I am trying to make sense of the introduction to Section 2 of the following paper.
Let $R$ be a $\mathbb{Z}$-graded ring and $\text{Mod}_R$ be the 'category of graded $R$-modules'. Let $D$ be the contravariant Hom functor $\text{Hom}_R(-,R)$. I am confused about the precise details of the category $\text{Mod}_R$, which I assume must be abelian however it is defined since the authors claim that self-injectivity of $R$ implies that $D$ is exact.
I first assumed the morphisms in $\text{Mod}_R$ were the homogeneous $R$-linear maps of degree zero i.e. those for which $f(M_i)\subseteq N_i$ for all $i$. However it seems that this does not make $\text{Hom}_0(M,N)$ an $R$-module, just an abelian group.
One other possibility would be $\sum_i\text{Hom}_i(M,N)$ i.e. those with $f=\sum_if_i$, $f_i$ homogeneous of degree $i$. This is a graded module, but the kernel of such morphisms do not appear to be submodules.
What exactly is $DM$ and what is the correct understanding of $\text{Mod}_R$? And is the correct choice of $\text{Mod}_R$ an abelian category?
Best Answer
The morphisms in the category $\text{Mod}_R$ are just the degree $0$ homomorphisms. However, in this context, $\operatorname{Hom}_R(M,R)$ is referring not to the Hom-set of the category $\text{Mod}_R$, but to the internal hom of graded objects, which is the graded $R$-module whose degree $i$ part is the degree $i$ homomorphisms.
Consequently, it does not just follow immediately from the definition that $D$ is exact if $R$ is self-injective, since self-injectivity is about only the degree $0$ homomorphisms. However, it follows easily since degree $i$ homomorphisms $M\to R$ are the same thing as homomorphisms $\Sigma^{-i}M\to R$, so self-injectivity of $R$ implies that the degree $i$ part of $DM$ is an exact functor $\text{Mod}_R\to\text{Ab}$ for each $i$ and thus $D$ itself is exact.
(A good way to think about this is that $\text{Mod}_R$ is not just an ordinary category but a category enriched in graded abelian groups, with the enriched Hom being the graded abelian group whose degree $i$ part is the degree $i$ homomorphisms. The forgetful functor to sets which turns this enriched category back into an ordinary category does not take a graded abelian group to the direct sum of its parts of each degree, but instead takes it to only the degree $0$ part.
Relatedly, in this context, you should always think of graded objects not as direct sums but simply as sequences, so a graded abelian group $A$ is not an abelian group with a direct sum decomposition $A=\bigoplus A_n$ but is simply a sequence $(A_n)$ of abelian groups. This is virtually always the right way to think about graded objects in the context of algebraic topology and homological algebra.)