3. By (62) and (88), we get $\mathrm{Ext}^3_{D(\bar{U}_i), \bar{U}_i}(\mathbb{C}, \mathbb{C}) \cong \mathbb{C}^3$. On the other hand by finding a usual projective resolution of $\mathbb{C}$ one can compute that $\mathrm{Ext}^3_{D(\bar{U}_i)}(\mathbb{C}, \mathbb{C}) = 0$. Hence the two allowable 3-fold exact sequences constructed above form a free family in $\mathrm{YExt}^3_{D(\bar{U}_i), \bar{U}_i}(\mathbb{C}, \mathbb{C}) \cong \mathbb{C}^3$ but are equal to 0 in $\mathrm{YExt}^3_{D(\bar{U}_i)}(\mathbb{C}, \mathbb{C}) = 0$.

4. It is straightforward to compute a usual projective resolution of $\mathbb{C}$ over $(U_i^*)^{\mathrm{op}}$ by using the defining relations (85). Thanks to Lemma 5.1 we get $H_{\mathrm{DY}}^3(U) = 0$, where $U : \bar{U}_i$-mod $\to \mathrm{Vect}_{\mathbb{C}}$ is the forgetful functor. This means that the map $\iota_3 : H_{\mathrm{DY}}^3(\bar{U}_i$-mod) $\to H_{\mathrm{DY}}^3(U)$ defined in Remark 5.3 is equal to 0. In other words, there exist elements $b_i \in \bar{U}_i^{\otimes 2}$ such that $\iota_3(c_i) = \delta^2(b_i)$ for $i = 1, 2, 3$. This is in contrast to the previous two examples, the algebras $B_k$ and the Taft algebras, where the DY cohomologies of the identity functor equal those of the forgetful functor.

## 6.4 Small quantum group $u_q(\mathfrak{s}\mathfrak{l}_2)$

Let $q$ be a primitive root of unity of odd order $p \ge 3$. The small quantum group $u_q = u_q(\mathfrak{s}\mathfrak{l}_2)$ is the $\mathbb{C}$-algebra generated by $E, F, K$ modulo

$$ KE = q^2EK, \quad KF = q^{-2}FK, \quad EF - FE = \frac{K - K^{-1}}{q - q^{-1}}, \quad E^p = F^p = 0, \quad K^p = 1. $$

The Hopf structure is given by the same formulas as in (84).

The Hopf algebra $u_q$ is known to be factorizable [Lyu95, Cor. A.3.3], so the results of §5.5 apply and we will not need to compute the Drinfeld double of $u_q$.

**Proposition 6.8.** $H_{\mathrm{DY}}^2(u_q(\mathfrak{s}\mathfrak{l}_2)$-mod) = 0.

*Proof.* Let $Q_1$ be as in §5.5, $K_1 = \ker(\varepsilon : Q_1 \to \mathbb{C})$ and $T = Q_1/(\mathbb{C} \boxtimes \mathbb{C})$. Then by Corollary 5.8:

$$ \dim H_{\mathrm{DY}}^2(u_q\text{-mod}) = \dim \mathrm{Hom}_{u_q \otimes u_q}(K_1, T) - \dim \mathrm{Hom}_{u_q \otimes u_q}(Q_1, T). $$

Recall that there are $p$ isomorphism classes of simple $u_q$-modules, denoted by $\mathcal{X}(s)$ with $1 \le s \le p$. $\mathcal{X}(s)$ has dimension $s$ and is generated by a vector $v_s$ such that $Ev_s = 0$ and $Kv_s = q^{s-1}v_s$. In particular $\mathcal{X}(1) = \mathbb{C}$. By the same analysis as in [FGST06, §4.4.2], the structure of $Q_1$ can be depicted by diagrams similar to the ones given below [FGST06, Prop. 4.4.2] (one has to take $s = 1$ and to erase the signs on the simple modules). Then $K_1$ is obtained by ignoring the node $\mathcal{X}(1) \boxtimes \mathcal{X}(1)$ at the top of the diagrams of $Q_1$ and the arrows out of it. Using the diagrammatic representation of the structure of these modules, we easily find

$$ \dim \mathrm{Hom}_{u_q \otimes u_q}(K_1, T) = 2, \quad \dim \mathrm{Hom}_{u_q \otimes u_q}(Q_1, T) = 2. \quad \square $$

For higher cohomology groups, the computation of the $\mathrm{Hom}$ spaces appearing in Corollary 3.5 becomes much more difficult because it requires to find the structure of $K_1^{\otimes (n-1)}$. If the root of unity $q$ has a small order, one can use a computer with the same strategy as in (87), but with the formula of Corollary 5.8. For instance if $q^3 = 1$,

$$ \dim H_{\mathrm{DY}}^3(u_q\text{-mod}) = \dim \mathrm{Inv}(T^{\otimes 3}) - \dim \mathrm{Inv}(T^{\otimes 2} \otimes Q_1) + \dim \mathrm{Inv}(T^{\otimes 2}) \\ = 8 - 9 + 2 = 1 $$

where $\mathrm{Inv}(V) = \mathrm{Hom}_{u_q \otimes u_q}(\mathbb{C} \boxtimes \mathbb{C}, V)$ and $T = Q_1/(\mathbb{C} \boxtimes \mathbb{C})$. For the first equality we used that $Q_1$ is self-dual and $K_1^\vee \cong T$ (Proposition 5.7) while the second equality is obtained by a program<fnref n="9" /> written in GAP4.

<footnote n="9">⁹https://github.com/mfaitg/DY_cohomology/blob/main/dimension formula DY uj.g</footnote>
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