P ROOF OF T HEOREM 1 ( SKETCH ) D EFINITION A bidga B is a dA - PowerPoint PPT Presentation

DG- ALGEBRAS AND DERIVED A ∞ - ALGEBRAS Steffen Sagave Universitetet i Oslo March 2008 http://folk.uio.no/sagave 1 / 16

I NITIAL Q UESTION Let A be a differential graded algebra over a commutative ring k , possibly unbounded and with homological grading. Its homology H ∗ ( A ) is a graded k -algebra. Q UESTION Is there some additional structure on H ∗ ( A ) which allows us to recover the quasi-isomorphism type of A from H ∗ ( A ) ? If k is a field (or, more generally, H ∗ ( A ) is k -projective), a minimal A ∞ -algebra structure on H ∗ ( A ) provides an answer. 2 / 16

A ∞ - ALGEBRAS D EFINITION (S TASHEFF ) An A ∞ -algebra is a graded k -module A together with a unit element 1 A ∈ A 0 and k -linear maps m j : A ⊗ j → A [ 2 − j ] for j ≥ 1, satisfying � ( − 1 ) rs + t m r + 1 + t ( 1 ⊗ r ⊗ m s ⊗ 1 ⊗ t ) = 0 r + s + t = n for n ≥ 1 (and a unit condition). • A map f : A → B of A ∞ -algebras is a family of k -linear maps f j : A ⊗ j → B [ 1 − j ] satisfying appropriate relations. • An A ∞ -algebra is minimal if m 1 = 0. • dgas are A ∞ -algebras with m 1 the differential, m 2 the multiplication, and all other m j = 0. 3 / 16

M INIMAL MODELS T HEOREM (K ADEISHVILI ) Let A be a dga with H ∗ ( A ) k-projective. There exist a minimal A ∞ -algebra structure on H ∗ ( A ) and a quasi-isomorphism f : H ∗ ( A ) → A, where the m 2 of H ∗ ( A ) is the algebra multiplication. One can recover the quasi-isomorphism type of A from the minimal A ∞ -algebra H ∗ ( A ) . In other words: The higher multiplications m j , j ≥ 3, on the graded algebra ( H ∗ ( A ) , m 2 ) provide the data we are asking for. 4 / 16

R ESOLUTIONS The statement of Kadeishvili’s theorem does in general not hold if H ∗ ( A ) is not k -projective. W HAT TO DO INSTEAD ? Look for higher multiplications on a resolution of H ∗ ( A ) ! We consider ( N , Z ) -bigraded k -modules with the N -grading the ‘horizontal’ direction and the Z -grading the ‘vertical’ direction. For E and F bigraded k -modules, • E [ st ] ij = E i − s , t − j � • ( E ⊗ F ) uv = E ij ⊗ k E pq i + p = u j + q = v 5 / 16

D EFINITION OF dA ∞ - ALGEBRAS D EFINITION A derived A ∞ -algebra (or dA ∞ -algebra ) is a ( N , Z ) -bigraded k -module E with a unit element 1 E ∈ E 0 , 0 and structure maps m ij : E ⊗ j → E [ i , 2 − ( i + j )] with i ≥ 0 , j ≥ 1 satisfying � ( − 1 ) rq + t + pj m ij ( 1 ⊗ r ⊗ m pq ⊗ 1 ⊗ t ) = 0 i + p = u r + q + t = v r + 1 + t = j for all u ≥ 0 and v ≥ 1 (and a unit condition). A dga may be viewed as a dA ∞ -algebra concentrated in horizontal degree 0, with m 01 the differential, m 02 the multiplication, and all other m ij = 0. 6 / 16

S TRUCTURE MAPS & F ORMULAS FOR dA ∞ - ALGEBRAS A dA ∞ -algebra has structure maps starting with: m 01 : E → E [ 0 , 1 ] m 11 : E → E [ 1 , 0 ] m 21 : E → E [ 2 , − 1 ] m 02 : E ⊗ 2 → E m 12 : E ⊗ 2 → E [ 1 , − 1 ] m 03 : E ⊗ 3 → E [ 0 , − 1 ] The first six relations are: m 01 m 01 = 0 m 01 m 02 = m 02 ( 1 ⊗ m 01 ) + m 02 ( m 01 ⊗ 1 ) m 01 m 11 = m 11 m 01 m 02 ( m 02 ⊗ 1 ) = m 01 m 03 + m 02 ( 1 ⊗ m 02 ) + m 03 ( m 01 ⊗ 1 ⊗ 2 ) + m 03 ( 1 ⊗ m 01 ⊗ 1 ) + m 03 ( 1 ⊗ 2 ⊗ m 01 ) m 11 m 02 = m 01 m 12 + m 12 ( 1 ⊗ m 01 ) + m 12 ( m 01 ⊗ 1 ) + m 02 ( 1 ⊗ m 11 ) + m 02 ( m 11 ⊗ 1 ) m 11 m 11 = m 01 m 21 + m 21 m 01 7 / 16

S OME TERMINOLOGY FOR dA ∞ - ALGEBRAS • A map of dA ∞ -algebras is a family of k -module maps f ij : E ⊗ j → E [ i , 1 − ( i + j )] satisfying appropriate relations. • A dA ∞ -algebra is minimal if m 01 = 0. • A map f : E → F is an E 2 -equivalence if it induces isomorphisms in the iterated homology with respect to m 01 and m 11 . This is possible since we require m 01 m 01 = 0 and m 01 m 21 + m 21 m 01 = m 11 m 11 . 8 / 16

M AIN THEOREMS T HEOREM 1 Let A be a dga over a commutative ring k. There exists a k-projective minimal dA ∞ -algebra E together with an E 2 -equivalence E → A of dA ∞ -algebras. • This minimal dA ∞ -algebra model E of A is well defined up to E 2 -equivalences between k -projective minimal dA ∞ -algebras. • ( E , m 11 , m 02 ) is a k -projective resolution of the graded k -algebra H ∗ ( A ) . T HEOREM 2 The quasi-isomorphism type of A can be recovered from E. 9 / 16

P ROOF OF T HEOREM 1 ( SKETCH ) D EFINITION A bidga B is a dA ∞ -algebra with m B ij = 0 if i + j ≥ 3. Maps of bidgas have f ij = 0 for i + j ≥ 2. • Equivalently: Monoids in the category of bicomplexes. • dgas are bidgas concentrated in horizontal degree 0 1 ST STEP OF PROOF Given A , there is a bidga B and an E 2 -equivalence B → A of bidgas such that H ∗ ( B , m B 01 ) is k -projective. 2 ND STEP OF PROOF Set E = H ∗ ( B , m B 01 ) . There exist a minimal dA ∞ -structure on E and an E 2 -equivalence E → B . 10 / 16

A PPLICATION AND EXAMPLE : Ext - ALGEBRAS Let M be a k -module and P be a k -projective resolution of M . The endomorphism dga A = Hom k ( P , P ) of P has homology H ∗ ( A ) = Ext −∗ k ( M , M ) . A minimal dA ∞ -algebra model of A is a resolution of the Yoneda algebra Ext ∗ k ( M , M ) together with structure maps m ij . This data encodes the quasi-isomorphism type of the endomorphism dga. E XAMPLE Let k = Z and M = Z / p . Then H ∗ ( A ) = Λ ∗ Z / p ( w ) with | w | = − 1. 11 / 16

� � E XAMPLE : Ext ∗ Z ( Z / p , Z / p ) Let E = Λ ∗ Z ( a , b ) with | a | = ( 0 , − 1 ) and | b | = ( 1 , 0 ) . 0 1 · p Z / p { ι } Z { ι } Z { b } 0 � � � � � · p Z / p { w } Z { a } Z { ab } − 1 � � � � • The given data specifies the m 11 and m 02 of a minimal dA ∞ -algebra. • m 12 satisfies m 12 ( a ⊗ b ) = ι , m 12 ( a ⊗ ab ) = a , m 12 ( ab ⊗ b ) = − b , m 12 ( ab ⊗ ab ) = − ab and vanishes on the other generators of E ⊗ E . • All other m ij vanish. This is a complete description of a minimal dA ∞ -model for A . 12 / 16

D ERIVED H OCHSCHILD COHOMOLOGY CLASS Let A be a dga with minimal dA ∞ -model E . The complex � C qt ( E ) = Hom k ( E ⊗ r , E [ s , t ]) r + s = q has a differential C qt → C q + 1 , t induced from m 11 and a Hochschild differential. Its cohomology is the derived Hochschild cohomology dHH qt ( H ∗ ( A )) . P ROPOSITION γ A := [ m 03 + m 12 + m 21 ] ∈ dHH 3 , − 1 ( H ∗ ( A )) is a well defined cohomology class depending only on the quasi-isomorphism type of A. If H ∗ ( A ) = 0 for ∗ < 0, then dHH 3 , − 1 ( H ∗ ( A )) → dHH 3 ( H 0 ( A ) , H 1 ( A )) maps γ A to the first k -invariant of A . 13 / 16

T WISTED CHAIN COMPLEXES D EFINITION A twisted chain complex E is an ( N , Z ) -graded k -module with differentials d E i : E → E [ i , 1 − i ] for i ≥ 0 satisfying � ( − 1 ) i d i d p = 0 for u ≥ 0 . i + p = u Maps are families of k -module maps f i : E → F [ i , − i ] satisfying � p = � i + p = u ( − 1 ) i f i d E i + p = u d F i f p . Composition of maps: ( gf ) u = � i + p = u g i f p S LOGAN A ∞ -algebras ↔ chain complexes dA ∞ -algebras ↔ twisted chain complexes If E is a dA ∞ -algebra, then ( E , m i 1 , i ≥ 0 ) is a twisted chain complex. 14 / 16

dA ∞ - STRUCTURES AND THE TENSOR COALGEBRA Let E be a dA ∞ -algebra, SE = E [ 0 , 1 ] , and TSE = � j ≥ 1 SE ⊗ j ij : SE ⊗ j → SE [ i , 1 − i ] m 1 � m ij ↔ � m q 1 ⊗ r ⊗ � is ⊗ 1 ⊗ t : SE ⊗ j → SE ⊗ q [ i , 1 − i ] m 1 � ↔ ij = r + 1 + t = q r + s + t = j m q m i : TSE → TSE [ i , 1 − i ] with components � � ↔ ij . L EMMA dA ∞ -relations ⇔ ( TSE , � m i ) is a twisted chain complex L EMMA dA ∞ -algebra maps E → F correspond to maps m E m F ( TSE , � i ) → ( TSF , � i ) of twisted chain complexes 15 / 16

P ROOF OF T HEOREM 2 ( SKETCH ) • The category of modules over a dA ∞ -algebra E is enriched in twisted chain complexes. • The endomorphism object Hom E ( E , E ) is a monoid in tCh k . • Tot Hom E ( E , E ) is a dga. • If E has E 2 -homology concentrated in horizontal degree 0, there is an E 2 -equivalence E → Tot Hom E ( E , E ) . • If E and F are E 2 -quasi-isomorphic, then Tot Hom E ( E , E ) and Tot Hom F ( F , F ) are quasi-isomorphic as dgas. P ROOF OF T HEOREM 2. Apply the last statement to E → A . I N OTHER WORDS : The dA ∞ -algebras with E 2 -homology concentrated in horizontal degree 0 model quasi-isomorphism types of dgas. 16 / 16