Homological algebra

Consider the chain complex $0 \to \mathbb{Z} \xrightarrow{d_2} \mathbb{Z}^2 \xrightarrow{d_1} \mathbb{Z} \to 0$ with $d_2(n) = (2n, 4n)$ and $d_1(a, b) = 2a - b$. How many elements does $H_1 = \ker d_1 / \operatorname{im} d_2$ have?

Which statements about chain maps are true in general? (Choose all that apply.)

How many elements does $\mathrm{Tor}_1^{\mathbb{Z}}(\mathbb{Z}/12, \mathbb{Z}/18)$ have?

How many equivalence classes of extensions $0 \to \mathbb{Z}/5 \to E \to \mathbb{Z}/5 \to 0$ of abelian groups are there?

In the existence part of the comparison theorem, why does $f_n d : P_{n+1} \to Q_n$ land in $\ker(d : Q_n \to Q_{n-1})$ (for $n \geq 1$)?

Apply $- \otimes \mathbb{Z}/2$ and its derived functors to $0 \to \mathbb{Z} \xrightarrow{2} \mathbb{Z} \to \mathbb{Z}/2 \to 0$. The long exact sequence reads $\mathrm{Tor}_1(\mathbb{Z}, \mathbb{Z}/2) \to \mathrm{Tor}_1(\mathbb{Z}/2, \mathbb{Z}/2) \xrightarrow{\partial} \mathbb{Z}/2 \xrightarrow{2} \mathbb{Z}/2 \to \mathbb{Z}/2 \to 0$. What is $\mathrm{Tor}_1(\mathbb{Z}/2, \mathbb{Z}/2)$?

Which Lean statement says that the extension $0 \to \mathbb{Z} \xrightarrow{2} \mathbb{Z} \to \mathbb{Z}/2 \to 0$ does not split (as proved in not_split)?

A chain complex $0 \to C_2 \xrightarrow{d_2} C_1 \xrightarrow{d_1} C_0 \to 0$ of free abelian groups is given by integer matrices d2 (shape $\dim C_1 \times \dim C_2$) and d1 (shape $\dim C_0 \times \dim C_1$), as lists of rows. Write betti_mod_p(d2, d1, p) returning [b0, b1, b2], the dimensions of $H_n(C \otimes \mathbb{F}_p)$ for a prime $p$, using $b_n = \dim C_n - \operatorname{rank} d_n - \operatorname{rank} d_{n+1}$ over $\mathbb{F}_p$ (with $d_0 = d_3 = 0$).