Is there a Mayer–Vietoris-type sequence for the homology of a coproduct of two Hopf algebras over an ideal? The definition of the coproduct can be found in [Agore - Categorical Constructions for Hopf Algebras](https://arxiv.org/abs/0905.2613v3). Maybe the question should be: what does the spectral sequence that computes this homology look like?


Edit: As was requested I specify what kind of homology I am talking about. I was mainly thinking about homology of Lie algebras and discrete groups. So, I am interested in $\operatorname{Tor}^H(k,M)$ for an $H$-module $M.$



Mayer–Vietoris sequence for coproduct of Hopf algebras
In the case of discrete groups, the MV sequence follows from the following argument: if you have $G=H\*K$ then you have a short exact sequence of the form:
$$0\to \mathbb{Z}G\stackrel{i}{\to} \mathbb{Z}G/H\oplus \mathbb{Z}G/K\stackrel{p}{\to} \mathbb{Z}\to 0,$$ where $i$ sends $g$ to $(gH,gK)$ and $p$ sends $gH$ and $gK$ to 1.
The fact that this sequence is exact can be proven combinatorially using the fact that if you take the bases $\{h\}\_{h\in h}$ and $\{k\}\_{k\in K}$ for $\mathbb{Z}H$ and $\mathbb{Z}K$ then a basis for $\mathbb{Z}G$ is given by the set of all alternating $\textit{words}$ in these bases. This argument can be generalized also the the definition of coproduct of two Hopf algberas given by Agore. If you have Hopf algebras $H\_1$ and $H\_2$ and $H=H\_1\*H2$, then choose bases $\{a\}\_{a\in A}$ for $H\_1$ and $\{b\}\_{b\in B}$ for $H\_2$ that contain 1. You then get a basis for $H$ by words in these bases:
Write $A' = A\backslash \{1\}$ and $B'=B\backslash\{1\}$. We have the following basis for $H$:
$$\{1\} \cup \{a\}\_{a\in A'}\cup \{b\}\_{b\in B'}\cup \{ab\}\_{a\in A',b\in B'}\cup\cdots $$


and you can prove in the same way that you get a short exact sequence of $H$-modules given by
$$0\to H\to \text{Ind}\_{H\_1}^{H}k\oplus \text{Ind}\_{H\_2}^H k\to k.$$ Applying now the relevant functors will give you a long exact sequence, which is the Mayer-Vietoris sequence you need here, using the fact that for a right module $M$ it holds that $\text{Tor}^H\_\*(\text{Ind}\_{H\_1}^H k,M)\cong \text{Tor}^{H\_1}\_\*(k,M)$.


I assume here that $k$ is a field for this construction to work.


