Derived categories

Which of these chain maps are quasi-isomorphisms? (Complexes of abelian groups; a single group means a complex concentrated in degree $0$. Choose all that apply.)

Compute the order of $\operatorname{Ext}^1_{\Z}(\Z/12, \Z/18)$.

The derived tensor product $\Z/8 \otimes^{L}_{\Z} \Z/12$ is an object of $D(\mathrm{Ab})$. What is the order of its cohomology group in degree $-1$?

Which of these Hom-groups in $D(\mathrm{Ab})$ are non-zero? (Choose all that apply.)

Let $R = \Z/4$. What is the order of $\operatorname{Hom}_{D(R)}(\Z/2, \Z/2[5]) = \operatorname{Ext}^5_R(\Z/2, \Z/2)$?

In the proof of Lemma 2 ($\operatorname{Hom}_K(P, Z) = 0$ for $P$ a bounded-above complex of projectives and $Z$ acyclic), we put $g = f^n - h^{n+1} d_P$. Where is the projectivity of $P^n$ used?

Let $p : P = (\Z \xrightarrow{2} \Z) \to \Z/2$ be the resolution map. Which statement is correct?

“Every object $X$ of $D^b(\mathcal{A})$ is isomorphic to $\bigoplus_n H^n(X)[-n]$.” Which hypothesis on $\mathcal{A}$ makes this true, together with an example showing that some hypothesis is needed?

Which Lean statement formalises “homotopic chain maps become equal in the derived category”? (Here f g : K ⟶ L are maps of cochain complexes and Q is DerivedCategory.Q.)

Write ext_orders(a, b) returning the list [|Ext^0|, |Ext^1|, |Ext^2|] for $\operatorname{Ext}^n_{\Z}(\Z/a, \Z/b)$, computed from the resolution $0 \to \Z \xrightarrow{a} \Z \to \Z/a \to 0$: apply $\operatorname{Hom}(-, \Z/b)$ to get $\Z/b \xrightarrow{a} \Z/b$ and count the kernel and the cokernel of $x \mapsto ax \bmod b$ (do not use gcd).