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$\newcommand\A{\mathcal{A}}$Let $\A$ be an additive category, and $D(\A)$ be its derived category (i.e. the category of chain complexes of $\A$ localized at quasi-isomorphisms). It is easy to show that if $X \simeq Y$ in $D(\A)$, then such equivalence induces a graded isomorphism on their homologies $H(X) \simeq H(Y)$. Therefore, $H(-)$ is an invariant for objects in $D(\A)$.

If $\A$ is abelian and hereditary (i.e. $\operatorname{Ext}_{\A}^2(-,-) \equiv 0$; e.g. the category of abelian groups), then any bounded $X \in D(\A)$ is isomorphic to $H(X)$ in $\A$ (see [1][2]). (The converse seems to be true as well, according to [1].) Therefore, homology is a complete invariant in this case.

Question

How about if $\A$ is abelian but non-hereditary (say those with $\operatorname{Ext}_{\A}^{3}(-,-) \equiv 0$ and so on)?

  1. What are some examples of a pair of non-isomorphic objects that have graded-isomorphic homologies?
  2. Are there other (complete) invariants?

Remark

I asked a similar but perhaps harder question "Invariants of objects in $\operatorname{Ch}(\mathrm{Ab})$ up to chain homotopy" on MathOverflow before localizing at the quasi-isomorphisms. Turned out that may be too hard, so I try my luck here.

Reference

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    $\begingroup$ Although it would be fine with TeX's practice of ignoring whitespace, in MathJax you cannot separate your command definitions from the body by blank lines; doing so will force literal blank lines into the rendered post. Unfortunately, though it makes the source ugly, it is necessary to put the $ that ends the command definitions on the same line as the beginning of the body. I have edited accordingly (also on MO). $\endgroup$
    – LSpice
    Nov 28, 2022 at 23:14

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An example for the first question

A simple way to construct an example for your first question is by truncating projective resolutions of objects, so in this answer I will assume that $\mathcal{A}$ is abelian and has enough projectives. As an example, you could take $\mathcal{A}$ to be the module category over an algebra.

Assume that $\mathcal{A}$ is not hereditary. Then there is an object in $\mathcal{A}$ of projective dimension at least $2$; let $M$ be such an object, and let $$ \ldots \xrightarrow{f_3} P_2 \xrightarrow{f_2} P_1 \xrightarrow{f_1} P_0 \xrightarrow{f_0} M \to 0 $$ be a projective resolution of $M$. Consider now the object $X$ in the derived category $D(\mathcal{A})$ given by the complex $$ \ldots \to 0 \to P_1 \xrightarrow{f_1} P_0 \to 0 \to \ldots. $$ Then the homology $H(X)$ is non-zero in degree 0 (since $H_0(X) \cong M$) and in degree 1 (otherwise $f_1$ would be injective and $M$ would have projective dimension at most 1). Finally, consider the object $X'$ of $D(\mathcal{A})$ given by $$ \ldots \to 0 \to H_1(X) \xrightarrow{0} M \to 0 \to \ldots. $$ Then $X$ and $X'$ have isomorphic graded homology, but they are not isomorphic objects of $D(\mathcal{A})$. To see that they are not isomorphic, one could note that, in $D(\mathcal{A})$, the functor $\operatorname{Ext}^2(X',-)$ does not vanish on $\mathcal{A}$ (since $M$ has projective dimension at least 2 and is a direct summand of $X'$) while $\operatorname{Ext}^2(X,-)$ does.

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