Let $(C, d_0)$ be some cochain complex over commutative ring $R$ and $R$ is an algebra over a field $k$, then I could consider complex $(C[t], d_0+td_1)$ over ring $R \otimes k[t]$ where $d_1$ is some perturbation such that $(d_0+td_1)^2=0$.

What conditions guarantees that this deformation has the same cohomology groups? In other words I want $H^i(C, d_0+td_1)$ to be "constant in t" i.e. $k[t]$-module $H^i(C, d_0+td_1)$ is free. I also could hope for quasi-isomorphism $\phi_t$between $(C, d_0)$ and $(C[t], d_0+td_1)$, but it might be to much to ask.

General deformation theory suggest that if I take $d_1$ that is trivial in $Ext^1(C,C)$ then I get a complex isomorphic to $C$ but this is to restrictive.

This question reminds me situation of homological perturbation lemma, but in homological perturbation lemma we start with some rich data and produce some explicit formulas and here I want only to know that cohomology groups are the same.

Let $(C, d_0)$ be some cochain complex over commutative ring $R$ and $R$ is an algebra over a field $k$, then I could consider complex $(C[t], d_0+td_1)$ over ring $R \otimes k[t]$ where $d_1$ is some perturbation such that $(d_0+td_1)^2=0$.

What conditions guarantees that this deformation has the same cohomology groups? In other words I want $H^i(C, d_0+td_1)$ to be "constant in t" i.e. $k[t]$-module $H^i(C, d_0+td_1)$ is free. I also could hope for quasi-isomorphism $\phi_t$between $(C, d_0)$ and $(C[t], d_0+td_1)$, but it might be to much to ask.

General deformation theory suggest that if I take $d_1$ that is trivial in $Ext^1(C,C)$ then I get a complex isomorphic to $C$ but this is too restrictive.

This question reminds me situation of homological perturbation lemma, but in homological perturbation lemma we start with some rich data and produce some explicit formulas and here I want only to know that cohomology groups are the same.