Minimal complexes and the perverse truncations in the real type-A Soergel category

The category and its radical. Let $\mathcal{B}={\mathrm{S}\mathrm{B}\mathrm{i}\mathrm{m}}_n^R$ be the real graded type-A Soergel category of The real type-A Soergel category by scalar extension, with degree-zero morphisms; it is a graded additive idempotent-complete category (Additive category, The idempotent completion of a preadditive category) whose objects are finite direct sums of shifts of the indecomposables $B_w^R$ (Indecomposable labels, characters and filtration ranks agree after real scalar extension). Its categorical radical is the two-sided ideal $\mathrm{rad}\mathcal{B}:=\{f:M\to N:\text{ for every }g:N\to M,1_M-gf\text{ is invertible}\},$ and $q:\mathcal{B}\to {\mathcal{B}}^{ss}:=\mathcal{B}\mathrm{/}\mathrm{rad}\mathcal{B}$ is the quotient functor. The radical is defined by the displayed invertibility criterion and is not claimed to be nilpotent in general; on objects with local endomorphism rings (as here, by the comparison lemma) it consists of the non-units in the relevant endomorphism algebras.

Minimal complexes. Let $K^b(\mathcal{B})$ be the homotopy category of bounded cochain complexes of graded bimodules with degree-zero differentials (Bounded graded bimodule complexes and signed tensor totalization, The homotopy category of chain complexes). A bounded complex $F$ is minimal when $q(d_F)=0$ in every degree, i.e. every differential of $F$ lies in the radical. Homological shifts are written $[j]$ (The shift of a chain complex); mapping cones are as in The mapping cone of a chain map; contractibility and homotopies are as in Complexes, homotopies and contractibility in an additive category; and the two-term cancellation used on this page is Gaussian elimination splits a contractible two-term complex with the block calculus of An invertible cochain differential block and its candidate reduction.

Perverse bimodules. Let ${\underset{‾}{H}}_x$ be the Kazhdan-Lusztig basis of the Hecke algebra of the frontier Hecke page, transported to this page by the comparison lemma. A Soergel bimodule is perverse when its character is a nonnegative integer combination of the Kazhdan-Lusztig basis elements ${\underset{‾}{H}}_x$ (not of the standard basis elements $H_x$); it is p-split when its indecomposable summands are shifts of perverse ones. Thus if $\mathrm{S}(x)$ holds then $B_x^R$ is perverse, since its character is ${\underset{‾}{H}}_x$; no other perversity of a normalized indecomposable is asserted before $\mathrm{S}(x)$.

The perverse cuts. Let $M$ be p-split and fix a decomposition $M={⨁}_{x,k}{B}_x^R(k{)}^{m_{x,k}}$ with the normalized $B_x^R$ perverse (such a decomposition exists by the comparison lemma, and the cuts below are proved intrinsic by the structural lemma). For $j\in \mathbb{Z}$ set ${\tau }_{\le j}M:={⨁}_{k\ge -j}{B}_x^R(k{)}^{m_{x,k}},{\tau }_{<j}M:={\tau }_{\le j-1}M,{\tau }_{\ge j}M:=M\mathrm{/}{\tau }_{<j}M,$ and define the associated graded $H^j(M):=\left({\tau }_{\le j}M\mathrm{/}{\tau }_{<j}M\right)(j),$ so that the $j$-th perverse piece is normalized by the internal shift $(j)$. Internal shifts $(d)$ change module degrees and homological shifts $[j]$ change complex degrees; the two are kept distinct throughout.

The perverse subcategories of complexes. For a bounded complex $F=(F^i,d^i)$ define $F\in p{K}_{\ge 0}^b⟺F\text{ is isomorphic in }{K}^b(\mathcal{B})\text{ to a complex with p-split terms and }{\tau }_{<-i}(F^i)=0\text{ for all }i,$ $F\in p{K}_{\le 0}^b⟺F\text{ is isomorphic in }{K}^b(\mathcal{B})\text{ to a complex with p-split terms and }{F}^i={\tau }_{\le -i}(F^i)\text{ for all }i\mathrm{.}$ Equivalently, in the termwise decompositions the internal shifts $k$ satisfy $k\le i$ for $p{K}_{\ge 0}^b$ and $k\ge i$ for $p{K}_{\le 0}^b$. The structural lemma of this page supplies the semisimplicity of ${\mathcal{B}}^{ss}$, the existence and uniqueness of minimal models, the intrinsicness and functoriality of the cuts, their preservation under direct summands and their extension closure in distinguished triangles; none of these properties is asserted by this definition. The objects $p{K}_{\ge 0}^b$ and $p{K}_{\le 0}^b$ are the classes inside which the Rouquier complexes of the next items are shown to lie, following the orientation of Elias-Williamson Section 6.3.

• The radical criterion is deliberately not strengthened. For a normalized indecomposable $B_x^R$ the radical of its endomorphism algebra vanishes only after $\mathrm{S}(x)$ makes the endomorphism algebra $\mathbb{R}$ (comparison lemma). Whenever its degree-zero endomorphism algebra is local with residue field $\mathbb{R}$, its image $q(B_x^R)$ is simple in the radical quotient: the quotient endomorphism algebra is the residue field, even if the original algebra has a nonzero radical. This does not make the original algebra one-dimensional before $\mathrm{S}(x)$.
• Perversity is not positivity. The definition of perverse uses only the existence of the Kazhdan-Lusztig basis and the nonnegativity of the coefficients of a given character; it does not assert that the coefficients of ${\underset{‾}{H}}_w$ in the standard basis are nonnegative, which is the positivity corollary proved at the end of this page.
• Small cases. If $M=0$ all cuts are zero and $H^j(M)=0$; if $M$ is a single shift $B_x^R(k)$ with $x$ satisfying $\mathrm{S}(x)$, then ${\tau }_{\le j}M=M$ for $j\ge -k$ and $0$ otherwise.