Derived Picard Groups Overview

Updated 14 October 2025

Derived Picard groups are invariants that capture derived autoequivalences in dg and triangulated categories, generalizing classical Picard groups.
They are computed via two-sided tilting complexes, braid group actions, and explicit semidirect product decompositions in examples like K3 surfaces and Brauer algebras.
Derived Picard groups connect algebraic symmetries to geometric moduli, stability conditions, and mirror symmetry, impacting both homotopical and motivic contexts.

A derived Picard group is an invariant associated to a triangulated or differential graded (dg) category, encodes its derived autoequivalences, and generalizes classical Picard groups of algebras or schemes to the context of derived categories. For a variety, algebra, stack, or dg algebra, the derived Picard group governs symmetries of its bounded derived category, often making explicit the relationships between classical invariants, categorical symmetries, stability conditions, and homological structures.

1. Formal Definition and Conceptual Framework

For a k-algebra A or a scheme X, the derived Picard group, frequently denoted DPic(A) or TrPic(A), is the group of standard/tensor triangulated autoequivalences of the bounded derived category Dᵇ(A) (or Dᵇ(Coh(X))) modulo natural isomorphism. In the dg or A∞-enhanced setting, it can more generally be viewed as the set of isomorphism classes of invertible objects ("two-sided tilting complexes") in the appropriate derived category of A-A^op bimodules. When A is commutative, the classical Picard group Pic(A)—the group of isomorphism classes of invertible A-modules—naturally embeds as a subgroup.

Derived Picard groups are stable under derived equivalence, making them homological invariants that capture the algebraic and categorical symmetries not seen on the level of abelian categories or module categories. In particular, for an algebra A,

$\mathrm{DPic}(A) := \{ [T] \mid T \text{ is a two-sided tilting complex in } D^b(A \otimes_k A^{op}) \},$

with group law given by derived tensor product.

For categories with enough enhancements (for instance, dg or A∞-categories), the notion extends to the group of quasi-equivalence classes of derived autoequivalences, often identified with invertible objects in the symmetric monoidal (triangulated) category of bimodules.

2. Structure and Generators in Key Examples

(a) K3 Surfaces of Picard Rank 1

For a K3 surface X of Picard rank 1, all exact autoequivalences of Dᵇ(Coh(X)) are of Fourier–Mukai type, and the derived Picard group fits into an exact sequence (see (Bayer et al., 2013)):

$1 \to (\text{even shifts} \times F) \to \mathrm{Aut}\ D(X) \xrightarrow{\pi} \mathrm{Aut}^+ H^*(X,\mathbb{Z}) \to 1,$

where F is the free group on the squares of spherical twist functors associated with μ-stable spherical vector bundles.

The "standard" symmetries (shifts, tensoring by line bundles, automorphisms of X) do not suffice: the squares of spherical twists, which act trivially on cohomology, generate the "exotic" derived symmetries unique to K3 type.

(b) Selfinjective Nakayama, Brauer star, and preprojective algebras

The derived Picard group for symmetric and selfinjective algebras, such as Brauer star algebras or preprojective algebras of Dynkin type, is generated by:

The shift functor,
The classical Picard group (automorphisms/bimodules),
Fundamental autoequivalences given by explicit tilting complexes (often associated with combinatorial operations, e.g., braid relations or spherical twists).

Example: For a Brauer star algebra with type (n, t>1), the derived Picard group is generated by the shift, Pic(A), and the family of equivalences {H_i}, which satisfy the relations of the affine braid group B_{Ã_{n-1}} (Zvonareva, 2014). In the multiplicity-free case, additional generators {Q_i} are required.

Example: For preprojective algebras of Dynkin type A or D, the group of two-sided tilting complexes is isomorphic to the braid group of the corresponding folded graph, and every derived autoequivalence decomposes into the product of an outer automorphism and a word in the fundamental tilting complexes (Mizuno, 2017).

(c) Affine Azumaya algebras

For an Azumaya algebra A over X = Spec(R), the derived Picard group is given by

$\mathrm{DPic}(A) \cong (T(X, \mathbb{Z}) \times \mathrm{Pic}(X)) \rtimes_\alpha \mathrm{Aut}(X)[A],$

where T(X, ℤ) is the group of locally constant integer-valued functions, Pic(X) is the classical Picard group, and Aut(X)[A] is the stabilizer of the Brauer class [A] in Aut(X); α is a 2-cocycle encoding composition twisting (Negron, 2016).

(d) Homologically Smooth Koszul DG Algebras

For a homologically smooth Koszul connected cochain dg algebra A, there is an isomorphism

$\mathrm{DPic}(A) \cong \mathbb{Z} \times \mathrm{Out}_k(E),$

where E = H(R\mathrm{Hom}_A(k,k)) is the Ext-algebra, typically a finite-dimensional local algebra (Mao et al., 2018).

3. Methods of Computation and Structural Theorems

Several techniques are used to compute or decompose derived Picard groups:

Braid Group Actions and Tilting Mutations: Many derived Picard groups of finite-dimensional algebras (Brauer tree/star, preprojective, Nakayama, etc.) are generated by tilting functors whose compositions encode braid group relations (Zvonareva, 2014, Mizuno, 2017, Volkov et al., 2015, Nordskova, 2021).
Orbit/Smash Product Techniques: For selfinjective algebras, passing to a smash product with a cyclic group or embedding into a larger category can facilitate the lifting of known group actions to derived autoequivalences (Volkov et al., 2015).
Explicit Formulas and Semidirect Product Decompositions: The derived Picard group sometimes takes the form of an explicit (semi)direct product of familiar groups (e.g., shifts, automorphism groups, free or braid groups corresponding to mutations/twists).
Spectral Sequences in Stable Homotopy: In ∞-categorical or spectral algebraic contexts (e.g., topological modular forms, higher real K-theories), descent spectral sequences and homotopy fixed point spectral sequences are used to compute derived Picard groups (Mathew et al., 2014, Heard et al., 2015).

4. Relationship to Stability Conditions and Moduli

On K3 surfaces of Picard rank 1, the derived Picard group carries geometric information about the moduli spaces of Bridgeland stability conditions. Wall-crossing phenomena, deformation retractions, and contractibility results for distinguished components of the stability space produce short exact sequences relating the derived Picard group to π₁ of certain moduli spaces and to the subgroup of cohomological autoequivalences (Bayer et al., 2013). This connects derived autoequivalences directly to moduli and period domains, with implications for mirror symmetry.

Similar phenomena are observed in derived categories of motives or stacks, where the derived Picard group structure is reflected in weight structures and the hearts of t-structures, linking categorical symmetries to motivic and cohomological invariants (Bondarko et al., 2015).

5. Connections, Consequences, and Broader Applications

Mirror Symmetry: Derived Picard groups of K3 surfaces and Fukaya categories play pivotal roles in mirror symmetry, where the group of derived autoequivalences matches (up to appropriate extensions) the fundamental group of the mirror moduli space; this correspondence is explicitly verified for K3 surfaces of Picard rank 1 (Bayer et al., 2013).
Derived Invariants and Obstruction Theory: Hochschild cohomology, particularly the positive-weight part of HH¹, controls infinitesimal symmetries; integrating these via pre-Lie/brace algebra structures yields explicit subgroups of the derived Picard group (e.g., via exponential/Baker–Campbell–Hausdorff maps), identifying the connected component of the identity in DPic as an “algebraic group” built from HH¹₊ (Opper, 23 May 2024).
Uniqueness of Lifts and Homotopical Rigidity: Nontriviality of HH¹₊ in A∞-categories provides a necessary condition for nonuniqueness of A∞-enhanced functor lifts, implying that DPic controls not only obvious geometric symmetries but also higher homotopical and infinitesimal obstructions (Opper, 23 May 2024).
Toric, Stacky, and Tropical Geometries: Derived Picard groups of toric varieties, stacky curves, and tropicalizations often reduce, in leading order, to the classical Picard group of the underlying combinatorial or coarse space, but with crucial extensions controlled by the action of stabilizer groups, gerbe classes, or log/tropical modifications (Lopez, 2023, Jun et al., 2017).
Blocks and Modular Representation Theory: Explicit computations for blocks with abelian defect groups confirm that their Picard and derived Picard groups are finite, and suggest the absence of "hidden" Morita or derived autoequivalences fixing the center (Picent trivial), adding structure to the classification and derived equivalence of such blocks (Eaton et al., 2018).

6. Impact in Algebraic, Motivic, and Homotopical Settings

Derived Picard groups form a unifying invariant in algebra, algebraic geometry, and stable homotopy theory.

In motivic and bootstrap categories, when a bounded weight structure with semisimple local heart exists, the derived Picard group admits a canonical splitting as Picard group of the heart × ℤ (shift), and derived autoequivalences can often be classified in terms of purity with respect to the weight (Bondarko et al., 2015).
In E∞-ring spectra, Picard spectra and descent methods relate classical Picard theory, homotopy theory, and spectral algebra: for TMF, Pic(TM F) ≅ ℤ/576 is explained by the periodicity generator, while for Tmf, Pic(Tmf) ≅ ℤ ⊕ ℤ/24 detects an exotic invertible module not a suspension (Mathew et al., 2014).
In combinatorial contexts, Picard and critical groups/Jacobians of graphs (possibly directed, possibly tropical) are computed as cokernels of Laplacians, revealing exotic torsion subgroups depending on orientation and graph operations, with analogs in chip-firing and sandpile dynamics (Jun et al., 2023).

7. Open Problems and Directions

Key open problems and ongoing areas of research include:

Full classification of derived Picard groups beyond the cases above, especially for general (possibly wild) finite-dimensional algebras, singular varieties, and higher stacks.
Understanding the relations between derived Picard groups, Hochschild cohomology, and deformation theory in positive characteristic or arithmetic settings.
Investigating the interaction of derived Picard groups with wall-crossing phenomena, birational geometry, and birational invariants in mirror symmetry and homological algebra.
Extending descent and spectral techniques to non-Noetherian, stacky, or logarithmic settings, and connecting to logarithmic/tropical Picard groups.
Clarifying the structure of the identity component of DPic for broad classes and the role of higher Hochschild or motivic cohomology, including implications for uniqueness of enhancements and derived invariance of various invariants.

Derived Picard groups serve as a fundamental bridge joining concrete algebraic, geometric, and homotopical symmetries, and their explicit computation and structural analysis continue to deepen the understanding of symmetries, moduli, and derived invariants across mathematics.