Relative and absolute Ext groups

Question

Given a homomorphism of rings $S \rightarrow R$, for a pair of $R$-modules $M, N$ the machinery of relative homological algebra defines relative $Ext$-groups

$Ext_{R, S}(M, N)$.

These can be defined, for example, by mapping the relatively projective resolution

$\ldots \rightarrow R \otimes_{S} R \otimes_{S} M \rightarrow R \otimes_{S} M \rightarrow M$

into $N$ and taking cohomology. I have two questions:

1. Is there a general framework for relating these relative $Ext$-groups with the absolute ones, for example through a long exact sequence or a spectral sequence?
2. Can these relative $Ext$-groups be identified with homotopy classes of maps in some appropriate relative derived $\infty$-category? Can the latter be expressed in terms of the absolute derived $\infty$-categories?

Answer 1

Let $K(S)$ be the category of chain complexes of $S$-modules; this category has Hom-complexes $hom_S(-,-)$ making it a dg-category, and thus produces a quasicategory via the construction in Higher Algebra section 1.3.1.

Take $f: S \to R$ a ring homomorphism, $M \in K(S)$, and $N \in K(R)$. Then there is a natural adjunction isomorphism between hom-complexes $$ hom_S(M, f^* N) \cong hom_R(R \otimes_S M, N) $$ which therefore induces an adjunction $K(S) \rightleftarrows K(R)$, and a monad $f_* f^*$ on $K(S)$ whose underlying functor is $R \otimes_S (-)$. This gives us a lift of the functor $f^*$ to the category $Alg(f_* f^*)$ of algebras for this monad (HA, 4.7.3.3).

For any object $M$ and $N$ in $K(R)$, we thus get a simplicial resolution $$ M \leftarrow R \otimes_S M \leftleftarrows \dots $$ The resulting complex $$ Hom_R(R \otimes_S M,N) \to Hom_R(R \otimes_ R \otimes_S M, N) \to \dots $$ can therefore be interpreted as calculating $Hom_{Alg(f^* f_*)}(f^* M, f^* N)$, because that's how one calculates homotopy classes of maps in the category of algebras over a monad. In the case where $M$ and $N$ are discrete complexes concentrated in fixed degrees, this recovers the relative Ext-groups you were interested in.

This adjunction between $K(S)$ and $K(R)$ is very unlikely to be monadic except in special circumstances. But there is a localizing subcategory $V \subset K(R)$ consisting of those complexes whose images under the forgetful functor are contractible, and in total a factorization of $f^*$ into maps of stable $\infty$-categories as follows: $$ K(R) \to K(R) / V \to Alg(f^* f_*) \to K(S) $$ Here $K(R) / V$ is the Bousfield localization / Verdier quotient / nullification of $V$. This is roughly a coimage / image factorization and one could ask whether the middle map is an equivalence; this I don't know.

Whether there are useful spectral sequences I also do not know (quotient maps $S \to S / I$ indicate that there is a cap on how much nice behavior we should expect). We can certainly construct filtrations of $R$ as an object of $K(S)$ and try to run with that, but it is typically much easier to reduce calculations in the derived category $D(S)$ to calculations in $K(S)$ and not the other way around.

Answer 2

This is only a partial answer to question 1. In case $S=\mathbb{Z}H$ and $R=\mathbb{Z}G$ are group rings, and $S\to R$ is induced by an inclusion $H\leq G$ of groups, there is a spectral sequence relating $H^*(G,H;M)=\operatorname{Ext}^*_{R,S}(\mathbb{Z},M)$ and $H^*(G;M)=\operatorname{Ext}^*_R(\mathbb{Z},M) $ for any $R$-module $M$. Let $\mathcal{F}$ be the family of subgroups of $G$ generated by $H$ (i.e. all subconjugates of $H$). There is a spectral sequence with $$ E^{p,q}_2=H^p_{\mathcal{F}}(G;H^q(-;M)), $$ the Bredon cohomology of $G$ with coefficents in the $\operatorname{Or}_\mathcal{F}(G)$-module $G/K\mapsto H^q(K;M)$, converging to $H^*(G;M)$. The relative cohomology appears on the bottom row of the $E_2$-page, since $$ H^p_{\mathcal{F}}(G;M^{(-)})\cong H^p(G,H;M). $$ The edge homomorphism of this spectral sequence is the map $H^*(G,H;M)\to H^*(G;M)$ from relative to absolute group cohomology.

Details of this spectral sequence appear in the papers

Pamuk, Semra; Yalçın, Ergün, Relative group cohomology and the orbit category., Commun. Algebra 42, No. 7, 3220-3243 (2014). ZBL1322.20041 (where it is stated as Theorem 1.3) and

Martínez-Pérez, Conchita, A spectral sequence in Bredon (co)homology, J. Pure Appl. Algebra 176, No. 2-3, 161-173 (2002). ZBL1017.18009 (where it appears in a slightly more general form as Theorem 6.1).

This spectral sequence can be derived as in Pamuk-Yalçın as a Grothendieck spectral sequence of derived limit functors. I wouldn't be surprised if somthing similar can be done for more general $R$ and $S$ but I haven't seen the details anywhere.