Ind-Coherent Sheaves Overview

• Ind-coherent sheaves are defined as the ind-completion of the derived category of coherent sheaves, offering a robust framework for handling derived and singular spaces.
• They support a full six-functor formalism that aligns with quasi-coherent sheaves while preserving a compatible t-structure, duality, and base-change properties.
• This formalism underpins advancements in categorical representation theory and geometric Langlands duality, especially for ind-schemes, stacks, and infinite-dimensional spaces.

Ind-coherent sheaves are a categorical enhancement of coherent sheaves on singular, derived, or infinite-type geometric objects—schemes, stacks, ind-schemes, and ind-stacks—designed to achieve robust functoriality, duality, and compatibilities with D-module and categorical representation theories. The foundational construction is the ind-completion of the bounded derived category of coherent sheaves, leading to a cocomplete $\infty$-category equipped with a compatible $t$-structure and admitting a full six-functor formalism. The passage from quasi-coherent to ind-coherent sheaves—often essential in derived and categorical contexts—underlies recent advances in higher representation theory, categorical (and spectral) Langlands duality, and geometric representation theory of infinite-dimensional and singular spaces.

1. Definition and Basic Formalism

For a quasi-compact, locally Noetherian derived (DG) scheme or stack $X$ over a field $k$ of characteristic zero, the DG category of ind-coherent sheaves, denoted $\operatorname{IndCoh}(X)$, is defined as the ind-completion of the bounded derived category of coherent sheaves $\operatorname{Coh}(X)$: $$\operatorname{IndCoh}(X) := \operatorname{Ind}(\operatorname{Coh}(X))$$ This cocomplete (presentable, stable) $\infty$-category is generated under filtered colimits by the subcategory $\operatorname{Coh}(X)$. For generalized spaces (e.g., ind-schemes, ind-stacks), one sets $$\operatorname{IndCoh}(X) \simeq \varinjlim \operatorname{IndCoh}(X_i)$$ where $t$0 is a suitable colimit of geometric objects via closed immersions, as in the ind-geometric framework (Cautis et al., 2023, Gaitsgory et al., 2011).

The canonical $t$1-structure on $t$2 is the unique one characterized by:

• $t$3-exactness of the inclusion $t$4,
• Colimit preservation of truncation functors,
• For stacks: left (anti)completion of the $t$5-structure on $t$6 in the sense of Lurie (Cautis et al., 2023).

A continuous, $t$7-exact comparison functor

$t$8

identifies coherent objects and induces an equivalence on bounded-below subcategories in reasonable situations. $t$9 is (under mild hypotheses) the left completion of $X$0 for the $X$1-structure (Gaitsgory, 2011).

2. Functoriality and Six-Functor Formalism

The category $X$2 admits the full formalism of Grothendieck's six operations, suitably extended to derived and infinite-type contexts (Sun, 30 Apr 2025, Cautis et al., 2023, Gaitsgory et al., 2011):

• *-Pullback $X$3 exists for morphisms of finite Tor-dimension,
• !-Pullback $X$4 is defined for ind-(quasi-)proper morphisms and is right adjoint to $X$5,
• External tensor product, internal Hom, proper and smooth base-change isomorphisms, and projection formulas all hold in the expected forms.

These functors satisfy compatibilities under colimits, admit base-change, intertwine with the $X$6-theoretic operations via $X$7, and extend via Kan extension to ind-geometric and stacky settings (Cautis et al., 2023).

The formalism enables the construction of symmetric monoidal structures and convolution products, especially critical for the representation theory of categories such as the coherent Satake and double affine Hecke categories on infinite-type (ind)-stacks (Cautis et al., 2023).

3. Relations to Quasi-Coherent Sheaves and Duality

While $X$8 and $X$9 share the same compact objects when $k$0 is smooth (perfect complexes), their behavior diverges sharply on singular, derived, or infinite-type spaces:

• For smooth, proper $k$1, $k$2.
• On singular or non-coconnective $k$3, $k$4 is (the left) $k$5-completion of $k$6, and the canonical functor $k$7 is an equivalence only on bounded-below objects (Gaitsgory, 2011, Gaitsgory et al., 2011).
• For formally smooth DG indschemes or ind-geometric stacks, $k$8 can be an equivalence globally (Gaitsgory et al., 2011).

Ind-coherent sheaves support canonical self-duality: there is a symmetric monoidal equivalence (Serre duality)

$k$9

with the dualizing object being $\operatorname{IndCoh}(X)$0, supporting the correct functorial adjunctions and duality statements required for categorical representation theory (Gaitsgory, 2011, Cautis et al., 2023).

4. Singular Support, Hochschild Cohomology, and Operator Actions

For a quasi-smooth (i.e., $\operatorname{IndCoh}(X)$1-Tor amplitude cotangent complex) derived scheme $\operatorname{IndCoh}(X)$2, $\operatorname{IndCoh}(X)$3 admits a singular-support theory—extending the classical Kashiwara-Schapira microlocal theory and Arinkin-Gaitsgory's singular support "crystal" (Beraldo et al., 2024). Objects can be stratified according to their singular support inside the derived space

$\operatorname{IndCoh}(X)$4

with

$\operatorname{IndCoh}(X)$5

for closed conical subsets $\operatorname{IndCoh}(X)$6.

A key categorical interaction is the action of the category of sheared $\operatorname{IndCoh}(X)$7-modules $\operatorname{IndCoh}(X)$8 (D-modules on the shifted cotangent bundle with twisted grading) on $\operatorname{IndCoh}(X)$9. This action commutes (up to coherent homotopy) with that of the Hochschild cochain algebra $\operatorname{Coh}(X)$0, leading to a Morita equivalence

$\operatorname{Coh}(X)$1

and an equivalence of module categories: $\operatorname{Coh}(X)$2 This structure underpins local and global duality phenomena, including extension to quasi-smooth Artin stacks via descent and sheafification on the smooth site (Beraldo et al., 2024).

5. Ind-Coherent Sheaves for (Ind)-Stacks and Stacks with Group Actions

For stacks, and especially in the context of moduli, loop groups, and equivariant infinite-dimensional quotients, the ind-coherent formalism is essential:

• On Artin (or more generally geometric) stacks, $\operatorname{Coh}(X)$3 is defined as the ind-completion of coherent sheaves, with descent under fpqc covers and compatibilities with the stack structure.
• For ind-geometric stacks—colimits of geometric stacks under closed inclusion—the cocomplete $\operatorname{Coh}(X)$4 category is constructed as the filtered colimit $\operatorname{Coh}(X)$5 with all compatibilities preserved (Cautis et al., 2023).
• For quotient stacks $\operatorname{Coh}(X)$6 arising from a group action, $\operatorname{Coh}(X)$7 is identified with the $\operatorname{Coh}(X)$8-invariants in $\operatorname{Coh}(X)$9, typically via a comodule structure and categorical Pontryagin duality (Sun, 30 Apr 2025).

Functoriality encompasses pushforward, pullback, and induction/restriction along stacky correspondences, including compatibilities with the corresponding constructible categories.

6. Coherent-Constructible Correspondence and Equivariant Mirror Symmetry

The coherent–constructible correspondence (CCC) is a striking manifestation of the flexibility of $$\operatorname{IndCoh}(X) := \operatorname{Ind}(\operatorname{Coh}(X))$$0, equating categories of ind-coherent sheaves on (possibly stacky or base-fibered) toric varieties (or stacks) to categories of constructible sheaves with prescribed singular support (FLTZ skeleton) on real tori or their covers: $$\operatorname{IndCoh}(X) := \operatorname{Ind}(\operatorname{Coh}(X))$$1 The correspondence is symmetric monoidal, functorial under toric morphisms and inclusions, and has explicit kernel-theoretic descriptions in terms of tensor and convolution products on both sides (Sun, 30 Apr 2025, Hu et al., 2023).

Variations for principal toric fibrations over a base $$\operatorname{IndCoh}(X) := \operatorname{Ind}(\operatorname{Coh}(X))$$2 generalize the correspondence to families, with the global ind-coherent category equivalent to the global sections of a Kashiwara–Schapira stack twisted by a local system of categories with stalk $$\operatorname{IndCoh}(X) := \operatorname{Ind}(\operatorname{Coh}(X))$$3 (Hu et al., 2023).

7. Applications and Structural Significance

$$\operatorname{IndCoh}(X) := \operatorname{Ind}(\operatorname{Coh}(X))$$4 is central in geometric representation theory, the categorical geometric Langlands correspondence, and the study of categories on singular/derived/infinite-type moduli spaces:

• The spectral side of categorical Langlands requires $$\operatorname{IndCoh}(X) := \operatorname{Ind}(\operatorname{Coh}(X))$$5 to handle non-left-completeness and to accommodate objects with unbounded cohomology ("spectral Eisenstein series"), with duality and parabolic induction functors constructed via $$\operatorname{IndCoh}(X) := \operatorname{Ind}(\operatorname{Coh}(X))$$6 (Gaitsgory, 2011).
• Convolution categories such as the coherent Satake and double affine Hecke categories are realized as monoidal subcategories of ind-coherent sheaves on ind-(geo)metric stacks, benefiting from the stabilization, dualizability, and descent properties of $$\operatorname{IndCoh}(X) := \operatorname{Ind}(\operatorname{Coh}(X))$$7 (Cautis et al., 2023).
• Singular support theory, microlocal and Hochschild structures, and functorial correspondences—both in algebraic and analytic contexts—critically leverage the properties of ind-coherent sheaves (Beraldo et al., 2024).

The development of $$\operatorname{IndCoh}(X) := \operatorname{Ind}(\operatorname{Coh}(X))$$8 unifies and extends tools from coherent and constructible sheaf theory, providing a robust framework for derived and categorical representation theory across algebraic geometry, topology, and mathematical physics.