The integral form of the cyclotomic Hecke-KLR block

The integer lattice and its spanning forms. The KLR presentation is defined over $\mathbb{Z}$ and its relations involve only integer coefficients [F3]; ordered dot monomials, crossing words and word idempotents give normal forms that straighten with integer structure constants. In the finite characteristic-zero quotient the dots are nilpotent (the relations force bounded dot powers in every position), so only finitely many normal forms are nonzero in each block; their $\mathbb{Z}$-span $H_{\alpha }^{\mathrm{\Lambda }}(\mathbb{Z})$ is therefore a finitely generated torsion-free $\mathbb{Z}$-module, closed under multiplication, and the homogeneous degrees of the generators induce its grading.

Base change from K. Put $L=H_{\alpha }^{\mathrm{\Lambda }}(\mathbb{Z})$ and let $C_{\mathbb{Q}}$ be the rational KLR presentation. Its generator map $C_{\mathbb{Q}}\twoheadrightarrow \mathbb{Q}{\otimes }_{\mathbb{Z}}L$ is surjective by [F3]. Extending to $K$ gives the composition $C_K\twoheadrightarrow K{\otimes }_{\mathbb{Z}}L\twoheadrightarrow H_{\alpha }^{\mathrm{\Lambda }}(K)$, with the second arrow induced by the inclusion of $L$. Both arrows are surjective because their named generators generate the targets, and their composition is the generator-matching all-sector isomorphism of [F2]; therefore both arrows are isomorphisms. Faithful scalar extension from $\mathbb{Q}$ to $K$ also proves the rational generator map is an isomorphism, without requiring a Hecke root of unity in $\mathbb{Q}$. Hence the lattice has rank ${\dim }_K{H}_{\alpha }^{\mathrm{\Lambda }}(K)$ and the first base-change map is an isomorphism of graded algebras.

Reduction modulo p is surjective, including zero sectors. Let $F$ be a field of sub-branch (CP) with a primitive $e$-th root of unity, and put $C_F=R_{\alpha }^{\mathrm{\Lambda }}(F)$ for the cyclotomic KLR presentation. Its generators map onto $F{\otimes }_{\mathbb{Z}}L$ because the generators of $L$ satisfy the integer presentation relations [F3]. The all-sector theorem [F2] identifies $H_{\alpha }^{\mathrm{\Lambda }}(F)$ with $C_F$, so composing gives a generator-matching graded surjection $H_{\alpha }^{\mathrm{\Lambda }}(F)\twoheadrightarrow F{\otimes }_{\mathbb{Z}}L$ for every content. If $H_{\alpha }^{\mathrm{\Lambda }}(F)=0$, that theorem gives $C_F=0$, and the surjection forces $F{\otimes }_{\mathbb{Z}}L=0$; since $L$ is free of rank ${\dim }_K{H}_{\alpha }^{\mathrm{\Lambda }}(K)$ by step 1.2, the characteristic-zero sector is also zero. In all cases ${\dim }_F{H}_{\alpha }^{\mathrm{\Lambda }}(F)\ge {\dim }_K{H}_{\alpha }^{\mathrm{\Lambda }}(K)$. Thus no nonzero-support premise or prior cross-characteristic support assertion is needed.

Dimension comparison. Both sides are graded algebras spanned by the finitely many normal forms of rank $d$, and the total dimension over all contents of rank $d$ equals $l^{d}d!$ in every field of the regime, by the PBW basis and dimension statement [F4]. Summing the individual inequalities of step 2.1 over the finitely many contents of rank $d$ therefore forces equality for every content; each surjection is an isomorphism of graded algebras. Its inverse is the generator-matching map from the tensor lattice asserted in the Definition. The result includes zero sectors and rank zero, where both sides are the ground field with their singleton unit, and level zero at positive rank, where both sides are the zero quotient.

No choice. The lattice is the span of explicit normal forms, and the base-change maps are checked by a finite dimension count over finitely many contents; no family of nonempty sets is selected. Hence no choice principle is used.