The type-A Soergel category ${\mathrm{S}\mathrm{B}\mathrm{i}\mathrm{m}}_n$

Ambient bimodules. Fix $R=\mathbb{Q}[x_1,\dots ,x_n]$ with $\mathrm{deg}{x}_i=2$ and the simple reflections $s_1,\dots ,s_{n-1}$ as in The standard type-A reflection realization and its polynomial ring. We work inside the category of graded $(R,R)$-bimodules and degree-zero bimodule maps, where "graded" means a $\mathbb{Z}$-indexed direct sum decomposition $M={⨁}_{d\in \mathbb{Z}}{M}_d$ with $R_i{M}_j{R}_k\subseteq M_{i+j+k}$, and where the internal shift $M\{r\}$ has $(M\{r\}{)}_d=M_{d-r}$; as on the rest of this page we also write $M(r):=M\{-r\}$ for the Elias–Williamson shift, so that $M(1{)}_d=M_{d+1}$.

Bott–Samelson objects. For a finite word $\underset{‾}{i}=(i_1,\dots ,i_r)$ let $B_{\underset{‾}{i}}=B_{i_1}{\otimes }_R\cdots {\otimes }_R{B}_{i_r}$ be the Bott–Samelson bimodule of The Bott–Samelson bimodule of a word, with $B_{\mathrm{\varnothing }}=R$. Let ${\mathrm{B}\mathrm{S}\mathrm{B}\mathrm{i}\mathrm{m}}_n$ be the full subcategory of the ambient category whose objects are the finite direct sums $B_{{\underset{‾}{i}}_1}(d_1)\oplus \cdots \oplus B_{{\underset{‾}{i}}_s}(d_s)$ of shifts of Bott–Samelson bimodules, with the biproduct structure on direct sums and degree-zero maps between them; the empty sum is the zero bimodule. This is an additive $\mathbb{Q}$-linear category with grading shifts. Its total graded morphism space ${\mathrm{Hom}}^{\bullet }(M,N)={⨁}_{d\in \mathbb{Z}}{\mathrm{Hom}}^0(M,N(d))$ is an $R$-module by left multiplication on the output: a homogeneous scalar of degree $a$ sends its degree-$d$ part to its degree-$(d+a)$ part. The categorical morphisms are its degree-zero part, which need not be an $R$-submodule. The tensor product over $R$ makes it a monoidal category with unit $R=B_{\mathrm{\varnothing }}$, because the tensor product of two finite sums of shifted Bott–Samelson products is again such a sum, it distributes over the direct sums in each variable, and $(M(d)){\otimes }_{R}N\cong M{\otimes }_R(N(d))\cong (M{\otimes }_{R}N)(d)$ naturally.

Idempotent completion. The type-A Soergel category is the idempotent completion, in the sense of The idempotent completion of a preadditive category, ${\mathrm{S}\mathrm{B}\mathrm{i}\mathrm{m}}_n:=\mathrm{K}\mathrm{a}\mathrm{r}({\mathrm{B}\mathrm{S}\mathrm{B}\mathrm{i}\mathrm{m}}_n)\mathrm{.}$ Its objects are the pairs $(M,e)$ with $M$ a finite sum of shifted Bott–Samelson products and $e\in \mathrm{End}(M)$ a degree-zero idempotent, and its morphisms are the degree-zero maps $\mathrm{Hom}((M,e),(N,f))=f\mathrm{Hom}(M,N)e$. The tensor product extends to the completion by $(M,e){\otimes }_R(N,f):=(M{\otimes }_{R}N,e\otimes f)$ and makes ${\mathrm{S}\mathrm{B}\mathrm{i}\mathrm{m}}_n$ a graded, additive, idempotent-complete monoidal category with the same unit $R$; equivalently, ${\mathrm{S}\mathrm{B}\mathrm{i}\mathrm{m}}_n$ is the smallest full subcategory of the ambient bimodule category that contains $R$ and the $B_i$, is closed under finite direct sums, internal shifts, tensor products over $R$, and direct summands. Every object of ${\mathrm{S}\mathrm{B}\mathrm{i}\mathrm{m}}_n$ is a finite direct sum of graded shifts of indecomposable objects.

Separation from the diagrammatic presentation. We write ${\mathrm{S}\mathrm{B}\mathrm{i}\mathrm{m}}_n$ for the bimodule category defined here and $\mathcal{D}$ (or ${\mathcal{D}}_n$) for the $k$-linear graded monoidal category presented by diagrams in The type-A diagrammatic Soergel category and its candidate bimodule functor; the two are related, but not identified, by the evaluation functor, which is proved to be an equivalence only later on this page. In particular no statement about $\mathcal{D}$ may be read as a statement about ${\mathrm{S}\mathrm{B}\mathrm{i}\mathrm{m}}_n$ before that equivalence is established. For $n\le 1$ there are no simple reflections, so the generating object $R$ is the only indecomposable up to shift and ${\mathrm{S}\mathrm{B}\mathrm{i}\mathrm{m}}_n$ is the closure of $R$ under finite direct sums, internal shifts and direct summands: its objects are the finite direct sums of graded shifts of $R$, its morphisms are the graded $R$-bimodule maps between them, and the summands are again finite sums of shifts of $R$. Indeed a finite graded projective module over the connected nonnegatively graded ring $R$ is graded free: lift a homogeneous basis modulo $R_{+}={⨁}_{d>0}{R}_d$, obtaining a surjection from a finite graded free module by graded Nakayama. Projectivity splits it; its kernel has zero reduction modulo $R_{+}$ and is bounded below, so graded Nakayama kills the kernel. This applies to every graded summand here, whose two $R$-actions agree.