The KLR category categorifies the negative quantum half

(Surjectivity.) By [F1] the classes $[P(Y_b)]$ form an $A$-basis of $K_0(R(\beta ))$ for a finite family of ordered divided-power words; each $Y_b$ is a word in divided powers of single colours, so its class equals $\gamma ({\theta }_{i_1}^{(n_1)}\cdots {\theta }_{i_r}^{(n_r)})$ by multiplicativity of $\gamma$ and the definition of $P_{\theta }$. Hence every class in $K_0(R(\beta ))$ is an $A$-linear combination of classes $[P_{\theta }]$ of divided-power projectives, and $\gamma$ is onto.

(Injectivity.) Suppose $\gamma (x)=0$ for some $x\in A{f}_{\nu }$. By [F2] and [F3], for every $y$ in the same weight component one has $B_L(x,y)=(\gamma (x),\gamma (y))=0$; since $B_L$ is a unit multiple of the nondegenerate form $B_K$ on the finite-dimensional $\mathbb{Q}(q)$-space $A{f}_{\nu }{\otimes }_A\mathbb{Q}(q)$, this forces $x=0$. The categorical and form-comparison steps of [F2] are choice-free; the Axiom of Choice is inherited only through the nondegeneracy supplier [F3].

(Compatibility and ranks.) The intertwining of multiplications is multiplicativity of $\gamma$ together with the induction formula of The KLR Grothendieck ring satisfies the divided-power quantum Serre relations; the intertwining of comultiplications is the Mackey comparison of Generic KLR induction and restriction adjunctions preserve projectives against the coproduct $r$ of the negative half; the bar compatibility is $P_{i,n}^{\mathrm{\#}}=P_{i,n}$ for the normalized divided projectives of Nil-Hecke idempotents and divided-power KLR projectives, and form preservation is [F2]. Since $\gamma$ is an isomorphism of $A$-modules by steps 1.1 and 2.1, the free $A$-rank of $K_0(R(\beta ))$ equals that of $A{f}_{\beta }$; the finite free basis supplied by [F1] transfers through $\gamma$ to a finite free basis of $A{f}_{\beta }$, and the infinite-dimensional algebra $R(\beta )$ itself is not compared with $f_{\beta }$.