Graded module and projective Grothendieck groups of the KLR and Hecke towers

Restricted sums are A-modules. Each summand $K_0^{\mathrm{g}\mathrm{r}}(R_{\alpha })$ is a module over $A$ by $q[P]=[P\{1\}]$, and shift by an integer $r$ sends a class concentrated in degree $\alpha$ to the same root degree $\alpha$, because internal shift changes only the internal grading and not the idempotent label [F1, F3]. Hence every finite $A$-combination of classes lives in finitely many root summands and the restricted direct sums are $A$-modules; the local units $e(\alpha )$ make each $R_{\alpha }$ a unital subalgebra and annihilate the other summands, so the restricted direct sum is well defined and no infinite-support class is introduced.

The maps c and zeta. On each block the ungraded forgetful map sends a graded projective class to its underlying projective class and a graded module class to its underlying module class; it is additive, well defined on the defining relations, and satisfies $\zeta (q[M])=\zeta ([M\{1\}])=[M{]}_{\mathrm{u}\mathrm{n}\mathrm{g}\mathrm{r}}=\zeta ([M])$ [F1]. The graded Cartan homomorphism is the sum of the blockwise maps $[P]\mapsto [P]$, which are $A$-linear because including a projective into the module category commutes with internal shift; compatibility with $\zeta$ is immediate from the definitions.

The KLR tower. For the standard level-one cyclotomic quotient every block $R^{\mathrm{\Lambda }}(\alpha )$ is finite dimensional and carries the homogeneous symmetrizing functional of [F5], so the shift-orbit bases of [F2] apply blockwise without any positivity hypothesis [F4, F5]. Since the root degree is additive, a finite combination of block classes has finite support and belongs to the restricted direct sum.

Transport to the Hecke tower. For each nonzero residue-content sector, [F5] gives an algebra isomorphism ${\phi }_{\alpha }:H_{\alpha }^{\mathrm{\Lambda }}\to R^{\mathrm{\Lambda }}(\alpha )$ for the fixed admissible normalising series, homogeneous for the transported grading. Restriction of scalars along ${\phi }_{\alpha }$ is an exact equivalence with exact inverse given by restriction along ${\phi }_{\alpha }^{-1}$; it preserves finite-dimensional modules, projectives and internal shifts. For every zero Hecke sector, [F6] makes the corresponding KLR sector zero as well. A unital module over the zero algebra satisfies $m=1m=0m=0$, so both Grothendieck groups of either zero sector vanish. Summing the nonzero-sector isomorphisms and these zero-group identifications over finite root supports gives $K_0^{\mathrm{g}\mathrm{r}}(H^{\mathrm{\Lambda }})\cong K_0^{\mathrm{g}\mathrm{r}}(R^{\mathrm{\Lambda }})$ and $G_0^{\mathrm{g}\mathrm{r}}(H^{\mathrm{\Lambda }})\cong G_0^{\mathrm{g}\mathrm{r}}(R^{\mathrm{\Lambda }})$. These are $A$-linear since transport commutes with internal shift, and they intertwine the Cartan maps and the forgetful maps $\zeta$ with the ungraded identifications.

The groups remain distinct and no choice is used. The identification $K_0^{\mathrm{g}\mathrm{r}}(H^{\mathrm{\Lambda }})\cong K_0^{\mathrm{g}\mathrm{r}}(R^{\mathrm{\Lambda }})$ of step 1.4 transports projective classes between the specified Hecke and KLR towers; it asserts no identification of the arbitrary tower $R$ with $R^{\mathrm{\Lambda }}$. The Cartan homomorphism $c$ and the forgetful $\zeta$ are not inverted or identified, and no statement of the definition equates $K_0^{\mathrm{g}\mathrm{r}}$ with $G_0^{\mathrm{g}\mathrm{r}}$. All constructions use the finite block decompositions, the local units, and the explicit homogeneous algebra isomorphisms, so no choice principle is invoked.