Ordinary middle perverse t structure, recollement and intersection complexes

The recollement and the glued t-structure. Let $U\stackrel{j}{\to }X\stackrel{i}{\leftarrow }Z$ be an algebraic open/closed decomposition. By [F1] the six-operation supplier gives $j_{!}⊣j^{\ast }⊣R{j}_{\ast }$ and $i^{\ast }⊣i_{\ast }⊣i^{!}$, full faithfulness of $i_{\ast }$ and $j_{!}$, the vanishing $j^{\ast }{i}_{\ast }=0$ and the two localization triangles $j_{!}{j}^{\ast }K\to K\to i_{\ast }{i}^{\ast }K\to j_{!}{j}^{\ast }K[1]$ and $i_{\ast }{i}^{!}K\to K\to R{j}_{\ast }{j}^{\ast }K\to i_{\ast }{i}^{!}K[1]$. Given t-structures on the bounded constructible categories of $U$ and $Z$, define $T^{\le 0}$ by $j^{\ast }K\in D^{\le 0}$ and $i^{\ast }K\in D^{\le 0}$, and $T^{\ge 0}$ by $j^{\ast }K\in D^{\ge 0}$ and $i^{!}K\in D^{\ge 0}$. Orthogonality $\mathrm{Hom}(T^{\le 0},T^{\ge 1})=0$ follows from the first localization triangle: the two endpoint Hom groups into $B\in T^{\ge 1}$ vanish by adjunction and the two existing t-structures. For truncation, form $Y\to K\to R{j}_{\ast }{\tau }^{\ge 1}(j^{\ast }K)$ and then $A\to Y\to i_{\ast }{\tau }^{\ge 1}(i^{\ast }Y)$; octahedral completion [F3] produces $A\to K\to B$ with a triangle $i_{\ast }{\tau }^{\ge 1}(i^{\ast }Y)\to B\to R{j}_{\ast }{\tau }^{\ge 1}(j^{\ast }K)$. Applying $j^{\ast }$ shows $j^{\ast }A={\tau }^{\le 0}{j}^{\ast }K$ and $j^{\ast }B={\tau }^{\ge 1}{j}^{\ast }K$; applying $i^{\ast }$ to $A\to Y$ gives $i^{\ast }A={\tau }^{\le 0}{i}^{\ast }Y$, and applying $i^{!}$ to the $B$-triangle and using $i^{!}R{j}_{\ast }=0$ gives $i^{!}B={\tau }^{\ge 1}{i}^{\ast }Y$. Thus $A\in T^{\le 0}$, $B\in T^{\ge 1}$ and shift-closure is immediate, so $(T^{\le 0},T^{\ge 0})$ is a t-structure in the sense of [F2]. Ordinary truncations preserve finite-rank locally constant cohomology, so [F2] restricts to the stratum-constructible category and shifting by ${\dim }_{\mathbb{C}}S$ gives its required t-structure. Gluing a maximal open union of strata to its closed complement and inducting on their finite number proves the inequalities define such a t-structure and that it stays inside $D_c^b$, because the six operations preserve constructibility and boundedness by [F1] and the local dimension data are those of [F4].

Independence of compatible refinement and gluing. Let $T\subset S$ be a refinement of smooth complex strata of codimension $c$ along the same algebraic variety. Ordinary pullback preserves the upper bound $-{\dim }_{\mathbb{C}}S\le -{\dim }_{\mathbb{C}}T$, and on locally constant cohomology extraordinary pullback is ordinary pullback shifted by $-2c$: in local normal coordinates the closed-support complex of a real $2c$-ball relative to its puncture has one oriented class in degree $2c$, and tensoring with the local coefficient complex computes that shift, so its lower bound becomes $-{\dim }_{\mathbb{C}}S+2c\ge -{\dim }_{\mathbb{C}}T$; hence both old halves lie in the refined halves, and uniqueness of truncation proves equality on the old category. Taking common refinements gives independence of the chosen compatible stratification, and the same argument applied over each member of an algebraic open/closed decomposition shows the glued structure is independent of the decomposition. This uses the local conical and orientation suppliers of [F4] rather than presupposing stratification independence.

Duality exchanges the aisles. Let $S$ be a smooth complex stratum with $e={\dim }_{\mathbb{C}}S$. The exchange identities of [F1] give $i_S^{!}{D}_{X}A\cong D_S{i}_S^{\ast }A$ and $i_S^{\ast }{D}_{X}A\cong D_S{i}_S^{!}A$, and on bounded complexes with finite-rank locally constant cohomology on $S$, complex orientation and exact finite-dimensional duality give $H^j(D_{S}K)\cong (H^{-j-2e}K{)}^{\vee }$. Thus $H^t(i_S^{\ast }A)=0$ for $t>-e$ implies $H^j(i_S^{!}{D}_{X}A)=0$ for $j<-e$, and $H^t(i_S^{!}A)=0$ for $t<-e$ implies $H^j(i_S^{\ast }{D}_{X}A)=0$ for $j>-e$: the two opposite middle-perverse inequalities. Exact finite-dimensional duality holds over every field, and $2e$ is an integer cochain shift rather than a scalar being inverted. Applying the same argument to $D_{X}A$ and using the involutivity of [F1] gives equality of the exchanged aisles.

The heart is abelian. For a morphism $u:A\to B$ of heart objects, its cone $C$ has t-cohomology only in degrees $-1,0$, by the truncation bounds. Put $K={}^p{H}^{-1}C$ and $Q={}^p{H}^{0}C$. Applying octahedral completion to $K[1]\to C\to Q$ and $A\to B\to C$ produces one object $I$ with triangles $K\to A\to I\to K[1]$ and $I\to B\to Q\to I[1]$. The first bounds $I$ in degrees $[-1,0]$ and the second in $[0,1]$, so $I$ is in the heart. Applying Hom from and to heart objects and using orthogonality proves that $K$ is the kernel of $u$, $Q$ its cokernel, $I$ the cokernel of $K\to A$ and the kernel of $B\to Q$. Thus image equals coimage and the additive heart is abelian. For the glued structure $j^{\ast }$ preserves both aisles, hence truncation triangles and these kernels and cokernels, so open restriction is perverse exact.

Intermediate extension and its boundary characterization. For a heart object $M$ on an open $j:U\hookrightarrow Z$ with closed complement $i$, let $J_{!}={}^p{H}^0{j}_{!}M$ and $J_{\ast }={}^p{H}^{0}R{j}_{\ast }M$. Adjunction and orthogonality show that $J_{!}$ is left adjoint and $J_{\ast }$ right adjoint to exact $j^{\ast }$ on hearts; in particular $J_{!}$ is right exact. They restrict to $M$. Moreover $J_{!}$ has no nonzero quotient supported on the boundary, since $\mathrm{Hom}(J_{!},i_{\ast }N)=\mathrm{Hom}(M,j^{\ast }{i}_{\ast }N)=0$, and $J_{\ast }$ has no boundary subobject by the dual adjunction. Define $j_{!\ast }M=\mathrm{im}(J_{!}\to J_{\ast })$. It inherits both properties. If $P$ restricts to $M$ and has neither boundary subobjects nor boundary quotients, the adjunction maps $J_{!}\to P\to J_{\ast }$ have boundary-supported cokernel and kernel, so the first is epic and the second monic; their composite identifies $P$ with $j_{!\ast }M$. A subobject of $j_{!\ast }M$ for simple $M$ restricts either to zero or to $M$; its boundary kernel or the resulting boundary quotient must vanish. Hence $j_{!\ast }M$ is simple. Closed pushforward is exact and fully faithful on hearts by recollement, so this construction also defines intermediate extension from a locally closed dense open of a closed support. For a local system on a smooth open of dimension $e$, its input is $M=L[e]$.

Simple local systems give simple shifted objects on a smooth connected variety $S$ of complex dimension $e$. A path transports stalks by composing its local trivializations on a finite subdivision. Transition matrices are locally constant; a common subdivision proves independence of subdivision, and a finite rectangular subdivision of a path homotopy into trivializing opens proves homotopy invariance. Thus local subsystems correspond to invariant subspaces of the transported stalk. For a dense algebraic open $U\subseteq S$, every loop in $S$ can be moved off $S\setminus U$: triangulate the algebraic subset as in [F4], and perturb a finite polygonal representative in smooth coordinate balls off its finitely many pieces of real codimension at least two. Move its finitely many vertices into $U$ inside the overlapping balls. For a segment with endpoints in $U$, forbidden intermediate vertices lie on rays from either endpoint through boundary strata. Each ray family is locally parametrized by at most $2e-1$ real parameters; covering countably many compact parameter boxes by small cubes bounds the total $2e$-dimensional covering volume by an arbitrarily small number. Their union therefore cannot fill a coordinate ball. Choose an intermediate vertex outside both families and join by two straight segments. These replacements avoid the boundary and are homotopies in the balls, fixing the loop base point. Consequently transport on $U$ has the same invariant subspaces as on $S$, and a simple $L$ remains simple on $U$. Also $L[e]$ has no perverse subobject or quotient supported on a proper algebraic closed subset: on any smooth boundary stratum of codimension $c>0$, ordinary restriction has degree $-e<-\dim T$, and extraordinary restriction has degree $2c-e>-\dim T$, by the relative-ball calculation of step 2.1. These are the strict upper and lower boundary bounds; adjunction and orthogonality kill respectively boundary quotients and subobjects. A putative proper subobject is, on a sufficiently small dense algebraic open, a local subsystem of $L[e]$, since its cohomology there is locally constant and perversity on that smooth open is the shifted ordinary structure. Simplicity after restriction makes it zero or all of $L[e]$ there, and the preceding no-boundary properties then make it zero or all globally. Finite-rank local systems themselves have finite composition series, because every strict sub-local-system increases stalk dimension.

Classification and finite length. For a simple heart object choose a dense smooth open in one irreducible component of its support where it is a nonzero $L[e]$. A proper sub-local-system would, by the left adjunction in step 3.1, give a nonzero proper image subobject. Thus $L$ is simple. A simple object has neither boundary subobjects nor quotients, so step 3.1 identifies it with the closed pushforward of $j_{!\ast }L[e]$; step 4.1 and step 3.1 prove that every such intersection complex is simple. For finite length, induct on the dimension of an object's support, with the empty support giving zero. Choose a dense smooth open union of its components on which its restriction $M$ is a finite-rank local system with the appropriate shift. Each simple factor $M_r$ has $J_{!}{M}_r\twoheadrightarrow j_{!\ast }{M}_r$ with boundary-supported kernel. That kernel has finite length by the dimension induction, and the quotient is simple, so $J_{!}{M}_r$ has finite length. Right exactness of $J_{!}$ and a finite local-system composition series then show that $J_{!}M$ has finite length. The image of $J_{!}M\to P$ has finite length, and its cokernel is supported on the smaller-dimensional boundary, hence also has finite length by induction. Their short exact sequence proves finite length of $P$. This argument does not assume an arbitrary subobject chain stays constructible for an initially chosen stratification.

Negative Hom and division endomorphism algebras over every field. For $m<0$ the shift $B[m]$ lies in $T^{\ge -m}\subseteq T^{\ge 1}$ and $A$ lies in $T^{\le 0}$, so orthogonality gives $\mathrm{Hom}(A,B[m])=0$: heart objects have no negative Hom. If $u:T\to T$ is nonzero with $T$ simple, its kernel and image are subobjects with kernel zero and image $T$, so $u$ is invertible and the endomorphism ring of $T$ is a division $k$-algebra. Finite-rank local systems have finite length by the dimension of their underlying vector spaces, and the recollement and kernel/image arguments use only exactness and that finite length, so the ordinary perverse construction uses no characteristic-zero input and no equivariant, normalization, semisimplicity or noncharacteristic theorem.