q}$. If $R$ is a Noetherian $\mathbb{Q}$-algebra, we may conclude that finitely generated functors from $\mathrm{Rel}$ to $R$-modules are Noetherian. If $R$ is an Artinian $\mathbb{Q}$-algebra then these functors are actually finite length. I do not know if this example could also be completed using Sam-Snowden Groebner theory, but Gitlin's proof is sufficiently elegant that I'm sure his way is easier in this case.


