Cyclotomic induction is projective on the right

Comparison modules and their projectivity. Put $n=\mathrm{\mid }\beta \mathrm{\mid }$ and let $R_1(\beta )={\xi }_{n}R(\beta )$ act on positions $2,\dots ,n+1$. For $n>0$ set $J_0=Sa(x_1)R(\beta )e_R$ and $J_1=Sa(x_2)R_1(\beta )e_L$; for $n=0$ set $J_0=J_1=0$, since the old block has no cyclotomic relation. Put $K_0=S{e}_R\mathrm{/}{J}_0$, $K_1=S{e}_L\mathrm{/}{J}_1$. The right action of $A[t]$ on $K_0$ uses the first $n$ strands and $t=x_{n+1}$; on $K_1$ it uses ${\xi }_n$ and $t=x_1$. The labelled PBW cosets make $S{e}_R$ and $S{e}_L$ finite projective over the corresponding separated-block algebras $R(\beta )[t]$: each homogeneous coset summand is a shifted principal word-idempotent module. Tensoring these summands with $A[t]$ gives precisely $K_0,K_1$, so both are finite projective over $A[t]$. At $n=0$ both are $k[t]$.

Monic annihilators and projectivity, locally. Let $0\to P_1\stackrel{j}{\to }{P}_0\to M\to 0$ be an exact sequence of right $A[t]$-modules with finite projective $P_0,P_1$, and suppose a central monic $f$ annihilates $M$. Multiplication by $f$ is injective on a free $A[t]$-module by its unit leading coefficient, hence on its projective summands. On a finite free module monic division gives an $A$-linear operator $p_{\mathrm{f}\mathrm{r}\mathrm{e}\mathrm{e}}$ taking a polynomial to its quotient by $f$, with $p_{\mathrm{f}\mathrm{r}\mathrm{e}\mathrm{e}}(xf)=x$. Choose split inclusion $u:P_1\to A[t{]}^N$ and projection $v:A[t{]}^N\to P_1$; then $p=v{p}_{\mathrm{f}\mathrm{r}\mathrm{e}\mathrm{e}}u$ satisfies $p(xf)=x$ on $P_1$, since $u$ is $A[t]$-linear and $vu=1$. Since $fM=0$, multiplication by $f$ on $P_0$ has image in $j(P_1)$, defining $h:P_0\to P_1$ by $jh=f$. Injectivity of $j$ gives $hj=f$ on $P_1$, so $phj=1$. Thus $j$ has the $A$-linear retraction $ph$, and $M$ is a direct summand of the $A$-projective module $P_0$ and is $A$-projective. Equivalently, $P_r\mathrm{/}f{P}_r$ is a summand of $(A[t]\mathrm{/}fA[t]{)}^N\cong A^{mN}$ and is $A$-projective, which is the monic-quotient reason for the splitting. This argument uses finite projective presentations and division and does not assume $f$ invertible or the target $M$ projective.

The cyclotomic ideal and the map $P$. Throughout put ${\kappa }_a=-{\tau }_a$. The first-position PBW cosets give $S={\sum }_{r=0}^n(k^I[x_1]\otimes R_1(n)){\kappa }_1\cdots {\kappa }_r$. Since $a(x_1)$ commutes with this coefficient algebra, $Sa(x_1)S={\sum }_{r=0}^{n}Sa(x_1){\kappa }_1\cdots {\kappa }_r$. After multiplying by $e_R$, terms $r<n$ lie in $J_0$; the remaining term is the image of $\tilde{P}:S{e}_L\to K_0$, $u\mapsto ua(x_1){\kappa }_1\cdots {\kappa }_n$. Thus its image is the kernel of $K_0\to B{e}_R$, where $B=R^{\mathrm{\Lambda }}(\beta +{\alpha }_i)$. To check right linearity, move $x_{a+1}$ through $a(x_1){\kappa }_1\cdots {\kappa }_n$ to $x_a$ and move $x_1$ to $x_{n+1}$. Each mixed-relation correction omits one transfer crossing and has the form, up to its color projector and sign, $({\kappa }_{a+1}\cdots {\kappa }_n)a(x_1){\kappa }_1\cdots {\kappa }_{a-1}$, hence belongs to $J_0$. Moving ${\kappa }_{a+1}$ to ${\kappa }_a$ introduces only the braid correction $a(x_1){\kappa }_1\cdots {\kappa }_{a-1}{D}_a{\kappa }_{a+2}\cdots {\kappa }_n$, with $D_a$ a dot polynomial; the far crossings can be moved left and the last-dot powers commute with the shorter word, again putting it in $J_0$. Idempotents follow the cyclic transfer. These generator identities prove $\tilde{P}$ is right $R(\beta )[t]$-linear. It therefore kills $J_1$, and induces an $(S,A[t])$-map $P:K_1\to K_0$ with the same image.

Intertwiners and the map $Q$. Define $g_a={\kappa }_a$ on unequal adjacent colors and $g_a=(x_{a+1}-x_a)-(x_{a+1}-x_a{)}^2{\kappa }_a$ on equal colors. The source mixed relations give $x_{s_a(b)}{g}_a=g_a{x}_b$ for every $b$. They also give ${\kappa }_a{g}_{a+1}{g}_a=g_{a+1}{g}_a{\kappa }_{a+1}$: subtract the two sides and use the mixed relations to see that the difference intertwines $x_a$ with $x_{a+2}$ and fixes $x_{a+1}$. In the faithful localized polynomial realization [F4], its longest-permutation coefficients cancel, since both products cross the same three color pairs with the same factors. All remaining permutations have length at most two, and none interchanges $a$ and $a+2$ while fixing the middle variable. Independence of the permutation operators then forces every coefficient to vanish. Consequently right multiplication by $g_n\cdots g_1$ intertwines old dots/crossings with their shifted copies and $x_{n+1}$ with $x_1$. It descends to $Q:K_0\to K_1$: the first-position cosets write $J_0$ as the sum of the terms $Sa(x_1){\kappa }_1\cdots {\kappa }_r{e}_R$, $r<n$, and the intertwining identities send each such term to $S{g}_n\cdots g_{1}a(x_2){\kappa }_2\cdots {\kappa }_{r+1}{e}_L\subseteq J_1$. Thus $Q$ is an $(S,A[t])$-map; at $n=0$ it is the identity.

The comparison polynomial, including repeated colors. For a prefix $\nu =({\nu }_1,\dots ,{\nu }_n)$ set $C_{\nu }=a_i(x_1){\prod }_{a:{\nu }_a\mathrm{\ne }i}{Q}_{i{\nu }_a}(x_1,x_{a+1})$. We prove $a(x_1){\kappa }_1\cdots {\kappa }_n{g}_n\cdots g_{1}e(i,\nu )\equiv C_{\nu }e(i,\nu )(\mathrm{m}\mathrm{o}\mathrm{d}{J}_1)$ by induction on $n$. At $n=0$ it is multiplication by $a_i(t)$; at $n=1$ unequal colors use the quadratic relation. For equal colors, ${\kappa }_1{g}_1={\kappa }_1(x_2-x_1)$, and ${\kappa }_1(x_2-x_1)-1$ interchanges $x_1,x_2$; hence $a_i(x_1){\kappa }_1{g}_1\equiv a_i(x_1)$ modulo $S{a}_i(x_2)$. For $n>1$, if ${\nu }_n\mathrm{\ne }i$, the quadratic relation and the dot-intertwining identities peel off $Q_{i{\nu }_n}(x_1,x_{n+1})$, leaving the shorter comparison. If ${\nu }_n=i$ and ${\nu }_{n-1}=j\mathrm{\ne }i$, the braid correction splits the comparison into two terms whose reductions, for ${\nu }^{\mathrm{\prime }\mathrm{\prime }}=({\nu }_1,\dots ,{\nu }_{n-2})$, are $C_{{\nu }^{\mathrm{\prime }\mathrm{\prime }}}{Q}_{ji}(x_n,x_{n+1})$ and $-C_{{\nu }^{\mathrm{\prime }\mathrm{\prime }}}(Q_{ij}(x_{n+1},x_n)-Q_{ij}(x_1,x_n))$. Their sum is $C_{{\nu }^{\mathrm{\prime }\mathrm{\prime }}}{Q}_{ij}(x_1,x_n)=C_{\nu }$ by $Q$ symmetry. If the last two colors are $i$, expansion of $g_{n-1}$, the ordinary equal-color braid, and the shorter comparison reduce the expression to $C_{{\nu }^{\mathrm{\prime }}}[(x_{n+1}-x_1){\kappa }_n-{\kappa }_n(x_{n+1}-x_1){\kappa }_n(x_n-x_1)]$, where ${\nu }^{\mathrm{\prime }}={\nu }^{\mathrm{\prime }\mathrm{\prime }}i$. Here $C_{{\nu }^{\mathrm{\prime }}}$ commutes with ${\kappa }_n$, ${\kappa }_n(x_{n+1}-x_1){\kappa }_n={\kappa }_n$ and ${\kappa }_n(x_n-x_1)=(x_{n+1}-x_1){\kappa }_n-1$, so the bracket is $1$ and $C_{{\nu }^{\mathrm{\prime }}}=C_{\nu }$. All congruences remain valid under dot multiplication because the shifted cyclotomic ideal defining $J_1$ is a right dot submodule. These cases exhaust the induction and establish $QP=C(t)$, with $C(t)={\sum }_{\nu \in I^{\beta }}{a}_i(t){\prod }_{a:{\nu }_a\mathrm{\ne }i}{Q}_{i{\nu }_a}(t,x_a)e(\nu )\in A[t]$.

Injectivity and the exact resolution. The degree of $C(t)$ in $t$ is ${\lambda }_i+{\sum }_{j\mathrm{\ne }i}(-a_{ij}){\beta }_j$, and its leading coefficient is the nonzero scalar ${\prod }_{j\mathrm{\ne }i}{t}_{ij}^{{\beta }_j}$, independent of $\nu$. Its coefficients are central in $A$: unequal crossings merely permute the color-labelled factors, and on equal adjacent colors the factor product is symmetric, so its divided-difference correction vanishes. Normalize the leading coefficient to $1$. Multiplication by this central monic polynomial is injective on a free $A[t]$-module by its leading term, and hence on a projective summand. Step 4.1 therefore makes $QP$, and consequently $P$, injective. With step 2.1 this proves $0\to q^s{K}_1\stackrel{P}{\to }{K}_0\to B{e}_R\to 0$. In the standard homogeneous case $\mathrm{deg}P=2d_i{\lambda }_i-{\sum }_a{b}_{i{\nu }_a}=({\alpha }_i\mid 2\mathrm{\Lambda }-\beta )=s$, so this is a graded resolution; the monic branch is ungraded.

Projectivity of the cyclotomic bimodules. The appended dot on $M=B{e}_R$ is annihilated by a central monic scalar-coefficient polynomial by [F2]. Apply step 1.2 to the projective $A[t]$-resolution in step 5.1, with $M=B{e}_R$, to conclude that $B{e}_R$ is right $A$-projective. This module is finitely generated over $A$: its finite $A[t]$-generators together with dot powers below the degree of the annihilator generate it over $A$. In the homogeneous standard case the resolution and $M$ are graded; decompose the resulting $A$-linear splitting into homogeneous components on finite homogeneous generators, and take its degree-zero component to get a graded splitting. The generator-fixing anti-involution sends $B{e}_R$ to $e_{R}B$ and converts this right projectivity into left $A$-projectivity. The same monic argument is valid for arbitrary ungraded monic cyclotomic deformations.

Exactness and preservation. Tensoring by the right $A$-projective module $B{e}_R$ is exact, so $F_i^{\mathrm{\Lambda }}$ is exact, and idempotent restriction $E_i^{\mathrm{\Lambda }}$ is exact. The left $B$-module $B{e}_R$ is a direct summand of $B$, hence induction sends any projective $A$-module, presented as a summand of a free module, to a projective $B$-module; finite generators remain finite under induction. A finitely generated projective $B$-module is a summand of a finite direct sum of shifts of $B$, and its restriction is consequently a summand of the corresponding finite direct sum of shifts of $e_{R}B$. Left $A$-projectivity and finite generation of $e_{R}B$ from step 6.1 therefore show that restriction preserves finitely generated projectives. These arguments apply ungraded to monic deformations as well.

The exact sequence (c). Tensor the bimodule resolution $0\to q^s{K}_1\stackrel{P}{\to }{K}_0\to B{e}_R\to 0$ of step 5.1 over $A$ with a left $A$-module $M$. It remains exact because its cokernel $B{e}_R$ is right $A$-projective by step 6.1, so the original sequence is split as a sequence of right $A$-modules. By their definitions in step 1.1, $K_1{\otimes }_{A}M$ is the left-append generic induction ${\bar{F}}_{i}M$ and $K_0{\otimes }_{A}M$ is the right-append generic induction $F_{i}M$, while $B{e}_R{\otimes }_{A}M=F_i^{\mathrm{\Lambda }}M$. The map $P$ is left $R(\beta +{\alpha }_i)$-linear and has homogeneous degree $s=({\alpha }_i\mid 2\mathrm{\Lambda }-\beta )$, giving exactly the functorial sequence in (c) in the declared target category. On the ungraded monic-deformation branch the same sequence is interpreted without a grading shift.

Conclusion. The projectivity, preservation and exact-sequence arguments prove (a), (b) and (c) with the displayed shift, in the graded standard case with graded projectives and verbatim ungraded for a monic deformation; the comparison is monic and no filtered-colon citation is used.