Type-A graph-bimodule extension vanishing

Set $I:=I(Gr(x))=\mathrm{Ann}(R_x)$, generated by ${\mathrm{\ell }}_1,\dots ,{\mathrm{\ell }}_n$, and fix a homogeneous lift $e\in E$ of the generator $1_{R_x(a)}$, so that the sub identifies with $R_y(b)$ through a chosen generator $\iota$ of degree $-b$. Since every $h\in I$ kills the image of $e$ in $R_x(a)$, one has $h\cdot e\in R_y(b)$, so $h\cdot e=c(h)\iota$ for a unique homogeneous element $c(h)\in R$, and $E$ is generated over $R$ by $e$ and $\iota$ with the single family of relations $h\cdot e=c(h)\iota$.

The assignment $h\mapsto c(h)$ is additive and satisfies $c(h{h}^{\mathrm{\prime }})=(\text{action of }h\text{ on }{R}_y)\cdot c(h^{\mathrm{\prime }})$ because $(h{h}^{\mathrm{\prime }})\cdot e=h\cdot (c(h^{\mathrm{\prime }})\iota )=h\cdot (c(h^{\mathrm{\prime }})\iota )$ in the sub-bimodule $R_y(b)$; writing $u_i$ for the image of ${\mathrm{\ell }}_i$ under $R\otimes R\to R_y$, $h\mapsto h\cdot 1_{R_y}$, one has $u_{i}c({\mathrm{\ell }}_j)=u_{j}c({\mathrm{\ell }}_i)$ for all $i,j$, since ${\mathrm{\ell }}_i{\mathrm{\ell }}_j={\mathrm{\ell }}_j{\mathrm{\ell }}_i$ and $c({\mathrm{\ell }}_i{\mathrm{\ell }}_j)=u_{i}c({\mathrm{\ell }}_j)$. Writing $m_i:=c({\mathrm{\ell }}_i)$, the extension is thus encoded by elements $m_1,\dots ,m_n\in R$ with $u_i{m}_j=u_j{m}_i$ for all $i,j$, and nothing else.

The forms ${\mathrm{\ell }}_i$ lie in the linear span of the variables of $R\otimes R$, and their images are $u_i=x_i-x_{y{x}^{-1}i}$: acting on $1_{R_y}$ one has $x_i\cdot 1=x_i$ and $(1\otimes x_{x^{-1}i})\cdot 1=x_{y{x}^{-1}i}$. Hence the ideal generated by the $u_i$ is the ideal of the subspace $V^{y{x}^{-1}}$ of $V$, the rank of the family $(u_i)$ is $r=n-\dim V^{y{x}^{-1}}=n-\dim V^{x^{-1}y}$, and $Gr(x)\cap Gr(y)=\{(x\lambda ,\lambda ):x^{-1}y\lambda =\lambda \}$, of dimension $\dim V^{x^{-1}y}$ in either graph.

Splitting: the extension splits iff there is $m\in R$ with $m_i=u_{i}m$ for all $i$. Indeed a splitting of the sequence is exactly a homogeneous $R$-bimodule section of $E\to R_x(a)$, which has the form $1_{R_x(a)}\mapsto e^{\mathrm{\prime }}$ with $e^{\mathrm{\prime }}\in E$ lifting the generator; writing $e^{\mathrm{\prime }}=e-m\iota$ for some $m\in R$, the section is well defined iff $I\cdot e^{\mathrm{\prime }}=0$, i.e. iff ${\mathrm{\ell }}_i\cdot (e-m\iota )=0$ for all $i$, that is $m_i=u_{i}m$ for all $i$. Conversely such an $m$ makes $h\mapsto h\cdot e^{\mathrm{\prime }}$ a well-defined splitting.

Reduction of the $r\ge 2$ case to two coprime forms: view $\sigma :=y{x}^{-1}$ as a permutation of the coordinate set, so $u_i=x_i-x_{\sigma i}$. If some cycle of $\sigma$ has length $\ge 3$, two consecutive forms of that cycle, $x_i-x_{\sigma i}$ and $x_{\sigma i}-x_{{\sigma }^{2}i}$, are distinct irreducible elements of the polynomial ring and hence coprime in the UFD $R$; if all cycles have length $\le 2$ and $r\ge 2$, there are at least two transposition cycles and one form from each is a pair with disjoint supports, again coprime. In both cases pick such indices $p,q$.

Conclusion in the case $r\ge 2$: from $u_p{m}_q=u_q{m}_p$ and $\gcd (u_p,u_q)=1$ in the UFD $R$ we get $u_p\mid m_p$ and $u_q\mid m_q$; write $m_p=u_{p}m$ and $m_q=u_q{m}^{\mathrm{\prime }}$; substituting gives $u_p{u}_q{m}^{\mathrm{\prime }}=u_q{u}_{p}m$, hence $m^{\mathrm{\prime }}=m$ as $R$ is a domain. For every index $i$, $u_p{m}_i=u_i{m}_p=u_i{u}_{p}m$ forces $m_i=u_{i}m$, since $u_p\mathrm{\ne }0$ and $R$ is a domain. By step 3.1 the extension splits.

The case $r=1$: then $\sigma =y{x}^{-1}$ is a reflection, so it exchanges two coordinates $p\mathrm{\ne }q$ and fixes the rest; hence $u_p=x_p-x_q$, $u_q=-u_p$ and $u_i=0$ for all other $i$. The relations $u_i{m}_j=u_j{m}_i$ give $m_i=0$ for $i\mathrm{\notin }\{p,q\}$ and $m_q=-m_p$, and by step 3.1 the extension splits iff $u_p\mid m_p$. In the localization at $u_p$ the obstruction dies because $u_p$ becomes a unit; moreover $u_p$ is a unit multiple of ${\alpha }_{\sigma }$, the equation of the hyperplane of fixed points of $\sigma =y{x}^{-1}$, and for $y=xt$ that reflection is $xt{x}^{-1}$ with equation $x({\alpha }_t)$.

Both assertions follow: for $r\ge 2$ every extension splits by step 4.1, and for $r=1$ with $y=xt$ every extension splits in the stated localization by step 4.2. All constructions were made on homogeneous lifts, so the argument applies degree by degree in the graded category. ∎