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Closed-support splitting of a perverse sheaf is detected by its costalk-to-stalk map

Statement

Assume AC as inherited from the ordinary constructible suppliers, and let $i : Z ↪ Y$ be a closed complex algebraic inclusion with complementary open inclusion $j : U ↪ Y$ . Let $P$ be an ordinary constructible perverse sheaf over a characteristic-zero field. Set $A =^{p} H^{0} i^{!} P$ and $B =^{p} H^{0} i^{*} P$ . The adjunction maps define a natural morphism $α : A \to B$ . It is an isomorphism if and only if $P ≃ i_{*} A \oplus j_{! *} j^{*} P .$ More precisely, $i_{*} A$ is the largest subobject of $P$ supported on $Z$ , $i_{*} B$ is its largest quotient supported there, and their composite equals $i_{*} α$ . If $α$ is invertible, these canonical maps give the displayed splitting. For $Z = {y}$ , $A = H^{0} (i_{y}^{!} P)$ and $B = H^{0} (i_{y}^{*} P)$ are finite vector spaces, so the criterion is exactly invertibility of the degree-zero costalk-to-stalk map. This criterion does not assert that that map is invertible for projective direct images.

Facts & Assumptions

Given: The closed inclusion $i : Z ↪ Y$ with open complement $j : U ↪ Y$ , the ordinary constructible perverse sheaf $P$ over a characteristic-zero field, the objects $A =^{p} H^{0} i^{!} P$ , $B =^{p} H^{0} i^{*} P$ and $Q = j^{*} P$ , and the adjunction morphisms between the six functors for the pair $(i, j)$ .

[F1]

Ordinary perverse recollement, its adjunctions and intermediate extension are supplied by Ordinary middle perverse t structure, recollement and intersection complexes. In particular $i_{*}$ and $j^{*}$ are perverse exact, $i^{*}$ preserves the upper aisle $\leq$ and $i^{!}$ the lower aisle, $R j_{!}$ preserves the upper aisle and $R j_{*}$ the lower aisle; the fully faithful functor $i_{*}$ identifies the perverse sheaves on $Z$ with the perverse sheaves on $Y$ supported on $Z$ . Intermediate extension $j_{! *} Q$ is the image of the natural perverse morphism $^{p} H^{0} R j_{!} Q \to^{p} H^{0} R j_{*} Q$ , and it has no nonzero subobject and no nonzero quotient supported on $Z$ . The perverse heart on a point is the category of finite-dimensional vector spaces, in ordinary degree zero.

[F2]

The ordinary localization triangles $i_{*} i^{!} P \to P \to R j_{*} j^{*} P \overset{+ 1}{\to}$ , $R j_{!} j^{*} P \to P \to i_{*} i^{*} P \overset{+ 1}{\to}$ and their compatible adjunction maps are supplied by Actual analytic constructible sheaves admit coherent six operations. The Axiom of Choice is inherited solely from those constructed ordinary categories and functors; no additional selection is used.

Proof

1.1givenF1F2

Put $Q = j^{*} P$ . The localization triangle $i_{*} i^{!} P \to P \to R j_{*} Q \to$ has both outer terms $i_{*} i^{!} P$ and $R j_{*} Q$ in the lower perverse aisle: $i^{!}$ preserves it, $i_{*}$ is perverse exact and $R j_{*}$ preserves it by [F1]. Its perverse cohomology exact sequence therefore begins $0 \to^{p} H^{0} (i_{*} i^{!} P) \to^{p} H^{0} P$ , and $^{p} H^{0} (i_{*} i^{!} P) = i_{*} A$ because $i_{*}$ is perverse exact; so the degree-zero adjunction map is a monomorphism $a : i_{*} A ↣ P$ . Dually, the triangle $R j_{!} Q \to P \to i_{*} i^{*} P \to$ has its outer terms in the upper perverse aisle, so its perverse cohomology sequence ends with $^{p} H^{0} P \to^{p} H^{0} (i_{*} i^{*} P) \to 0$ , that is, the adjunction map $b : P ↠ i_{*} B$ is an epimorphism. Full faithfulness of $i_{*}$ identifies the composite $b a : i_{*} A \to i_{*} B$ with $i_{*} α$ for the induced map $α : A \to B$ .

2.1F1step 1.1

A morphism $i_{*} T \to P$ from a supported perverse sheaf corresponds by adjunction to $T \to i^{!} P$ . Since $i^{!} P \in^{p} D^{\geq 0}$ , its truncation triangle is $A \to i^{!} P \to C \to A [1]$ , with $C =^{p} τ_{\geq 1} i^{!} P$ . Orthogonality gives $Hom (T, C) = 0 = Hom (T, C [- 1])$ , because $C \in^{p} D^{\geq 1}$ and $C [- 1] \in^{p} D^{\geq 2}$ . The Hom exact sequence therefore identifies $Hom (T, A)$ with $Hom (T, i^{!} P)$ , so every supported morphism into $P$ factors uniquely through $a$ . Thus $i_{*} A$ is the largest supported subobject. Dually, the truncation of $i^{*} P \in^{p} D^{\leq 0}$ identifies $Hom (B, T)$ with $Hom (i^{*} P, T)$ , so every supported quotient factors uniquely through $b$ , and $i_{*} B$ is the largest supported quotient.

3.1F1step 1.1step 2.1algebra

Suppose $α$ is invertible, and define $r = (i_{*} α)^{- 1} b : P \to i_{*} A$ . Then $r a = (i_{*} α)^{- 1} b a = 1_{i_{*} A}$ . As $a$ is a monomorphism and $r$ a retraction in the abelian heart, $P = i_{*} A \oplus C$ with $C = \ker r$ , and $j^{*} C = Q$ because $j^{*}$ is exact, $j^{*} i_{*} A = 0$ and $j^{*} P = Q$ . A subobject $S \subseteq C$ supported on $Z$ is also a supported subobject of $P$ , so by step 2.1 its inclusion into $P$ factors through $a$ ; applying $r$ makes that factorization zero, while $S \subseteq \ker r$ , so $S = 0$ . A supported quotient $C ↠ i_{*} T$ composed with the projection $P ↠ C$ is a morphism $P \to i_{*} T$ ; by step 2.1 it factors as $h b$ with $h : i_{*} B \to i_{*} T$ , and it kills $i_{*} A$ , so $0 = h b a = h i_{*} α$ and $h = 0$ ; since $P ↠ C$ is an epimorphism, the quotient map $C \to i_{*} T$ is zero. Thus $C$ has no nonzero supported subobject and no nonzero supported quotient.

4.1F1F2step 1.1step 3.1

Let $C$ be a perverse extension of $Q$ with no nonzero supported subobject or quotient. The counit of $R j_{!} ⊣ j^{*}$ and the unit of $j^{*} ⊣ R j_{*}$ induce $f :^{p} H^{0} R j_{!} Q \to C$ and $g : C \to^{p} H^{0} R j_{*} Q$ . Their restrictions to $U$ are identities, so the cokernel of $f$ and the kernel of $g$ are supported on $Z$ and must vanish. Thus $f$ is epic and $g$ is monic. Their composite is the natural comparison: by the heart adjunction of [F1], a morphism $^{p} H^{0} R j_{!} Q \to^{p} H^{0} R j_{*} Q$ is determined by its restriction $Q \to Q$ , which here is the identity. Consequently $C ≃ im (g f) = j_{! *} Q$ . By [F1], this intermediate extension itself has neither supported subobjects nor supported quotients.

5.1F1step 2.1step 4.1∎

Suppose $P ≃ i_{*} T \oplus j_{! *} Q$ for a perverse $T$ on $Z$ . By step 4.1, $j_{! *} Q$ has no nonzero subobject or quotient supported on $Z$ , so $^{p} H^{0} i^{!} j_{! *} Q = 0 =^{p} H^{0} i^{*} j_{! *} Q$ ; and $i^{!} i_{*} = 1 = i^{*} i_{*}$ exhibits $A = B = T$ on the $i_{*} T$ summand, with the adjunction maps the identities. Additivity of the exact functors $i^{!}$ , $i^{*}$ on the displayed direct sum therefore identifies $A = B = T$ and $α = 1_{T}$ , an isomorphism. Combined with steps 3.1 and 4.1 this proves the equivalence: $α$ is invertible if and only if $P ≃ i_{*} A \oplus j_{! *} j^{*} P$ . For $Z = {y}$ the perverse heart on the point consists of finite-dimensional vector spaces in degree zero by [F1], so $A = H^{0} (i_{y}^{!} P)$ and $B = H^{0} (i_{y}^{*} P)$ are finite vector spaces and the criterion reduces to invertibility of the costalk-to-stalk map. Nothing in the argument asserts invertibility of $α$ for projective direct images; that is the separate intersection-form obligation.

Depends on

Used by

Simultaneous projective Lefschetz, Hodge and perverse induction step Lemma

Dependency tree · two levels

24 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

M. A. A. de Cataldo and L. Migliorini, «The Hodge theory of algebraic maps», arXiv:math/0306030v2 (complete arXiv PDF) (standard reference, not scraped)
A. A. Beilinson, J. Bernstein and P. Deligne, «Faisceaux pervers», Astérisque 100 (1982) (complete Numdam PDF) (standard reference, not scraped)
M. Goresky, «Lecture Notes on Sheaves and Perverse Sheaves» (version of June 26, 2026) (standard reference, not scraped)