Faithful perversities

1. Problem Statement and Context

The paper investigates the relationship between the algebraic structure of categories of perverse sheaves and the topological properties of the underlying stratified spaces.

• Background: Perverse sheaves, introduced by Beilinson, Bernstein, and Deligne (BBD), are the heart of a bounded $t$-structure on the constructible derived category $D_c(X)$ of a stratified space $X$. They play a crucial role in singular topology, representation theory, and Hodge theory.
• The Core Question: Under what conditions is the category of perverse sheaves, denoted $p\text{Perv}(X)$, a faithful heart?
  • A heart $\mathcal{H} \subset \mathcal{D}$ is faithful if the realization functor $D^b(\mathcal{H}) \to \mathcal{D}$ is an equivalence.
  • Significance: If $p\text{Perv}(X)$ is faithful, the topological information contained in the derived category $D_c(X)$ is fully accessible via the algebra of the abelian category $p\text{Perv}(X)$ (specifically, as modules over a finite-dimensional algebra). This allows topological invariants to be computed algebraically and vice versa.
• Gap: While faithfulness is known in specific cases (e.g., simplicial complexes, smooth complex varieties with affine stratifications), a general characterization for topologically stratified spaces and a systematic understanding of the global dimension and cohomological computations were lacking.

2. Methodology

The authors employ a synthesis of categorical algebra and stratified topology:

1. Categorical Framework:

   • They utilize the theory of highest weight categories and exceptional collections.
   • They introduce the notion of serially faithful hearts, defined via an admissible thick filtration where each step is a faithful heart.
   • They leverage the concept of glued hearts (recollements) and dual pairs of exceptional collections (standard and costandard objects).
   • They rely on the equivalence between the constructible derived category and the derived category of modules over a finite-dimensional algebra when the heart is faithful.

2. Topological Tools:

   • They analyze topologically stratified spaces (in the sense of Goresky-MacPherson) with finitely many strata.
   • They utilize pairwise links ($L_{S<T}$) between strata to characterize local topological obstructions.
   • They apply hypercohomology vanishing theorems and induction on the depth of the stratification.

3. Synthesis:

   • The authors translate topological conditions (vanishing of cohomology on links) into categorical conditions (exactness of functors, exceptional collections) to characterize faithfulness.
   • They use projective resolutions of the constant sheaf to compute hypercohomology, bridging the gap between algebraic resolutions and topological invariants.

3. Key Contributions and Results

A. Characterization of Faithful Highest Weight Hearts (Theorem A)

The paper establishes a fundamental equivalence for algebraic triangulated categories $\mathcal{D}$. For a heart $\mathcal{H} \subset \mathcal{D}$, the following are equivalent:

1. $\mathcal{H}$ is faithful and highest weight.
2. The sequence of standard objects $(\Delta_0, \dots, \Delta_n)$ forms a full exceptional collection, and the sequence of costandard objects $(\nabla_n, \dots, \nabla_0)$ is its left dual pair.
3. $\mathcal{H}$ contains a dual pair of full exceptional collections.
4. $\mathcal{H}$ is serially faithful and glued along an exceptional collection.

Significance: This provides a purely algebraic criterion for faithfulness in the context of perverse sheaves, linking it directly to the existence of specific exceptional collections.

B. Topological Characterization of Faithful Perverse Sheaves (Theorem B)

For a topologically stratified space $X$ with finitely many contractible strata, the category $p\text{Perv}(X)$ is a faithful highest weight heart if and only if:

1. The functors $i_{S*}$ and $i_{S!}$ (inclusion of a stratum) are $t$-exact.
2. Cohomological Vanishing: For any pair of distinct strata $S \subset T$, the pairwise link $L_{S<T}$ satisfies:
   $$H^{>p(S)-p(T)}(L_{S<T}) = 0 \quad \text{and} \quad H^{<p(S)-p(T)-1}_c(L_{S<T}) = 0$$
   where $p$ is the perversity function.

Significance: This gives a concrete topological test for faithfulness. It generalizes previous results (e.g., by Polischuk and Beilinson-Ginzburg-Soergel) and clarifies that faithfulness depends on the vanishing of specific cohomology groups of the links between strata.

C. Bounds on Global Dimension (Theorem C)

The authors prove bounds on the global dimension ($\text{gldim}$) of the category of perverse sheaves:

• General Bound: For any $E, F \in p\text{Perv}(X)$, $\text{Ext}^k_{D_c(X)}(E, F) = 0$ if $k < 0$ or $k > \dim(X)$.
• Faithful Case: If $p\text{Perv}(X)$ is faithful, then $\text{gldim}(p\text{Perv}(X)) \leq \dim(X)$.
• Faithful Highest Weight Case: If $p\text{Perv}(X)$ is faithful and highest weight, then $\text{gldim}(p\text{Perv}(X)) \leq 2 \cdot \text{depth}(X)$.

Significance: This resolves the question of the finiteness of the global dimension for faithful perverse categories and provides explicit bounds related to the geometry of the space.

D. Computation of Hypercohomology via Projective Resolutions (Theorem D)

Assuming $p\text{Perv}(X)$ is faithful and strata have finite fundamental groups:

1. Hypercohomology from Resolutions: The hypercohomology of a perverse sheaf $E$ can be computed as the cohomology of the complex $\text{Hom}(P_\bullet, E)$, where $P_\bullet$ is a projective resolution of the constant sheaf $k_X$.
2. Multiplicities as Intersection Betti Numbers: If the constant sheaf (shifted) is perverse, the multiplicities of indecomposable projectives in its minimal projective resolution correspond to intersection Betti numbers (intersection cohomology groups).
3. Simplicial Case: For a simplicial complex $X$, the constant sheaf admits a finite projective resolution indexed by the perverse dimension of simplices. This recovers the theory of cellular perverse sheaves (Goresky), providing a combinatorial method to compute hypercohomology.

4. Significance and Impact

1. Unification of Algebra and Topology: The paper rigorously demonstrates that when a perverse heart is faithful, the complex topological data of a stratified space is encoded entirely in the representation theory of a finite-dimensional algebra.
2. Computational Power: The results provide a practical algorithm for computing hypercohomology and intersection cohomology using projective resolutions, which are often easier to handle algebraically than topological complexes.
3. Generalization: The characterization of faithfulness via pairwise link cohomology extends known results from specific geometric settings (like complex algebraic varieties) to general topologically stratified spaces.
4. Counterexamples and Nuance: The authors provide examples showing that not all faithful glued hearts are highest weight, and not all highest weight hearts are faithful, clarifying the precise boundaries of these concepts.
5. Foundational for Future Work: The identification of faithful perverse hearts with serially faithful glued hearts opens the door to applying techniques from tilting theory and silting objects (via the König-Yang correspondence) to the study of perverse sheaves on stratified spaces.

In summary, Cipriani and Woolf provide a comprehensive structural theory for faithful perverse sheaves, establishing precise topological criteria for their existence, bounding their homological complexity, and offering new algebraic methods for computing topological invariants.