Moduli problems for operadic algebras joint with D. Calaque, R. Campos Joost Nuiten Universit´ e de Montpellier Operad Pop-Up August 11, 2020
Goal Throughout: fix k field of characteristic zero. Classical principle in deformation theory Every deformation problem over k is controlled by a dg-Lie algebra g . Question Suppose that g admits additional algebraic structure. How can this additional structure be understood in terms of deformation problems?
� � � Classical example: deforming complex varieties X - proper smooth variety over C . Study infinitesimal deformations of X along an Artin local C -algebra A : ˜ X X ⌟ Spec ( C ) � Spec ( A )
� � � Classical example: deforming complex varieties X - proper smooth variety over C . Study infinitesimal deformations of X along an Artin local C -algebra A : ˜ X X ⌟ Spec ( C ) � Spec ( A ) Kodaira–Spencer: infinitesimal deformations controlled by tangent bundle T X . H 0 ( X , T X ) ↔ first order automorphisms of X .
� � � Classical example: deforming complex varieties X - proper smooth variety over C . Study infinitesimal deformations of X along an Artin local C -algebra A : ˜ X X ⌟ Spec ( C ) � Spec ( A ) Kodaira–Spencer: infinitesimal deformations controlled by tangent bundle T X . H 0 ( X , T X ) ↔ first order automorphisms of X . H 1 ( X , T X ) ↔ deformations of X over C [ ǫ ]/ ǫ 2 .
� � � � � � Classical example: deforming complex varieties X - proper smooth variety over C . Study infinitesimal deformations of X along an Artin local C -algebra A : ˜ X X ⌟ Spec ( C ) � Spec ( A ) Kodaira–Spencer: infinitesimal deformations controlled by tangent bundle T X . H 0 ( X , T X ) ↔ first order automorphisms of X . H 1 ( X , T X ) ↔ deformations of X over C [ ǫ ]/ ǫ 2 . H 2 ( X , T X ) controls obstructions to extending deformations: ˜ ˜ X n X n + 1 ⌟ ⇐ ⇒ ob ( X n ) = 0 ∈ H 2 ( X , T X ) . ∃ Spec ( k [ ǫ ]/ ǫ n ) � Spec ( k [ ǫ ]/ ǫ n + 1 )
Example: deforming complex varieties H ∗ ( X , T X ) computed by the Dolbeault complex Ω 0 , ∗ ( T X ) = [ Ω 0 , 0 ( X C , T 1 , 0 X ) � → Ω 0 , 1 ( X C , T 1 , 0 X ) � → Ω 0 , 2 ( X C , T X ) → ... ] . ∂ ∂ This is a dg-Lie algebra, from commutator of vector fields and multiplication of forms.
Example: deforming complex varieties H ∗ ( X , T X ) computed by the Dolbeault complex Ω 0 , ∗ ( T X ) = [ Ω 0 , 0 ( X C , T 1 , 0 X ) � → Ω 0 , 1 ( X C , T 1 , 0 X ) � → Ω 0 , 2 ( X C , T X ) → ... ] . ∂ ∂ This is a dg-Lie algebra, from commutator of vector fields and multiplication of forms. Idea. Ω 0 , ∗ ( T X ) controls deformations of X via the Maurer–Cartan equation . More precisely, for Artin algebra A with maximal ideal m A X over Spec ( A )} ≃ { x ∈ m A ⊗ Ω 0 , 1 ( T X ) { deformations ˜ } dx + 1 2 [ x , x ] = 0 automorphisms exp ( m A ⊗ Ω 0 , 0 ( T X ))
Example: deforming complex varieties H ∗ ( X , T X ) computed by the Dolbeault complex Ω 0 , ∗ ( T X ) = [ Ω 0 , 0 ( X C , T 1 , 0 X ) � → Ω 0 , 1 ( X C , T 1 , 0 X ) � → Ω 0 , 2 ( X C , T X ) → ... ] . ∂ ∂ This is a dg-Lie algebra, from commutator of vector fields and multiplication of forms. Idea. Ω 0 , ∗ ( T X ) controls deformations of X via the Maurer–Cartan equation . Higher cohomology groups: control derived deformations of X over dg -Artin algebra A . Definition An augmented commutative dg-algebra A over k is called Artin if: H ∗ ( A ) finite-dimensional and in nonpositive degrees. H 0 ( A ) → k has nilpotent kernel.
Example: deforming complex varieties H ∗ ( X , T X ) computed by the Dolbeault complex Ω 0 , ∗ ( T X ) = [ Ω 0 , 0 ( X C , T 1 , 0 X ) � → Ω 0 , 1 ( X C , T 1 , 0 X ) � → Ω 0 , 2 ( X C , T X ) → ... ] . ∂ ∂ This is a dg-Lie algebra, from commutator of vector fields and multiplication of forms. Idea. Ω 0 , ∗ ( T X ) controls deformations of X via the Maurer–Cartan equation . Higher cohomology groups: control derived deformations of X over dg -Artin algebra A . Definition An augmented commutative dg-algebra A over k is called Artin if: H ∗ ( A ) finite-dimensional and in nonpositive degrees. H 0 ( A ) → k has nilpotent kernel. Negative cohomology groups of a dg-Lie algebra: control homotopies between automorphisms.
Formal moduli problems Definition A formal moduli problem is a functor of ∞ -categories F ∶ Art → Spaces from Artin commutative dg-algebras to spaces, such that: F ( k ) ≃ ∗ . Schlessinger condition: for A 1 ↠ A 0 ↞ A 2 surjective on H 0 : F ( A 1 × h A 0 A 2 ) � F ( A 1 ) × h F ( A 0 ) F ( A 2 ) ∼ ‘gluing along Spec ( A 1 ) ∪ Spec ( A 0 ) Spec ( A 2 ) ’
Formal moduli problems Definition A formal moduli problem is a functor of ∞ -categories F ∶ Art → Spaces from Artin commutative dg-algebras to spaces, such that: F ( k ) ≃ ∗ . Schlessinger condition: for A 1 ↠ A 0 ↞ A 2 surjective on H 0 : F ( A 1 × h A 0 A 2 ) � F ( A 1 ) × h F ( A 0 ) F ( A 2 ) ∼ Theorem (Pridham, Lurie) There is an equivalence of ∞ -categories between formal moduli problems and dg-Lie algebras ≃ � Alg Lie . FMP
Example: deforming modules B - associative algebra. V - (left) B -module (concentrated in cohomological degrees ≤ 0). Deformations of V form a formal moduli problem Def V ∶ Art → Spaces Mod B { V } = { A ⊗ B -modules V A Def V ( A ) = Mod A ⊗ B × h → V } . with k ⊗ A V A � ∼ This is classified by RHom B ( V , V ) , endowed with the commutator bracket.
Example: deforming modules B - associative algebra. V - (left) B -module (concentrated in cohomological degrees ≤ 0). Deformations of V form a formal moduli problem Def V ∶ Art → Spaces Mod B { V } = { A ⊗ B -modules V A Def V ( A ) = Mod A ⊗ B × h → V } . with k ⊗ A V A � ∼ This is classified by RHom B ( V , V ) , endowed with the commutator bracket. Explicit model: bar construction [ Hom k ( V , V ) → Hom ( B ⊗ V , V ) → Hom ( B ⊗ B ⊗ V , V ) → ... ] with commutator bracket B B V B B V ... ... [ α,β ] = − ● β ● α B B B B ... ... ● α ● β
Example: deforming associative algebras B - associative algebra over k . Deformations of B form a formal moduli problem Def B ( A ) = Alg A × h Alg k { B } = { A -linear associative algebras B A } . with k ⊗ A B A � → B ∼ This is classified by the (reduced) Hochschild cochains HH ( B , B ) = [ Hom ( B , B ) → Hom ( B ⊗ 2 , B ) → ... ] ,
Example: deforming associative algebras B - associative algebra over k . Deformations of B form a formal moduli problem Def B ( A ) = Alg A × h Alg k { B } = { A -linear associative algebras B A } . with k ⊗ A B A � → B ∼ This is classified by the (reduced) Hochschild cochains HH ( B , B ) = [ Hom ( B , B ) → Hom ( B ⊗ 2 , B ) → ... ] , with Lie structure given by the Gerstenhaber bracket: ● β [ α,β ] = α ○ β − β ○ α, α ○ β = ∑ ( ± ) i ... ... i ● α Differential: d = [ ● µ B , − ]
� � Adding algebraic structure Question Let Lie → P be a map of k -linear (dg-) operads. If g arises from a P -algebra, what structure does the corresponding formal moduli problem have? � ? Alg P forget � FMP . Alg Lie ∼
Example 0: linear deformation problems ǫ ∶ Lie → k the augmentation. ǫ ∗ ∶ Mod k → Alg Lie takes the trivial Lie algebra.
� � � � Example 0: linear deformation problems ǫ ∶ Lie → k the augmentation. ǫ ∗ ∶ Mod k → Alg Lie takes the trivial Lie algebra. Proposition ⇐ ⇒ Lie algebra arises as corresponding formal moduli problem arises as g ≃ triv ( V ) Art Spaces ≃ A ↦ m A ‘linear FMP’ Perf ≤ 0 ( ⇔ reduced excisive) k
� � � � � � Example 0: linear deformation problems ǫ ∶ Lie → k the augmentation. ǫ ∗ ∶ Mod k → Alg Lie takes the trivial Lie algebra. Proposition ⇐ ⇒ Lie algebra arises as corresponding formal moduli problem arises as g ≃ triv ( V ) Art Spaces ≃ A ↦ m A ‘linear FMP’ Perf ≤ 0 ( ⇔ reduced excisive) k More precisely, there is a commuting square ∼ � FMP lin Mod k triv restrict � FMP . Alg Lie ∼
Example 1: deforming modules V a B -module. (1) The Lie algebra RHom B ( V , V ) arises from an associative algebra . (2) The corresponding formal moduli problem Def V ( A ) = { A ⊗ B -modules V A → V } with k ⊗ A V A � ∼ arises from a functor defined on Artin associative algebras A .
� � Example 1: deforming modules V a B -module. (1) The Lie algebra RHom B ( V , V ) arises from an associative algebra . (2) The corresponding formal moduli problem Def V ( A ) = { A ⊗ B -modules V A → V } with k ⊗ A V A � ∼ arises from a functor defined on Artin associative algebras A . In fact: the associative extensions (1) and (2) correspond to each other via ∼ � FMP As Alg As forget restrict � FMP Com . Alg Lie ∼
Example 2: deforming the trivial algebra Suppose ( B ,µ = 0 ) is a trivial associative algebra. Then HH ( B , B ) = [ Hom ( B , B ) � → Hom ( B ⊗ 2 , B ) � → ... ] 0 0 together with the operation ● β α ○ β = ∑ ( ± ) ... ... ● α form a pre-Lie algebra : α ○ ( β ○ γ ) − ( α ○ β ) ○ γ = α ○ ( γ ○ β ) − ( α ○ γ ) ○ β. Question: interpretation in terms of formal moduli problems?
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