# Bounds on homological dimension of functor categories

Let $$A$$ be a Grothendieck abelian category. I will say that $$A$$ is of global dimension less or equal to $$n$$ if $$Ext^{k}_{A}(a, b) = 0$$ for $$k > n$$ and all $$a, b \in A$$. This is equivalent to saying that any object of $$A$$ admits an injective resolution of length at most $$n$$.

Let $$I$$ be a small diagram category, so that the category of functors $$Fun(I, A)$$ is again Grothendieck. Are there any general bounds on the dimension of this functor category in terms of the dimension of $$A$$ and some invariant of the diagram category?

Note that I do not expect this dimension would be finite for arbitrary $$I$$, I am rather looking for examples of "nice" $$I$$ for which we have such a bound for arbitrary Grothendieck category. In the simplest possible case, I would like to know the answer even if $$A$$ is a category of modules over a ring.

• Suppose the simplicial nerve of $I$ has simplicial dimension $m$. Maybe I am confused, but it seems to me that in this case Fun$(I, A)$ has global dimension $m+n$. Apr 8, 2021 at 15:09
• So you want something like a cohomological bound on $Tw(I)$, namely you want the derived limit functors to always vanish above some degree. It will also be related to $|I|$, which you want to have finite cohomological dimension, and the derived limit functors on $I$, which you also wanto vanish above some degree.For the latter two, some bound on the dimension of $I$ is enough, but I'm not sure about the first one Apr 8, 2021 at 15:09

Claim: If the simplicial nerve of $$I$$ has dimension $$m$$ and $$A$$ has global dimension $$n$$, then $$\operatorname{Fun}(I, A)$$ has global dimension (at most) $$m+n$$.
To prove it, you can use the standard simplicial resolution of a functor $$F\colon I \to A$$ by representable functors.
$$F(-)\leftarrow \bigoplus_{c_0\in Ob(I)} F(c_0)\times \operatorname{mor}_I(c_0,-) \Leftarrow \bigoplus_{c_0\to c_1} F(c_0)\times \operatorname{mor}_I(c_1,-)\cdots$$
Taking $$\operatorname{nat}(-, G)$$ we obtain a cosimplicial resolution of $$\operatorname{nat}(F, G)$$. The cosimplicial object has dimension $$m$$, and each term has homological dimension $$n$$. It follows that $$\operatorname{Ext}^k(F, G)=0$$ for $$k>m+n$$.