Koszul dual 2-functors and extension algebras of simple modules for G L 2
aa r X i v : . [ m a t h . R T ] J u l KOSZUL DUAL -FUNCTORS AND EXTENSION ALGEBRAS OFSIMPLE MODULES FOR GL VANESSA MIEMIETZ AND WILL TURNER
Abstract.
Let p be a prime number. We compute the Yoneda extension alge-bra of GL over an algebraically closed field of characteristic p by developing atheory of Koszul duality for a certain class of 2-functors, one of which controlsthe category of rational representations of GL over such a field. Contents
1. Introduction. 22. Example. 73. Homological duality. 103.1. Grading conventions. 103.2. The 2-category T . 113.3. Keller equivalence. 133.4. The operator E . 143.5. Koszul duality. 153.6. The operator ! O and O . 164.1. The operator O . 164.2. The operator O . 184.3. Comparing O and O . 195. Operators O respect dualities. 205.1. Operators O and dg equivalence. 205.2. Reversing gradings. 215.3. Koszul duality for operators O GL . 257. Reduction. 277.1. The pair ( Ω , Ψ ) . 277.2. A chain of isomorphisms. 298. An explicit dg algebra. 298.1. The homology H ( Ψ ) . 298.2. The homology H ( Ψ ⊗ Ω i ) . 348.3. Proof of Theorem 32. 389. Computing y . 39References 39 Mathematics Subject Classification.
Key words and phrases.
Koszul duality, extension algebra, GL , dg algebra, 2-functor. Introduction.
Let F be an algebraically closed field of characteristic p >
0. Let G = GL ( F ) denote the group of 2 × F . Let L denote a completeset of irreducible objects in the category G -mod of rational representations of G .The object of this paper is to give an explicit description of the Yoneda extensionalgebra Y = ⊕ L,L ′ ∈L Ext ● G -mod ( L, L ′ ) of G . The strongest previous results in this direction were obtained by A. Parker,who outlined an intricate algorithm to compute the dimension of Ext nG -mod ( L, L ′ ) for L, L ′ ∈ L and n ≥ O which we introduced pre-viously in our study of the category of rational representations of G . Heuristicallyspeaking, these operators (which are indeed 2-functors on a certain 2-category) forma way of ”applying” one algebra to another to produce a bigger algebra (draggingcertain bimodules around). While G -mod is not Koszul itself, it is obtained as aniterated application of Koszul algebras to the ground field, and in this article weshow that iteratively applying their Koszul duals (together with the appropriatebimodules) will produce a dg algebra whose homology is the desired extension alge-bra. We use this to give a combinatorial description of Y as an algebra, includingthe multiplicative structure, described by explicit formulae how to multiply basiselements. While in the end, we do arrive at an explicit combinatorial descriptionof Y , we would like to stress that the iterated dg algebra, as whose homologywe obtain the extension algebra, keeps track of all the homotopical information,and hence in principle incorporates the (nontrivial) A ∞ -structure of the Yonedaextension algebra of GL . To investigate this A ∞ -structure would in itself be aninteresting problem.The category G -mod has countably many blocks, all of which are equivalent. There-fore, the algebra Y is isomorphic to a direct sum of countably many copies of y ,where y is the Yoneda extension algebra of the principal block of G . Our problem isto compute y . In the remainder of this section, we will introduce the main objectsin the explicit computations, so we can state our main theorem, Theorem 1. Wewill then deduce the description of a multiplicative (up to prescribed sign) basisfrom this, and finish with some remarks. In Section 2, we will give an explicit de-scription of an example illustrating the monomial basis. In Section 3, we introducethe 2-category in which our theory is developed and extend notions of homologicalduality to this setting. In Section 4, we introduce our 2-functors, before studyingtheir behaviour under the homological dualities from Section 3 in Section 5. Wethen recall the results needed from the representation theory of GL ( F ) in Section6. Section 7 then uses the developed theory to reduce the computation to the com-putation of the homology of an explicit tensor algebra. The latter is then computedin Section 8, which is largely independent of the preceding sections. OSZUL DUAL 2-FUNCTORS AND EXTENSION ALGEBRAS OF SIMPLE MODULES FOR GL Let Π denote the preprojective algebra of bi-infinite type A : it is the path algebraof the quiver ... x ( ( ● y i i x ( ( ● y h h x ( ( ● y h h ⋯ p − ● x ( ( p ● y j j x ' ' ... y h h , modulo relations xy − yx , and is naturally Z + -graded so that paths are homogeneouswith degree given by their length.This algebra has a natural basis given by B Π = {( s, α, β ) ∣ s ∈ Z , α, β ∈ N } . Here, basis elements of Π correspond to paths in the quiver of Π; the element ( s, α, β ) corresponds to the element x α y β e s in the path algebra. The target t ofsuch a path can be computed via the formula t − s = α − β . The product in Π oftwo such basis elements is defined by the following formula: ( s, α, β ) ⋅ ( s ′ , α ′ , β ′ ) = { ( s, α + α ′ , β + β ′ ) if s ′ = s + α − β ;0 otherwise . In the following, we will call such a basis for an algebra monomial, meaning thebasis contains a complete set of primitive idempotents and the product of any twoof its elements is either equal to ( ± ) another basis element or equal to zero.For b = ( s, α, β ) ∈ B Π , we define the degree of b to be ∣ b ∣ = α + β .Let Π ≤ p denote the subalgebra of Π generated by arrows which begin and end atvertices indexed by i ≤ p . Let Ω denote the quotient of Π ≤ p by the ideal generatedby idempotents e i corresponding to vertices i ≤
0, that is Ω is given by quiver andrelations as Ω = F ( ● x ( ( ● y k k x ( ( ● y h h ⋯ p − ● x p − , , p ● )/ I ⊥ y p − j j , where I ⊥ = ( x l y l − y l + x l + , y x ∣ ≤ l ≤ p − ) ..The Loewy structure of the p th projective Ω e p of this algebra is given by p ✇✇ p − rr ❋❋ p − rr ▲▲ p ①① p − ✈✈ ▼▼▼ p − qqq ❋❋❋ ⋮ ⋮ ☎☎ ✿✿ ☎☎ ❁❁ ⋯ p − rr ❋❋ ✿✿ ☎☎ ✿✿ ⋯ p − rrr ▲▲ p ①① ✿✿✿ ☎☎☎ ❂❂ ⋯ p − qqq ❋❋❋ ⋮ ⋮ p − ▲▲ p − rr ❋❋ p − ▲▲ p ①① p − ●● p VANESSA MIEMIETZ AND WILL TURNER and the remaining projectives are the corresponding submodules of this module.A monomial basis for Ω is given by the subset B Ω = {( s, α, β ) ∈ B Π ∣ ≤ s ≤ p, α ≤ β + p − s, β ≤ s − } of B Π . When multiplying elements of B Ω , we use the multiplication rule for B Π ,with the caveat that products of basis elements are zero if their product in B Π doesnot belong to B Ω .Let Θ denote the quotient of Π by the ideal generated by idempotents correspondingto vertices i ≤ i ≥ p . Alternatively, we can describe Θ as the quotient of Ωby the ideal generated by vertex p , i.e. Θ = Ω / Ω e p Ω. We will denote the naturalquotient map Ω → Θ by π . If p =
5, the Loewy structure is given by ✁✁ ❂❂ ❂❂ ✁✁ ❂❂ ❂❂ ✁✁ ✁✁ ❂❂ ✁✁ ❂❂ ✁✁ ❂❂ ✁✁ which conveys the general picture.A monomial basis for Θ is given by the subset B Θ = {( s, α, β ) ∈ B Π ∣ ≤ s ≤ p − , α ≤ p − s − , β ≤ s − } of B Π .We denote by σ the involution of Π which sends vertex i to vertex p − i and exchanges x and y . This induces an action of σ on Θ which, on a basis element ( s, α, β ) ∈ B Θ ,is given by the formula σ ( s, α, β ) = ( p − s, β, α ) . Let Θ σ denote the Ω-Ω bimodule given by lifting along π the Θ-Θ-bimodule ob-tained by twisting the regular Θ-bimodule on the right by σ .We now define Λ = T Ω ( Θ σ ) ⊗ F [ ζ ] , to be the tensor product over F of the tensor algebra over Ω of Θ twisted on theright by σ , with a polynomial algebra in a single variable. Note that as a vectorspace ( Θ σ ) ⊗ n is just isomorphic to Θ.A monomial basis B Λ for Λ is given by B Λ = {( b, n, h )∣ n, h ∈ N , b ∈ B Ω if n = , b ∈ B Θ if n > } . The product is given by ( b, n, h )( b ′ , n ′ , h ′ ) = { ( bb ′ , n + n ′ , h + h ′ ) if n is even; ( bσ ( b ′ ) , n + n ′ , h + h ′ ) if n is odd . Here, a basis element ( b, n, h ) belongs to the component ( Θ σ ) ⊗ n ⊗ ζ h of Λ.We define a trigrading Λ = ⊕ i,j Λ ijk on Λ as follows: the i -grading is defined byplacing Ω in i -degree 0 and Θ σ and ζ in i -degree 1; the j -grading is defined bygrading elements of Ω and Θ σ according to path length and placing ζ in j -degree p ;the k -grading is defined by grading elements of Ω and Θ σ according to path length OSZUL DUAL 2-FUNCTORS AND EXTENSION ALGEBRAS OF SIMPLE MODULES FOR GL and placing ζ in k -degree p −
1. For a basis element β = ( b, n, h ) this yields the ijk -degree (∣ β ∣ i , ∣ β ∣ j , ∣ β ∣ k ) = ( n + h, ph + ∣ b ∣ , ∣ b ∣ + ( p − ) h ) . Now suppose Γ = ⊕ i,j,k ∈ Z Γ ijk is a Z -trigraded algebra. We have a combinatorialoperator O Γ which acts on the collection of bigraded algebras ∆ after the formula O Γ ( ∆ ) ik = ⊕ j,k + k = k Γ ijk ⊗ ∆ jk , where ⊗ denotes the super tensor product i.e. multiplication is given by ( γ ⊗ δ )( γ ⊗ δ ) = ( − ) ∣ γ ∣ k ∣ δ ∣ k ( γ γ ⊗ δ δ ) where γ , γ ∈ Γ , δ , δ ∈ ∆ and ∣ ⋅ ∣ k denotes the k -degree of the correspondingelement.We consider the field F as a trigraded algebra concentrated in degree ( , , ) .We have a natural embedding of bigraded algebras F → Λ, which sends 1 to theidempotent e ⊗
1, where e ∈ Ω is the idempotent corresponding to vertex 1. Thisembedding lifts to a morphism of operators O F → O Λ . We have O F = O F . Puttingthese together, we obtain a sequence of operators O F → O F O Λ → O F O → ... which, applied to the bigraded algebra F [ z ] with z placed in jk -degree ( , ) , givesa sequence of algebra embeddings λ → λ → λ → ..., where λ q = O F O q Λ ( F [ z ]) . Taking the union of the algebras in this sequence givesus an algebra λ . Our main theorem is the following: Theorem 1.
The algebra y is isomorphic to λ as a bigraded algebra. The proof of this theorem will occupy the rest of the article. For now, we wouldlike to use the monomial basis given for Λ to describe a monomial basis for λ . Wewill do this by describing a monomial basis for each λ q .Firstly, we put B q = {[ β , ..., β q , l ]∣ β i ∈ B Λ , l ∈ N } , which, identifying the tuple [ β , ..., β q , l ] with β ⊗ ⋯ ⊗ β q ⊗ z l forms a monomialbasis of the super tensor product Λ ⊗ q ⊗ F [ z ] , where again multiplication is givenby [ β , ..., β q , l ][ β ′ , ..., β ′ q , l ′ ] = ( − ) ∑ s ′< s ∣ β s ∣ k ∣ β ′ s ′ ∣ k [ β β ′ , ..., β q β ′ q , l + l ′ ] . By construction, our algebra λ q = O F O q Λ ( F [ z ]) is a subalgebra of Λ ⊗ q ⊗ F [ z ] consisting of linear combinations of those tuples where the i -degree of β is zero,the i -degree of β s equals the j -degree of β s − for s = , . . . , q , and l = ∣ β q ∣ j . Hencedefining a height function on B q via [ β , ..., β q , l ] ↦ (∣ β ∣ i , ∣ β ∣ i − ∣ β ∣ j , ∣ β ∣ i − ∣ β ∣ j , . . . , ∣ β q ∣ i − ∣ β q − ∣ j , l − ∣ β q ∣ j ) we obtain a basis B λ q of λ q as the subset of B q of height ( , . . . , ) . VANESSA MIEMIETZ AND WILL TURNER
Letting ˆ1 = (( , , ) , , ) denote the element of ijk-degree ( , , ) in B Λ of whichcorresponds to the idempotent in Ω indexed by the vertex 1, we obtain an embed-ding of B q in B q + which takes ( β , ..., β q , l ) to ( ˆ1 , β , ..., β q , l ) . This embedding ismultiplicative and height preserving, and restricts to the embedding of λ q into λ q + described earlier.Hence, letting B denote the union of the sequence of embeddings B → B → B → ... and B λ the subset of B consisting of elements of height 0, we arrive at the following: Theorem 2.
The basis B λ , with product obtained by restriction from B , forms amonomial basis for λ . We conclude this section with some remarks.
The Yoneda grading.
While our algebras are multigraded, the grading of mostinterest for representation theorists is the grading by extension degree, or Yonedagrading. An element of B represents an extension of degree d if its total k -degreeis d . Note that the total k -degree of an element [ β , ..., β q , l ] for β i = ( b i , n i , h i ) isgiven by d = q ∑ i = ∣ b i ∣ + ( p − ) h i . Since for 1 ≤ i ≤ q − ∣ b i ∣ + ph i = n i + + h i + , the sum rearranges to d = ∣ b q ∣ + h q + q ∑ i = n i = l + q ∑ i = n i where we have used that n = Polytopes.
The basis B λ is infinite. However, it is a union of monomial bases B λ q for λ q , whose elements are in natural one-one correspondence with elements of finitelattice polytopes P q of dimension 4 q . We can build this up as follows: by definitionelements of B Ω and B Θ are indexed by lattice elements of a finite polytope in Z ;elements of B Λ are indexed by lattice elements of a polytope in Z ; elements of B q are indexed by lattice elements of an infinite nonconvex polytope in Z q + ; elementsof B q of height 0 are indexed by elements of a polytope in Z q + −( q + ) since theyare the intersection of B q with the kernel of a linear surjection from Z q + to Z q + ;elements of a monomial basis B λ q for λ q are therefore indexed by lattice elementsof a polytope P q in Z q ; the polytope P q is finite because λ q is finite dimensional.Computing dimensions Ext-groups now amounts to counting points in a polytopewith a certain boundary condition (the total k -degree). Recipe for application to GL . Suppose you want to know the dimension ofExt d ( L ( ν ) , L ( µ )) , where ν, µ are two highest weights of simple modules for GL ( F ) ,each given by a non-negative integer. First you need to determine whether they arein the same block. Weights in a block are linearly ordered, so we have a naturalorder-preserving bijection between weights in a given block and the natural numbers1 , , . . . . The natural number associated to a weight ν will be denoted by m ν . Thealgorithm to determine whether two weights are in the same block is sketched in[8, Section 1], and we repeat and refine it here for the reader’s convenience. Write OSZUL DUAL 2-FUNCTORS AND EXTENSION ALGEBRAS OF SIMPLE MODULES FOR GL ν = ∑ ri = a i p i , µ = ∑ ri = b i p i with 0 ≤ a i , b i ≤ p −
1, and r the greater one of the p -adicvaluations of ν and µ . If at least one of a , b is not equal to p −
1, the weights ν, µ are in the same block if and only if either ∑ ri = ( a i − b i ) p i − ≡ a = b or ∑ ri = ( a i − b i ) p i − ≡ a = p − − b . If ν, µ are in the sameblock, the natural numbers to which they correspond in the linear order are givenby m ν ∶ = + ∑ ri = ( a i ) p i − and m µ ∶ = + ∑ ri = ( b i ) p i − respectively. If both a , b equal p −
1, take the first index (say s ) such that one of a s , b s differs from p −
1, andrepeat the above algorithm with ¯ ν = ∑ ri = s a i p i − s , ¯ µ = ∑ ri = b i p i − s .If ν and µ are in the same block corresponding to natural numbers m ν and m µ respectively, we then find the number of occurrences of the m ν th simple in the m µ th projective for λ r , which have k -degree d .For example, simple representations in the principal bock have highest weights ofthe form ν n = ( n − ) p or µ n = ( n − ) p + p − n ≥
1. These get assignedvalues m ν n = n − m µ n = n respectively in our notation. Then we writethese numbers as a + ∑ qi = ( a i − ) p i − with 1 ≤ a i ≤ p and the idempotent atthe corresponding vertex (i.e. the identity morphism on the correponding simple)is given by the basis element [(( a q , , ) , , ) , . . . , (( a , , ) , , ) , ] ∈ B Λ q . Anyelement of Ext will be given by a basis element of the form [(( a q , , ) , , ) , . . . , (( a , , ) , , ) , (( a , , ) , , ) , ] , [(( a q , , ) , , ) , . . . , (( a , , ) , , ) , (( a , , ) , , ) , ] ∈ B λ q (between neighbouring simples, here a has to be strictly greater than 1 in thesecond case) or [(( a q , , ) , , ) , . . . , (( a i + , , ) , , ) , (( p − a i , , ) , , ) , . . . , (( a , , ) , , ) , ] , [(( a q , , ) , , ) , . . . , (( a i + , , ) , , ) , (( p − a i , , ) , , ) , . . . , (( a , , ) , , ) , ] ∈ B λ q for some i , between simples L ( µ ) and L ( ν ) with m µ − m ν = ( p − a i ) p i − in the firstand m µ − m ν = − a i p i − in the second case, which again needs a i + > Example.
The algebra λ q is isomorphic to the Yoneda extension algebra of a block of Schuralgebra S ( , r ) with p q simple modules. Here, and throughout this paper, by theYoneda extension algebra of an abelian category, we mean the Yoneda extensionalgebra of a complete set of nonisomorphic simple objects in that category; by theYoneda extension algebra of an algebra, we mean the Yoneda extension algebra ofthe category of finite dimensional modules for that algebra. To place our feet onthe ground, let us describe an example of such an algebra.Let p =
3. Let y denote the Yoneda extension algebra of a block of a Schuralgebra S ( , r ) with 9 simple modules. We label the m th simple module by a , a if ( a − ) + a = m . The algebra y is isomorphic to F Q / J , where Q is the quiver ( , ) x / / α (cid:15) (cid:15) f ! ! ❉❉❉❉❉❉❉❉❉❉❉❉❉❉❉❉❉❉❉ ( , ) f } } ③③③③③③③③③③③③③③③③③③③ y o o α (cid:15) (cid:15) x / / ( , ) y o o α (cid:15) (cid:15) ( , ) α (cid:15) (cid:15) β O O x / / f ! ! ❉❉❉❉❉❉❉❉❉❉❉❉❉❉❉❉❉❉❉ g = = ③③③③③③③③③③③③③③③③③③③ ( , ) g a a ❉❉❉❉❉❉❉❉❉❉❉❉❉❉❉❉❉❉❉ f } } ③③③③③③③③③③③③③③③③③③③ y o o α (cid:15) (cid:15) β O O x / / ( , ) y o o α (cid:15) (cid:15) β O O ( , ) β O O x / / g = = ③③③③③③③③③③③③③③③③③③③ ( , ) g a a ❉❉❉❉❉❉❉❉❉❉❉❉❉❉❉❉❉❉❉ y o o β O O x / / ( , ) y o o β O O and J is the ideal generated by the following relations ● xy − yx, f g − gf, αβ − βα ● xβ − βx, xα − αxyβ − βy, yα − αy,f β − βf, f α − αf,gβ − βg, gα − αgyf − gx, yg − f x, gy − xf, f y − xg, ● yxe ( i, ) , gf e ( ,i ) , βαe ( ,i ) , βf e ( ,i ) , gαe ( ,i ) for 1 ≤ i ≤ ● e ( i, ) gye ( i + , ) , e ( i + , ) xf e ( i, ) , e ( i + , ) f ye ( i, ) , e ( i, ) xge ( i + , ) for 1 ≤ i ≤ e ( i,j ) denotes the idempotent at vertex ( i, j ) .The solid arrows have Yoneda degree 1, whilst the dotted arrows have Yoneda degree3. This presentation was obtained by direct computation, however, the filtration ofprojectives by certain graded subquotients is directly visible from our construction.For example, the projective indecomposable module indexed by vertex ( , ) has abasis given by paths B ∶ = { e, xe, x e,f e, yf e,f e, xf e,αe, xαe, x αe,f αe, yf αe,α e, xα e, x α e } where e is the idempotent corresponding to vertex ( , ) . OSZUL DUAL 2-FUNCTORS AND EXTENSION ALGEBRAS OF SIMPLE MODULES FOR GL It’s composition structure is given by1 ✂✂ ◆◆◆◆◆ ✂✂ ◆◆◆◆◆ ✂✂ ◆◆◆◆◆ ◆◆◆◆◆ ✂✂ ✂✂ ◆◆◆◆◆ ✂✂ ◆◆◆◆◆ ✂✂ ✂✂ ✂✂ d in row d . Writingthe basis elements into the picture, we obtain e rrrr ❱❱❱❱❱❱❱❱❱❱ xe rrrr ❚❚❚❚❚❚❚❚❚ f e ttt ❚❚❚❚❚❚❚❚❚ x e yf e ❚❚❚❚❚❚❚❚ f e ✇✇ αe ✉✉✉✉ ❚❚❚❚❚❚❚❚❚ xf exαe rrr ❚❚❚❚❚❚❚❚ f αe ttt x αe yf αeα e tt xα e qqq x α e and the subquotients given by basis elements { e, xe, x e } , { αe, xαe, x αe } as wellas { α e, xα e, x α e } are each isomorphic to Ω e , the subquotients given by basiselements { f e, yf e } and { f αe, yf αe } are isomorphic to Θ σ e and the subquotientgiven by basis elements { f e, xf e } is isomorphic to Θ e .We now describe the monomial basis for this projective indecomposable module.Let { a, b, c } be a basis of the projective indecomposable Ω-module indexed by vertex1, i.e. a = ( , , ) , b = ( , , ) , c = ( , , ) ∈ B Ω , lying in degrees 0 , , { η = ( , , ) , θ = ( , , )} ⊂ B Θ and { ξ = ( , , ) , ς = ( , , )} ⊂ B Θ be the basis elements of Θ e and Θ σ e respectively,which are not annihilated by the idempotent at the vertex 1 from the right. Themonomial basis B is the set {[( γ, , h ) , ( γ ′ , , h ′ ) , l ] , [( γ, , h ) , ( δ, n, h ′ ) , l ] , [( δ, n, h ) , ( γ, , h ′ ) , l ] , [( δ, n, h ) , ( δ ′ , n ′ , h ′ ) , l ]} where h, h ′ , l ∈ N , n, n ′ ∈ N > , γ ∈ { a, b, c } and δ ∈ { η, θ } or { ξ, ς } depending onwhether n is even or odd.The elements of height zero in here are {[( γ, , ) , ( γ ′ , , ∣ γ ∣) , p ∣ γ ∣ + ∣ γ ′ ∣] , [( γ, , ) , ( δ, n, ∣ γ ∣ − n ) , ∣ δ ∣ + p ∣ γ ∣ − pn ]} where γ, δ are before and n = , . . . , ∣ γ ∣ . So the full set is B ′ ∶ = {[( a, , ) , ( a, , ) , ] , [( a, , ) , ( b, , ) , ] , [( a, , ) , ( c, , ) , ] , [( b, , ) , ( ξ, , ) , ] , [( b, , ) , ( ς, , ) , ] , [( c, , ) , ( η, , ) , ] , [( c, , ) , ( θ, , ) , ][( b, , ) , ( a, , ) , ] , [( b, , ) , ( b, , ) , ] , [( b, , ) , ( c, , ) , ][( c, , ) , ( ξ, , ) , ] , [( c, , ) , ( ς, , ) , ] , [( c, , ) , ( a, , ) , ] , [( c, , ) , ( b, , ) , ] , [( c, , ) , ( c, , ) , ]} We can identify the bases B and B ′ for y , as ordered sets. The Yoneda degree ofan element is given by summing the n s (i.e. the middle entries in the tuples) and l ,giving 0 , , , , , , , , , , , , , , Homological duality.
There are a number of approaches to homological duality in representation theory.Here we describe two. One is a general approach via differential graded algebras,due to Keller. The other is a special approach for Koszul algebras, which is easierto work with explicitly.3.1.
Grading conventions.
In order to fix our sign conventions, we now give abrief introduction to dg algebras and modules. A differential graded vector spaceis a Z -graded vector space V = ⊕ k V k with a graded endomorphism d of degree 1.We write ∣ v ∣ k for the degree of a homogeneous element of V . We assume d canact both on the left and the right of V , with the convention d ( v ) = ( − ) ∣ v ∣ k ( v ) d .A differential graded algebra is a Z -graded algebra A = ⊕ k A k with a differential d such that d ( ab ) = d ( a ) .b + ( − ) ∣ a ∣ k a.d ( b ) , or equivalently ( ab ) d = a. ( b ) d + ( − ) ∣ b ∣ k ( a ) d.b. If A is a differential graded algebra then a differential graded left A -module is agraded left A -module M with differential d such that d ( a.m ) = d ( a ) .m + ( − ) ∣ a ∣ k a.d ( m ) ;a differential graded right A -module is a graded right A -module M with differential d such that d ( m.a ) = d ( m ) .a + ( − ) ∣ m ∣ k m.d ( a ) . If A and B are dg algebras then a dg A - B -bimodule is a graded A - B -bimodule witha differential which is both a left dg A -module and a right dg B -module. OSZUL DUAL 2-FUNCTORS AND EXTENSION ALGEBRAS OF SIMPLE MODULES FOR GL If A M B and B N C are dg bimodules where A , B , and C are dg algebras, then M ⊗ B N is a dg A - C -bimodule with differential d ( m ⊗ n ) = d ( m ) ⊗ n + ( − ) ∣ m ∣ k m ⊗ d ( n ) . Speaking about morphisms of dg algebras and dg (bi-)modules we mean homoge-neous morphisms. However, if A M B and A N C are dg bimodules where A , B , and C are dg algebras, then Hom A ( M, N ) , the space of all A -module homomorphismsfrom M to N , is a dg B - C -bimodule with differential d ( φ ) = d ○ φ − ( − ) ∣ φ ∣ k φ ○ d. If A M is a left dg A -module, then End A ( M ) is a differential graded algebra whichacts on the right of M , giving M the structure of an A -End A ( M ) -bimodule, thedifferential on End A ( M ) being given by ( φ ) d = φ ○ d − ( − ) ∣ φ ∣ k d ○ φ . If M B is aright dg A -module, then End A ( M ) is a differential graded algebra which acts onthe left of M , giving M the structure of an End B ( M ) - B -bimodule, the differentialon End B ( M ) being given by d ( φ ) = d ○ φ − ( − ) ∣ φ ∣ k φ ○ d .A differential bi- (tri-)graded vector space is a vector space V with a Z - respec-tively Z -grading whose coordinates we denote by ( j, k ) respectively ( i, j, k ) and anendomorphism d of degree ( , , ) , ie.e d is homogeneous with respect to the i, j -gradings and has degree 1 in the k -grading, which we will also denote the homologi-cal grading. We denote by ⟨ ⋅ ⟩ a shift by 1 in the j -grading, meaning ( V ⟨ n ⟩) j = V j − n .Since we will often identify dg modules and complexes, we will stick to the complexconvention of [ ⋅ ] being a shift to the left, i.e. V [ n ] k = V k + n . Altogether ( V ⟨ n ⟩[ m ]) ijk = V i,j − n,k + m . All definitions above can be extended to the differential bi- (tri-)graded setting,defining differential bi- (tri-)graded algebras, differential bi- (tri-)graded (left andright) A -modules as well as bi- (tri-)graded A - B -bimodules as bi- (tri-)graded alge-bras resp. modules resp. bimodules which are differential graded algebras resp. mod-ules resp. bimodules with respect to the k -grading, i.e. with respect to an endomor-phism of degree ( , , ) . Speaking about morphisms of differential bi- (tri-)gradedalgebras and differential bi- (tri-)graded (bi-)modules we mean homogeneous mor-phisms with respect to all gradings. Similarly to the above, homomorphism spacestaken between A -modules (rather than differential (bi-) trigraded A -modules) willcarry a differential bi- (tri-)grading.For a dg algebra A , we denote by D dg ( A ) the dg derived category of A , whoseobjects are dg A -modules and where morphisms are given by the localisation of theset of dg module morphisms with respect to quasi-isomorphisms (see [5, Section3.1, 3.2]).We denote by H the cohomology functor, which takes a differential k -graded com-plex C to the k -graded vector space H C = H ● C .3.2. The -category T . Let T denote the 2-category given by the following data: ● objects of T are pairs ( A, M ) where A = ⊕ A k is a dg algebra and M = ⊕ M k is a dg A - A -bimodule; ● a 1-morphism ( X, φ X ) between two objects ( B, N ) and ( A, M ) in T is givenby a dg bimodule A X B together with a quasi-isomorphism of dg-bimodules φ X ∶ X ⊗ B N → M ⊗ A X ; ● a 2-morphism between two such 1-morphisms given by ( A X B , φ X ) and ( A Y B , φ Y ) respectively is a morphism f ∶ X → Y of dg bimodules such thatthe diagram X ⊗ B N φ X / / f ⊗ id (cid:15) (cid:15) M ⊗ A X id ⊗ f (cid:15) (cid:15) Y ⊗ B N φ Y / / M ⊗ A Y commutes. Definition 3.
We define a j -graded object of T to be an object ( a, m ) of T ,where a = ⊕ a jk is a differential bigraded algebra, and m = ⊕ m jk a differentialbigraded a - a -bimodule, and a jk = m jk = j < Definition 4.
We define a
Keller object of T to be an object ( A, M ) of T ,where A M A is in A -perf ∩ perf- A and the natural morphisms of dg algebras A → End A ( A M, A M ) and A → End A ( M A , M A ) are quasi-isomorphisms.We call such M a two-sided tilting complex. Definition 5.
We define a
Rickard object of T to be a Keller object ( A, M ) of T , where we have a quasi-isomorphism of dg algebras A → H A and H A is afinite-dimensional algebra of finite-global dimension.This terminology is motivated by Keller’s [5] and Rickard’s [9] Morita theory fordg derived categories and derived categories of algebras respectively, which identifyobjects with the properties asked of M in each case as objects inducing autoequiv-alences of the respective categories. Definition 6.
Let ( A, M ) and ( B, N ) be objects of T . A dg equivalence betweenobjects ( A, M ) and ( B, N ) of T is a pair ( A X B , φ X ) such that(i) A X belongs to A -perf, A X generates D dg ( A ) , and the natural map B → End A ( X ) is a quasi-isomorphism of dg algebras;(ii) φ X ∶ X ⊗ B N → M ⊗ A X is a quasi-isomorphism of dg A - B -bimodules.Note that Definition 6(ii) just implies that ( A X B , φ X ) is a 1-morphism in T . Ob-serve also that X is a dg A - B -bimodule which induces an equivalence between D dg ( A ) and D dg ( B ) via D dg ( A ) Hom A ( X, −) - - D dg ( B ) X ⊗ B − m m , OSZUL DUAL 2-FUNCTORS AND EXTENSION ALGEBRAS OF SIMPLE MODULES FOR GL by Keller [5, Lemma 3.10]. We then have a diagram D dg ( B ) X ⊗ B − / / N ⊗ B − (cid:15) (cid:15) D dg ( A ) M ⊗ A − (cid:15) (cid:15) D dg ( B ) X ⊗ B − / / D dg ( A ) which commutes up to natural isomorphism of functors X ⊗ B N ⊗ B − → M ⊗ A X ⊗ B − .If there is a dg equivalence between ( A, M ) and ( B, N ) , we write ( A, M ) ⋗ ( B, N ) . Definition 7.
Let ( A, M ) , ( B, N ) ∈ T . We define a quasi-isomorphism from ( B, N ) to ( A, M ) to be a pair ( ϕ B , ϕ N ) where ϕ B ∶ B → A is quasi-isomorphismof dg algebras, and ϕ N ∶ B N B → A M A is a quasi-isomorphism of vector spacescompatible with the bimodule structure and ϕ B in the sense that ϕ N ( b nb ) = ϕ B ( b ) ϕ N ( n ) ϕ B ( b ) . Similarly, we define an isomorphism from ( B, N ) to ( A, M ) by replacing the word quasi-isomorphism by isomorphism throughout.Note that this can be interpreted as a particular 1-morphism ( A, φ A ) from ( B, N ) to ( A, M ) in T where the right B -module structure on A is induced by ϕ B anddefining φ A ∶ A ⊗ B N → M ⊗ A A ∶ a ⊗ n ↦ aϕ N ( n ) ⊗
1. The compatibility conditionsin Definition 7 ensure that this is well-defined.3.3.
Keller equivalence.
Here we recall a general statement concerning extensionalgebras of simple modules viewed as dg algebras.Let A be a dg algebra, such that there is a quasi-isomorphism of dg algebras A → H A and H A is finite dimensional and of finite global dimension. Let S , . . . , S d be acomplete set of non-isomorphic simple H A -modules. Let P l = ⊕ k P kl be a projective A -resolution of S l , viewed as a dg A -module. Note that since A is assumed to finiteglobal dimension, we can choose P l to lie in A -perf. Denote by E ( A ) the dg algebra E ( A ) ∶ = ⊕ k,k ′ Hom A ( d ⊕ l = P kl , d ⊕ l = P k ′ l ) with the dg algebra structure as described in Section 3.1. Viewing these projectiveresolutions as complexes, the dg algebra differential translates into the total differ-ential on the Hom double complex and therefore H E ( A ) = Ext ● ( ⊕ dl = S l , ⊕ dl = S l ) . Then P = ⊕ l P l is a dg A - E ( A ) -bimodule. There are mutually inverse equivalences D dg ( A ) Hom A ( P, −) . . D dg ( E ( A )) P ⊗ E( A ) − m m , by [5, Lemma 3.10]. Since P is in A -perf, we have a natural isomorphism of functorsHom A ( P, − ) ≅ Hom A ( P, A ) ⊗ A − . Note that if ( A, M ) ∈ T is a Rickard object, the assumptions on A in the aboveparagraph are automatically satisfied. The operator E . Here we extend the duality described in the previous sectionto objects in T by defining an operator E on T . Definition 8.
Let ( A, M ) be a Rickard object in T and P be defined as in Section3.3. Denote by E ( M ) the dg E ( A ) - E ( A ) -bimodule E ( M ) = Hom A ( P, A ) ⊗ A M ⊗ A P. Furthermore, we define E ( A, M ) ∶ = ( E ( A ) , E ( M )) . Note that E ( M ) ≅ Hom A ( P, M ) ⊗ A P . Lemma 9.
Let ( A, M ) be a Rickard object in T . The bimodule P , defined asin Section 3.3, can be extended to a dg equivalence ( P, φ P ) between ( A, M ) and E ( A, M ) .Proof. We need to define a quasi-isomorphism of dg bimodules φ P ∶ P ⊗ E( A ) E ( M ) → M ⊗ A P , or in other words a quasi-isomorphism P ⊗ E( A ) Hom A ( P, A ) ⊗ A M ⊗ A P → M ⊗ A P. We obtain this via the natural morphism of dg A - A -bimodules P ⊗ E( A ) Hom A ( P, A ) → A, which is an isomorphism since P is a progenerator for A . (cid:3) In the setup of Lemma 9, we thus have a natural isomorphism of functors makingthe following diagram commute: D dg ( A ) Hom A ( P, −) / / M ⊗ A − (cid:15) (cid:15) D dg ( E ( A )) E( M )⊗ E( A ) − (cid:15) (cid:15) D dg ( A ) Hom A ( P, −) / / D dg ( E ( A )) . Lemma 10.
Let ( A, M ) be a Rickard object in T and P defined as in Section 3.3.We have an isomorphism of dg E ( A ) - E ( A ) -bimodules E ( M ) ⊗ E( A ) r → Hom A ( P A , M ⊗ A r ⊗ P A ) . Proof.
If we write out E ( M ) ⊗ E( A ) r in full, the internal occurences of the term P ⊗ E( A ) Hom A ( P, A ) cancel, thanks to the isomorphism P ⊗ E( A ) Hom A ( P, A ) → A of dg A - A -bimodules, leaving us withHom A ( P A , A ) ⊗ A M ⊗ A r ⊗ A P A ≅ Hom A ( P A , M ⊗ A r ⊗ A P A ) . (cid:3) OSZUL DUAL 2-FUNCTORS AND EXTENSION ALGEBRAS OF SIMPLE MODULES FOR GL Koszul duality.
In this subsection we summarise the results we need from[1]. We assume a = ⊕ a j is a graded finite dimensional algebra with finite globaldimension, with a semisimple, and a j = j <
0. We further assume that a isgenerated in degree 1 and quadratic. Then a = T a ( a )/ R, where R is the kernel of the multiplication map a ⊗ a a → a . We write x ∗ = Hom ( a x, a a ) , and ∗ x = Hom ( x a , a a ) if x is a left/right a -module.If x is concentrated in degree j , then by convention x ∗ and ∗ x are concentrated indegree − j . The quadratic dual algebra a ! is given by a ! = T a ( a ∗ )/ a ∗ , where a ∗ embeds in a ∗ ⊗ a a ∗ via the dual of the multiplication map, composedwith the inverse of the natural isomorphism a ∗ ⊗ a a ∗ → ( a ⊗ a a ) ∗ . We then have a ! j = j > a - a ! -bimodule K = a ⊗ a ∗ ( a ! ) , whose k -grading is given by K k = a ⊗ a ∗ ( a ! − k ) ≅ Hom ( a ! − ka , a a ) , and whose differential is given by the compositionHom ( a ! − k − a , a a ) → Hom ( a ! − k − a ⊗ a a , a a ⊗ a a ) → Hom ( a ! − ka , a a ) , obtained from the multiplication map a ⊗ a a → a, and the natural composition a ! − k = a ! − k ⊗ a a → a ! − k ⊗ a a ∗ ⊗ a a → a ! − k − ⊗ a a . Note that the differential has j -degree zero.The algebra a is said to be Koszul precisely when K → a is a projective resolutionof a a , in which case the map a ! → E ( a ) is a quasi-isomorphism. There are mutuallyinverse equivalences D b ( a -gr ) Hom a ( K, −) - - D b ( a ! -gr ) K ⊗ a ! − m m , Under Koszul duality, the irreducible a -module a e corresponds to the projective a ! -modules a ! e , for a primitive idempotent e ∈ a ; a shift ⟨ j ⟩ in D b ( a -gr ) correspondsto a shift ⟨ j ⟩[ − j ] in D b ( a ! -gr ) . Warning.
Our j -grading convention does not coincide with that in the literature.We assume that a is generated in degrees 0 and 1 whilst a ! is generated in degrees0 and −
1. Beilinson, Ginzburg and Soergel assume that a is generated in degrees 0and 1 but that a ! is also generated in degrees 0 and 1. We have changed conventionsin order to obtain gradings which behave well with respect to our operators. The operator ! . In this subsection, we generalise the notion of Koszul dualityfrom the previous subsection to objects of T . Definition 11.
Suppose that ( a, m ) is a j -graded Rickard object of T , such that a is a Koszul algebra. We then call the object ( a, m ) a Koszul object of T .Let ˜ K denote a projective resolution, viewed as a dg a - a ! -bimodule, of the complex K of graded a - a ! -bimodules introduced in Section 3.5. We define m ! to be thedifferential bigraded a ! - a ! -bimodule m ! ∶ = Hom a ( ˜ K, a ) ⊗ a m ⊗ a ˜ K. Lemma 12.
Let ( a, m ) be a Koszul object of T and ˜ K as in Definition 11.(i) The dg bimodule ˜ K can be extended a dg equivalence ( ˜ K, φ ˜ K ) between ( a, m ) and ( a ! , m ! ) .(ii) We have a quasi-isomorphism from ( a ! , m ! ) to E ( a, m ) .Proof. The proof of (i) is analogous to the proof Lemma 9. The action of a ! inducesa quasi-isomorphism a ! → E ( a ) , from which part (ii) follows. (cid:3) Operators O and O . The operator O . In this subsection, we introduce the operator O , which weshowed to control the rational representation theory of GL ( F ) in [7], cf. Section6.Let A = ⊕ A k be a dg algebra, and M = ⊕ M k a dg A - A -bimodule. The tensoralgebra T A ( M ) is differential bigraded with k -degree by the total k -grading on eachcomponent M ⊗ j , with A in j -degree zero, and M in j -degree 1.Let ( a, m ) be a j -graded object of T . The tensor algebra T a ( m ) is a differentialtrigraded algebra, where a has i -degree zero and m has i -degree 1, and the j - k -bigrading is given by the total j - and the total k -grading on each component m ⊗ i . Definition 13.
Given a j -graded object ( a, m ) of T , we define an operator O a,m ↻ T given by O a,m ( A, M ) = ( ⊕ a jk ⊗ F M ⊗ A j , ⊕ m jk ⊗ F M ⊗ A j ) . The algebra structure on ⊕ a jk ⊗ F M ⊗ A j is the restriction of the algebra structureon the super tensor product of algebras a ⊗ T A ( M ) . The k -grading and differentialon the complex ⊕ a jk ⊗ M ⊗ A j are defined to be the total k -grading and totaldifferential on the tensor product of complexes. The bimodule structure, gradingand differential on ⊕ m jk ⊗ M ⊗ A j are defined likewise.Explicit formulae for the multiplication and differential on a ⊗ T A ( M ) are given asfollows: For γ, δ ∈ a, θ , . . . , θ r , ζ , . . . , ζ s ∈ M , OSZUL DUAL 2-FUNCTORS AND EXTENSION ALGEBRAS OF SIMPLE MODULES FOR GL ( γ ⊗ F ( θ ⊗ A ⋅ ⋅ ⋅ ⊗ A θ r ))( δ ⊗ F ( ζ ⊗ A ⋅ ⋅ ⋅ ⊗ A ζ s )) = ( − ) ∣ δ ∣ k (∑ rt = ∣ θ t ∣ k ) γδ ⊗ F ( θ ⊗ A ⋅ ⋅ ⋅ ⊗ A θ r ⊗ ζ ⊗ A ⋅ ⋅ ⋅ ⊗ A ζ s ) and d ( γ ⊗ F ( θ ⊗ A ⋅ ⋅ ⋅ ⊗ A θ r )) = d ( γ ) ⊗ F ( θ ⊗ A ⋅ ⋅ ⋅ ⊗ A θ r ) + ( − ) ∣ γ ∣ k r ∑ u = ( − ) ∑ u − t = ∣ θ t ∣ k γ ⊗ F ( θ ⊗ A ⋅ ⋅ ⋅ ⊗ A d ( θ u ) ⊗ A ⋅ ⋅ ⋅ ⊗ A θ r ) . Similar formulae describe the bimodule structure on m ⊗ T A ( M ) by taking γ or δ in m .We again remark that this can be phrased in a 2-categorical language, making O a,m into a 2-endofunctor of T .We sometimes write O a,m ( A, M ) = ( a ( A, M ) , m ( A, M )) . If a and b are differential bigraded algebras, and a x b is differential bigraded a - b -bimodule, concentrated in nonnegative j -degrees, then we have a differential graded a ( A, M ) - b ( A, M ) -bimodule x ( A, M ) ∶ = ⊕ j,k x jk ⊗ M ⊗ A j . Lemma 14.
Let a, b, c be a differential bigraded algebras, b x a and a y c differentialbigraded modules, all concentrated in nonnegative j -degrees. Let ( A, M ) be an objectof T . Then x ( A, M ) ⊗ a ( A,M ) y ( A, M ) ≅ ( x ⊗ a y )( A, M ) as differential bigraded b ( A, M ) - c ( A, M ) -bimodules.Proof. We define a map x ( A, M ) ⊗ a ( A,M ) y ( A, M ) → ( x ⊗ a y )( A, M ) sending ho-mogeneous elements ( α ⊗ m ⊗ ⋯ ⊗ m j ) ⊗ ( β ⊗ n ⊗ ⋯ ⊗ n j ) (where α ∈ x j , β ∈ y j and m , . . . m j , n , . . . , n j ∈ M to ( α ⊗ β ) ⊗ ( m ⊗ ⋯ ⊗ m j ⊗ n ⊗ ⋯ ⊗ n j ) . It isstraightforward to check that this is an isomorphism of differential graded b ( A, M ) - c ( A, M ) -bimodules. (cid:3) Suppose ( A, M ) is a Keller object of T . Then M − = Hom A ( M, A ) is a two-sidedtilting complex such that M − ⊗ A − ≅ Hom A ( M, − ) induces an inverse equivalence to M ⊗ A − . The natural map q ∶ M ⊗ A M − → A which takes m ⊗ ξ to ξ ( m ) representsthe counit of the adjunction ( M ⊗ A − , M − ⊗ A − ) , and is a quasi-isomorphism ofdg A - A -bimodules.We previously assumed that ( a, m ) was a j -graded object of T such that a jk = m jk = j <
0. If we assume rather that a jk = m jk = j >
0, we define O a,m ( A, M ) = ( ⊕ a jk ⊗ F ( M − ) ⊗ A − j , ⊕ m jk ⊗ F ( M − ) ⊗ A − j ) for a Keller object ( A, M ) ∈ T . Given a differential bigraded a -module x , with com-ponents in positive and negative j -degrees, we define x ( A, M ) to be the a ( A, M ) -module given by x ( A, M ) = ( ⊕ j < x j ● ⊗ ( M − ) ⊗ A − j ) ⊕ ( x ● ⊗ A ) ⊕ ( ⊕ j > x j ● ⊗ M ⊗ A j ) , where the action is defined via the action of a on x , along with the quasi-isomorphism q . It is straightforward to check that this is well-defined. Lemma 15.
Let c be a differential bigraded algebra, x and y are differential bigraded c -modules, all concentrated in nonnegative j -degrees, and let ( A, M ) be a Rickardobject of T . Then we have a quasi-isomorphism of differential bigraded vector spaces Hom c ( x, y )( A, M ) → Hom c ( A,M ) ( x ( A, M ) , y ( A, M )) . Proof.
We established this in a previous paper [7, Proof of Theorem 13]. In thatpaper we only consider the case where ( c, x = y ) is a Rickard object. However,exactly the same proof works in this more general case. (cid:3) Lemma 16.
Suppose ( a, m ) is a j -graded Rickard object and ( A, M ) is a Rickardobject of T . Then O a,m ( A, M ) is Keller object of T .Proof. We need to check that m ( A, M ) is in a ( A, M ) -perf ∩ perf- a ( A, M ) , whichfollows directly from the definition of O a,m and that the natural maps a ( A, M ) → End a ( A,M ) ( a ( A,M ) m ( A, M )) and a ( A, M ) → End a ( A,M ) ( m ( A, M ) a ( A,M ) ) are quasi-isomorphisms of dg algebras. The latter follows since we have quasi-isomorphismsof dg algebras a ( A, M ) → Hom a ( a m, a m )( A, M ) → Hom a ( A,M ) ( a ( A,M ) m ( A, M ) , a ( A,M ) m ( A, M )) and similarly for the other side. Here the first quasi-isomorphism comes from thefact that ( a, m ) is a Rickard object, and the second follows from Lemma 15. (cid:3) The operator O . In this subsection, we introduce the operator O , whoserelationship to O we will analyse in the next subsection, and whose main advantageis that it is well-behaved with respect to taking homology. Definition 17.
Let Γ = ⊕ Γ ijk be a differential trigraded algebra. We have anoperator O Γ ↻ { ∆ ∣ ∆ = ⊕ ∆ jk a differential bigraded algebra } given by(1) O Γ ( ∆ ) ik = ⊕ j,k + k = k Γ ijk ⊗ ∆ jk . The algebra structure and differential are obtained by restricting the algebra struc-ture and differential from Γ ⊗ ∆.If we forget the differential and the k -grading, the operator O Γ is identical to theoperator O Γ defined in the introduction. OSZUL DUAL 2-FUNCTORS AND EXTENSION ALGEBRAS OF SIMPLE MODULES FOR GL Lemma 18.
We have H O Γ ≅ H O H Γ ≅ O H Γ H , for a differential trigraded algebra Γ .Proof. Both isomorphisms are straightforward and based on the facts that thetensor products in (1) are over F , so H ( Γ ijk ⊗ ∆ jk ) ≅ H ( Γ ijk ) ⊗ H ( ∆ jk ) , andthat HH = H . Using this we obtain H ( O Γ ( ∆ )) ik ≅ H ( ⊕ j,k + k = k Γ ijk ⊗ ∆ jk ) ≅ ⊕ j,k + k = k H ( Γ ijk ) ⊗ H ( ∆ jk ) . Similarly, we have H ( O H Γ ( ∆ )) ik ≅ H ( ⊕ j,k + k = k H ( Γ ijk ) ⊗ ∆ jk ) ≅ ⊕ j,k + k = k H ( Γ ijk ) ⊗ H ( ∆ jk ) and ( O H Γ H ( ∆ )) ik ≅ ⊕ j,k + k = k H ( Γ ijk ) ⊗ H ( ∆ jk ) , so they agree as graded vector spaces. Since all three multiplications are inducedfrom H Γ ⊗ H ∆, the resulting differential bigraded algebras and therefore the oper-ators are isomorphic. (cid:3) Comparing O and O . Here we describe relations between the operators O and O . Throughout this subsection, let ( a, m ) be a j -graded object of T and ( A, M ) ∈ T .Note that the algebra T a ( m )( A, M ) , formed with respect to the j -grading on T a ( m ) , is a differential bigraded algebra, with T a ( m )( A, M ) ik = ⊕ j T a ( m ) ijk ⊗ M ⊗ j . The algebra T a ( A,M ) ( m ( A, M )) is a differential bigraded algebra, with T a ( A,M ) ( m ( A, M )) ik = ( m ( A, M ) ⊗ a ( A,M ) i ) k . We write X i ◇● ≅ Y i ◇● to signify that X ijk ≅ Y ijk for all j, k . Lemma 19. (i) We have an isomorphism of objects of T O a,m ( A, M ) = ( O T a ( m ) ( T A ( M )) ◇● , O T a ( m ) ( T A ( M )) ◇● ) , where the k -grading on the components of O a,m ( A, M ) can be identified withthe k -grading on O T a ( m ) ( T A ( M )) . (ii) We have an isomorphism of differential bigraded algebras O T a ( m ) ( T A ( M )) ≅ T a ( m )( A, M ) . (iii) We have an isomorphism of differential bigraded algebras T a ( m )( A, M ) ≅ T a ( A,M ) ( m ( A, M )) . Proof.
The isomorphisms in (i) and (ii) follow directly from the definition, and theisomorphism in (iii) follows from Lemma 14, which gives m ( A, M ) ⊗ a ( A,M ) m ( A, M ) ≅ ( m ⊗ a m )( A, M ) . (cid:3) Suppose we are given j -graded objects ( a i , m i ) in T for 1 ≤ i ≤ n . Let us define ( A i , M i ) recursively via ( A i , M i ) = O a i ,m i ( A i − , M i − ) and ( A , M ) = ( A, M ) . Lemma 20. (i) We have an algebra isomorphism T A n ( M n ) ≅ O T an ( m n ) ... O T a ( m ) ( T A ( M )) . (ii) We have an isomorphism of objects of T O a n ,m n ... O a ,m ( A, M ) ≅ ( O T a ( m ) ... O T an ( m n ) ( T A ( M )) ◇● , O T a ( m ) ... O T an ( m n ) ( T A ( M )) ◇● ) . Proof.
The first part follows from Lemma 19(ii),(iii) by induction. Part (ii) thenfollows from Lemma 19(i) and the first part. (cid:3)
The application of Lemma 20 which will be relevant for our computation of y is: Corollary 21.
Let ( a, m ) be a j -graded object of T . Then we have an isomorphismof differential bigraded algebras O F, O na,m ( F, F ) ≅ O F O n T a ( m ) ( F [ z ]) . Operators O respect dualities. In this section, we analyse how the operators O introduced in Section 4 behaveunder the various incarnations of homological dualities that we have introduced inSection 3.5.1. Operators O and dg equivalence.Lemma 22. Let ( A, M ) and ( B, N ) be objects of T such that ( A, M ) ⋗ ( B, N ) .Let ( a, m ) be a j -graded Rickard object of T . Then O a,m ( A, M ) ⋗ O a,m ( B, N ) .Proof. Let X denote an A - B -bimodule inducing the dg equivalence ( A, M ) ⋗ ( B, N ) .We define x = a ( A, M ) ⊗ A X and claim that this is a dg a ( A, M ) - a ( B, N ) -bimodule which can be extended to adg equivalence O a,m ( A, M ) ⋗ O a,m ( B, N ) . Clearly x is a left differential bigraded OSZUL DUAL 2-FUNCTORS AND EXTENSION ALGEBRAS OF SIMPLE MODULES FOR GL a ( A, M ) -module. We have a sequence of canonical homomorphisms of differentialbigraded algebras a ( B, N ) ≅ B ⊗ B a ( B, N ) ≅ Hom A ( X, X ) ⊗ B a ( B, N ) ≅ Hom A ( X, X ⊗ B a ( B, N )) → Hom A ( X, a ( A, M ) ⊗ A X ) ≅ Hom A ( X, Hom a ( A,M ) ( a ( A, M ) , a ( A, M ) ⊗ A X )) ≅ Hom a ( A,M ) ( a ( A, M ) ⊗ A X, a ( A, M ) ⊗ A X ) = Hom a ( A,M ) ( x, x ) , whose composition give x the structure of a a ( A, M ) - a ( B, N ) -bimodule. Here thethird isomorphism is by virtue of X being in A -perf. Every term in this sequenceis a quasi-isomorphism, which means their composite is also a quasi-isomorphism,as we require.We have a sequence of quasi-isomorphisms of differential bigraded a ( A, M ) - a ( B, N ) -bimodules x ⊗ a ( B,N ) m ( B, N ) = a ( A, M ) ⊗ A X ⊗ a ( B,N ) m ( B, N ) → ( a ⊗ a m )( A, M ) ⊗ A X ≅ m ( A, M ) ⊗ A X ≅ m ( A, M ) ⊗ a ( A,M ) a ( A, M ) ⊗ A X ≅ m ( A, M ) ⊗ a ( A,M ) x where the quasi-isomorphism is constructed analogously to the isomorphism ofLemma 14. This defines our quasi-isomorphism φ x needed to turn ( x, φ x ) intoa dg equvalence and completes the proof of Lemma 22. (cid:3) Lemma 23.
Let ( A, M ) and ( B, N ) be quasi-isomorphic Keller objects of T . Let ( a, m ) be a j -graded object of T . Then O a,m ( A, M ) and O a,m ( B, N ) are quasi-isomorphic objects of T .Proof. Since by virtue of ( A, M ) and ( B, N ) being Keller objects, the endofunc-tors M ⊗ A − , − ⊗ A M, N ⊗ B − , − ⊗ B N are exact, tensoring the quasi-isomorphism f ∶ A M A → B N B together i times produces the necessary quasi-isomorphisms f i ∶ A M ⊗ A iA → B N ⊗ N iB in each j -graded component. It is immediate that this iscompatible with the multiplicative resp. bimodule structures. (cid:3) Reversing gradings.
Let us denote by R the sign reversing operator R on dif-ferential bigraded algebras, which sends a differential bigraded algebra a = ⊕ j,k a j,k to itself with reverse grading, R a j,k = a − j,k . Likewise, if m is a differential bigraded a - a -bimodule, we define R m by R m j,k = m − j,k . Lemma 24.
Let ( a, m ) and ( b, n ) be graded Rickard objects of T , and ( A, M ) aRickard object of T . Then there is a quasi-isomorphism of objects of T , O R a, R m O b,n − ( A, M ) → O a,m O b,n ( A, M ) . Proof.
For any ( B, N ) ∈ T , we have an isomorphism(2) O a,m ( B, N ) ≅ O R a, R m ( B, N − ) , since both sides are given by the object ( ⊕ j a j ⊗ F N ⊗ A j , ⊕ j m j ⊗ F N ⊗ A j ) of T .Note that ( b, n ) being Rickard implies ( b, n − ) being Rickard, so, by Lemma 16,both O b,n ( A, M ) = ( b ( A, M ) , n ( A, M )) and O b,n − ( A, M ) = ( b ( A, M ) , n − ( A, M )) are Keller objects.Since ( A, M ) is Rickard, Lemma 15 gives us a quasi-isomorphism of dg b ( A, M ) - b ( A, M ) -bimodules n − ( A, M ) → Hom b ( A,M ) ( n ( A, M ) , b ( A, M )) and ( b ( A, M ) , n ( A, M )) being a Keller object implies ( b ( A, M ) , n ( A, M ) − ) = ( b ( A, M ) , Hom b ( A,M ) ( n ( A, M ) , b ( A, M ))) is a Keller object. We therefore have a quasi-isomorphism between Keller objects(3) ( b ( A, M ) , n − ( A, M )) → ( b ( A, M ) , n ( A, M ) − ) in T . The operator O R a, R m respects quasi-isomorphisms between Keller objects byLemma 23. Applying this to (3) and recalling (2) for the special case ( B, N ) = O b,n ( A, M ) , we obtain a quasi-isomorphism O R a, R m O b,n − ( A, M ) = O R a, R m ( b ( A, M ) , n − ( A, M )) → O R a, R m ( b ( A, M ) , n ( A, M ) − ) ≅ O a,m ( b ( A, M ) , n ( A, M )) = O a,m O b,n ( A, M ) in T as required. (cid:3) Koszul duality for operators O . We now consider how Koszul duality be-haves towards the operators O . Lemma 25.
Let ( A, M ) be a Rickard object of T . Let ( a, m ) be a Koszul objectof T , with Koszul dual ( a ! , m ! ) . Then we have a dg equivalence O a,m ( A, M ) ⋗ O a ! ,m ! ( A, M ) .Proof. Consider the differential bigraded a ( A, M ) - a ! ( A, M ) -bimodule K ( A, M ) .We show that K ( A, M ) is can be extended to a dg equivalence O a,m ( A, M ) ∼ O a ! ,m ! ( A, M ) in T . By Lemma 14, we have an isomorphism of differential bigraded a ( A, M ) - a ! ( A, M ) -bimodules K ( A, M ) ≅ a ( A, M ) ⊗ a ( A,M ) ∗ ( a ! )( A, M ) . The action of a ! on K induces a quasi-isomorphism of differential bigraded algebras a ! → Hom a ( K, K ) . Consequently, using [7, Lemma 15], we have a quasi-isomorphism of differentialbigraded algebras a ! ( A, M ) → Hom a ( K, K )( A, M ) OSZUL DUAL 2-FUNCTORS AND EXTENSION ALGEBRAS OF SIMPLE MODULES FOR GL which, when composed with the quasi-isomorphismHom a ( K, K )( A, M ) → Hom a ( A,M ) ( K ( A, M ) , K ( A, M )) of Lemma 15, gives us a quasi-isomorphism of differential bigraded algebras a ! ( A, M ) → Hom a ( A,M ) ( K ( A, M ) , K ( A, M )) which is induced by the action of a ! ( A, M ) on K ( A, M ) .The object K ( A, M ) generates D dg ( a ( A, M )) ≅ D dg ( a ! ( A, M )) , because K ( A, M ) is quasi-isomorphic to a ( A, M ) ≅ a ⊗ F A , and A generates D dg ( A ) . All thatremains for us to do is to establish a quasi-isomorphism of differential bigraded a ( A, M ) - a ! ( A, M ) -bimodules K ( A, M ) ⊗ a ! ( A,M ) m ! ( A, M ) → m ( A, M ) ⊗ a ( A,M ) K ( A, M ) However, this follows from Lemma 14, and the fact that we have a quasi-isomorphismof differential bigraded a - a ! -bimodules K ⊗ a ! m ! → m ⊗ a K. (cid:3) A quasi-isomorphism of operators.
Here we show that, under good con-ditions, the operator EO a,m is quasi-isomorphic to O a ! ,m ! E . Theorem 26.
Let ( A, M ) be a Rickard object of T . Let ( a, m ) be a be a Koszulobject of T , such that O a,m ( A, M ) is again a Rickard object. We have a chain ofdg equivalences E ( O a,m ( A, M )) ⋖ O a,m ( A, M ) ⋗ O a ! ,m ! ( A, M ) ⋗ O a ! ,m ! ( E ( A, M )) . Proof.
This follows from Lemmas 9, 22 and 25. (cid:3)
Theorem 26 implies we have a dg equivalence between the objects E ( O a,m ( A, M )) and O a ! ,m ! ( E ( A, M )) of T . We strengthen this as follows: Theorem 27.
The chain of equivalences in Theorem 26 lifts to a quasi-isomorphismin T from O a ! ,m ! ( E ( A, M )) to E ( O a,m ( A, M )) .Proof. We need to prove that E ( a ( A, M )) is quasi-isomorphic to a ! ( E ( A, M )) as adg algebra and that there is a compatible quasi-isomorphism between E ( m ( A, M )) and m ! ( E ( A, M )) as in Definition 7. We start with the first assertion. By definition, E ( a ( A, M ) is obtained by the following recipe: first take a projective resolutionof the simple a ( A, M ) -modules, then take its endomorphism ring. It is thereforeenough to show that we can compute a ! ( E ( A, M )) by taking a projective resolutionof the simple a ( A, M ) -modules, and then taking the endomorphism ring.We denote by A a direct sum of a complete set of nonisomorphic simple A -modules.Then a ⊗ F A is a direct sum of a complete set of nonisomorphic simple a ( A, M ) -modules. Let P ● A denote a minimal projective resolution of A A , and P ● a a minimalprojective resolution of a a . We have E ( A ) = End ( P ● A ) . A projective resolution of the a ( A, M ) -module a ⊗ A is given by P a ( A, M ) ⊗ A P A . We can therefore write E ( a ( A, M )) = End ( P a ( A, M ) ⊗ A P A ) . We have a sequence of quasi-isomorphisms a ! ( E ( A ) , E ( M )) ≅ Hom A ( P A , a ! ( A, M ) ⊗ A P A ) by Lemma 10 , → Hom A ( P A , E ( a )( A, M ) ⊗ A P A ) by Koszul duality, and projectivity of P A , ≅ Hom A ( P A , Hom a ( P a , P a )( A, M ) ⊗ A P A ) by definition of E ( a ) , ≅ Hom A ( P A , Hom a ( A,M ) ( P a ( A, M ) , P a ( A, M )) ⊗ A P A ) by Lemma 15, ≅ Hom A ( P A , Hom a ( A,M ) ( P a ( A, M ) , P a ( A, M ) ⊗ A P A )) by projectivity and finite generation of P A , ≅ Hom a ( A,M ) ( P a ( A, M ) ⊗ A P A , P a ( A, M ) ⊗ A P A ) by adjunction, ≅ End ( P a ( A, M ) ⊗ A P A ) = E ( a ( A, M )) . We likewise have a sequence of quasi-isomorphisms of dg vector spaces m ! ( E ( A ) , E ( M )) ≅ Hom A ( P A , m ! ( A, M ) ⊗ A P A ) → Hom A ( P A , E ( m )( A, M ) ⊗ A P A ) ≅ Hom A ( P A , Hom a ( P a , m ⊗ a P a )( A, M ) ⊗ A P A ) ≅ Hom A ( P A , Hom a ( A,M ) ( P a ( A, M ) , m ( A, M ) ⊗ a ( A,M ) P a ( A, M )) ⊗ A P A ) ≅ Hom A ( P A , Hom a ( A,M ) ( P a ( A, M ) , m ( A, M ) ⊗ a ( A,M ) P a ( A, M ) ⊗ A P A )) ≅ Hom a ( A,M ) ( P a ( A, M ) ⊗ A P A , m ( A, M ) ⊗ a ( A,M ) P a ( A, M ) ⊗ A P A ) = E ( m ( A, M )) which are compatible with the bimodule structures in the sense of Definition 7. (cid:3) In light of our interest in the object E ( O q c , t ( F, F )) (see Section 6), we record thefollowing corollary. Corollary 28.
Let ( a, m ) be a be a Koszul object of T , such that O qa,m ( F, F ) isa Rickard object for all q ≥ . Then E ( O qa,m ( F, F )) and O qa ! ,m ! ( F, F ) are quasi-isomorphic in T .Proof. This is proved by induction on q . The base step is given by Lemma 12 (ii), sowe can inductively assume E ( O q − a,m ( F, F )) is quasi-isomorphic to O q − a ! ,m ! ( F, F ) . ByTheorem 27 and the fact that O q − a,m ( F, F ) is a Rickard object of T , the dg algebra E ( O qa,m ( F, F )) is quasi-isomorphic to O a ! ,m ! E ( O q − a,m ( F, F )) . The latter is in turn,by Lemma 23 and the induction hypothesis, quasi-isomorphic to O qa ! ,m ! ( F, F ) . (cid:3) OSZUL DUAL 2-FUNCTORS AND EXTENSION ALGEBRAS OF SIMPLE MODULES FOR GL Recollections of GL . Let Z denote the algebra given by the quiver ⋯ ● ξ ( ( ● η h h ξ ( ( ● η h h ξ ( ( ● η h h ⋯ , modulo relations ξ = η = ξη + ηξ =
0. We will call this the zigzag algebra andit or its various truncations show up in many guises in representation theory. Itis therefore a very well-studied little infinite dimensional algebra whose projectiveindecomposable modules have a Loewy structure given by L ( l ) qqq ▼▼▼ L ( l − ) ▼▼▼ L ( l + ) qqq L ( l ) where L ( l ) denotes the simple module at vertex l , and is well-known to have anumber of interesting homological properties, all of which are easily checked byhand. For example, it is quasi-hereditary, symmetric, and Koszul.We denote by τ the algebra involution of Z which sends vertex i to vertex p − i and exchanges ξ and η . Let e l denote the idempotent of Z corresponding to vertex l ∈ Z . Let t = ∑ ≤ l ≤ p, ≤ m ≤ p − e l Ze m . Then t admits a natural left action by the subquotient c of Z given by c = F ( ● ξ ( ( ● η k k ξ ( ( ● η h h ⋯ p − ● ξ p − + + p ● )/ I η p − j j , where I = ( ξ l + ξ l , η l η l + , ξ l η l + η l + ξ l + , ξ p − η p − ∣ ≤ l ≤ p − ) . By symmetry, t admits a right action by c , if we twist the regular right action by τ . In this way, t is naturally a c - c -bimodule. It is straightforward to see that the left restriction c t of t is a full characteristic tilting module for the quasi-hereditary algebra c . Thenatural homomorphism c → Hom ( c t , c t ) defined by the right action of c on t is anisomorphism, implying that c is Ringel self-dual.Note, in particular, that c and t both graded by path length. If we let ˜ t denotea graded projective resolution of t as a c - c -bimodule, then ˜ t is a two-sided tiltingcomplex, and ˜ t ⊗ c − induces a self-equivalence of the derived category D b ( c ) of c .Alternatively, we can (and will) interpret c as a differential bigraded algebra (withzero differential) with the j -grading given by path length, and by declaring c asconcentrated in k -degree 0. Then ˜ t is a differential bigraded c - c -bimodule and t is a differential bigraded c - c -bimodule concentrated in k -degree zero and with zerodifferential, making ( c , ˜ t ) and ( c , t ) into objects of T . In particular, since c asa quasi-hereditary algebra has finite global dimension, we have ˜ t ∈ A -perf and itbeing a 2-sided tilting complex means that ( c , ˜ t ) is a Keller object. Moreover, since c equals its homology and is finite-dimensional and of finite global dimension, ( c , ˜ t ) is Rickard object of T . The j -grading on both c and ˜ t together with Koszulity of c , which is easily checked by hand, now implies that ( c , ˜ t ) is even a Koszul object,which we will use extensively in the next section.We will now recall some facts about the rational representation theory of G = GL ( F ) . The category of polynomial representations of G of degree r is equivalentto the category S ( , r ) -mod of representations of the Schur algebra S ( , r ) [4]. Ablock of S ( , r ) is Ringel self-dual if and only if it has p q simple modules [3]. In,[6].[7] we developed a combinatorial way to describe these blocks, which we nowdescribe.The operator O c , t acts on T , which containts the pair ( F, F ) whose (dg) algebrais F and whose (dg) bimodule is the regular bimodule F . Since both c and t havezero differential, repeated application of O c , t to the pair ( F, F ) produces an elementof T where both components have zero differential and can hence regarded as analgebra and a bimodule respectively. The operator O F, takes an object ( A, M ) in T to the pair ( A, ) , which we will identify with the dg algebra A .We define b q to be the category of modules over the algebra O F, O q c , t ( F, F ) . Wehave an algebra homomorphism c → F which sends a path in the quiver to 1 ∈ F if it is the path of length zero based at 1, and 0 ∈ F otherwise. This algebrahomomorphism lifts to a morphism of operators O c , t → O F, . We have O F, = O F, ;We thus have a natural sequence of operators O F, ← O F, O c , t ← O F, O c , t ← ..., which, if we apply each term to ( F, F ) and take representations, gives us a sequenceof embeddings of abelian categories b → b → b → ... We denote by b the union of these abelian categories. In a previous paper [7], wehave proved the following theorem: Theorem 29. [7, Corollary 21, Corollary 27]
Every block of G -mod is equivalentto b . Every block of S ( , r ) -mod whose number of isomorphism classes of simpleobjects is p q is equivalent to b q . To compute the Yoneda extension algebra Y of G -mod, it is enough to computethe Yoneda extension algebra y of the principal block of G -mod and, thanks to theabove theorem, we can identify y with the Yoneda extension algebra of b . Since eachembedding b q → b q + corresponds to taking an ideal in the poset of irreducibles for b q + , the theory of quasi-hereditary algebras [2] gives us a sequence of fully faithfulembeddings of derived categories D b ( b ) → D b ( b ) → D b ( b ) → ... Therefore, if we define y q to be the Yoneda extension algebra of b q , we have asequence of algebra embeddings y → y → y → ... whose union is equal to y . OSZUL DUAL 2-FUNCTORS AND EXTENSION ALGEBRAS OF SIMPLE MODULES FOR GL Reduction.
The pair ( Ω , Ψ ) . Let c denote the quasi-hereditary algebra with p irreduciblemodules introduced in Section 6, and t its tilting bimodule and ˜ t a two-sided tiltingcomplex quasi-isomorphic to t . Lemma 30.
We have a quasi-isomorphism in T between O q c , ˜ t ( F, F ) and O q c , t ( F, F ) ,in particular O q c , ˜ t ( F, F ) is a Rickard object for any q ≥ .Proof. In [7, Lemma 22], we showed that for a quasi-hereditary Ringal self-dualalgebra A , its tilting bimodule T and a projective bimodule resolution t of T , thereis a quasi-isomorphism of differential bigraded algebras ϕ , ∶ c ( A, t ) → c ( A, T ) .The proof relied only on the fact that c is concentrated in j -degrees 0 , ,
2. As thesame is true for t , we obtain a quasi-isomorphism ϕ , ∶ t ( A, t ) → t ( A, T ) which iscompatible with ϕ , in the sense of Definition 7.Using this, we prove the lemma by induction on q , the case q = q −
1, i.e. we have a quasi-isomorphism O q − c , ˜ t ( F, F ) → O q − c , t ( F, F ) , given by a quasi-isomorphisms ϕ q − , ∶ c ( O q − c , ˜ t ( F, F )) → c ( O q − c , t ( F, F )) and ϕ q − , ∶ ˜ t ( O q − c , ˜ t ( F, F )) → t ( O q − c , t ( F, F )) . Since t ( O q − c , t ( F, F )) is, by [7, Proposition 20], the tilting bimodule for the quasi-hereditary algebra c ( O q − c , t ( F, F )) , tensoring with it is exact on standard filteredmodules (cf. [7, Lemma 22]), in particular on the tilting module itself. On the otherhand, ˜ t ( O q − c , ˜ t ( F, F )) is a two-sided tilting complex, hence tensoring with it is exact,so we can tensor the latter quasi-isomorphism once to obtain a quasi-isomorphism˜ t ( O q − c , ˜ t ( F, F )) ⊗ c ( O q − c , ˜ t ( F,F )) → t ( O q − c , t ( F, F )) ⊗ c ( O q − c , t ( F,F )) . Since c is concentrated in degrees 0 , ,
2, this suffices to obtain a quasi-isomorphismof differential bigraded algebras ϕ q, ∶ c ( O q − c , ˜ t ( F, F )) → c ( O q − c , t ( F, F )) . In order toobtain the necessary quasi-isomorphism ϕ q, ∶ ˜ t ( O q − c , ˜ t ( F, F )) → t ( O q − c , t ( F, F )) wecompose quasi-isomorphisms˜ t ( O q − c , ˜ t ( F, F )) → t ( O q − c , ˜ t ( F, F )) → t ( O q − c , t ( F, F )) where the second quasi-isomorphism is again constructed by virtue of t being con-centrated in degrees 0 , , ϕ q, is compatiblewith the bimodule structures as required in Definition 7, hence this shows that O q c , ˜ t ( F, F ) and O q c , t ( F, F ) are quasi-isomorphic in T .To see that O q c , ˜ t ( F, F ) = ( c ( O q − c , ˜ t ( F, F )) , ˜ t ( O q − c , ˜ t ( F, F ))) is Rickard, we note that itis Keller since ˜ t ( O q − c , ˜ t ( F, F )) is a two-sided tilting complex. Furthermore ϕ q, givesus a quasi-isomorphism between c ( O q − c , ˜ t ( F, F )) and its homology c ( O q − c , t ( F, F )) ,which is a finite-dimensional algebra of finite global dimension, as it is Moritaequivalent to a block of a Schur algebra (cf. Section 6). Therefore O q c , ˜ t ( F, F ) isRickard. (cid:3) We can hence apply our homological theory to the Koszul object ( c , ˜ t ) ∈ T (cf. Sec-tion 6) and thus define a differential bigraded c ! - c ! -bimodule t ! according to Defi-nition 11. In light of Corollary 28, we wish to compute (the homology of) iteratedapplications of the operator O c ! , t ! to the pair ( F, F ) . However, c ! is now negativelygraded so to evaluate this we need to work with the adjoint of t ! , which is given bythe differential bigraded c ! - c ! -bimodule t ! − = Hom c ! ( t ! , c ! ) . Since t ! is in c ! -perf (it is quasi-isomorphic to the tilting module for c ! , and thelatter, as a quasi-hereditary algebra, has finite global dimension), we have an iso-morphism of functors t ! − ⊗ c ! − ≅ Hom c ! ( t ! , − ) . For notational simplicity, we wish to describe algebras with positive gradings ratherthan negative gradings. Therefore rather than working with the negatively gradedalgebra c ! , we work with the positively graded algebra Ω, which is canonicallyisomorphic to c ! as an algebra, but whose j th homogeneous component is equalto the − j th homogeneous component of c ! . We define the differential bigraded Ω-Ω-bimodule Ψ to be the differential bigraded c ! - c ! -bimodule t ! − , whose ( j, k ) th homogeneous component is equal to the ( − j, k ) th homogeneous component of c ! .Thus Ω = R c ! and Ψ = R t ! − . Note that Ω = F ( ● x ( ( ● y k k x ( ( ● y h h ⋯ p − ● x p − , , p ● )/ I ⊥ y p − j j , where I ⊥ = ( x l y l − y l + x l + , y x ∣ ≤ l ≤ p − ) .Since we consider Ω as the extension algebra of c (with the j -grading multipliedby − k -degrees with zerodifferential. The generators x and y are in k -degree 1. For our computation of y we will use the following corollary of Lemma 24: Corollary 31.
We have a quasi-isomorphim of differential graded algebras O F, O q Ω , Ψ ( F, F ) → O F, O q c ! , t ! ( F, F ) . Proof.
We prove a slightly stronger statement, namely that O q Ω , Ψ ( F, F ) ≅ ( c ( O q − c ! , t ! ( F, F )) , t ! − ( O q − c ! , t ! ( F, F ))) = O c ! , t ! − ( O q − c ! , t ! ( F, F )) from which the Corollary follows by applying O F, .We proceed by induction on q . The case q = R c ! = Ωand R t ! − = Ψ. Now assume we have a quasi-isomorphism O q − , Ψ ( F, F ) → ( c ( O q − c ! , t ! ( F, F )) , t ! − ( O q − c ! , t ! ( F, F ))) . OSZUL DUAL 2-FUNCTORS AND EXTENSION ALGEBRAS OF SIMPLE MODULES FOR GL Then O q Ω , Ψ ( F, F ) = O R c ! , R t ! − ( O q − , Ψ ( F, F )) → O R c ! , R t ! − ( c ( O q − c ! , t ! ( F, F )) , t ! − ( O q − c ! , t ! ( F, F ))) → O c ! , t ! − ( c ( O q − c ! , t ! ( F, F )) , t ! ( O q − c ! , t ! ( F, F ))) = O c ! , t ! − ( O q − c ! , t ! ( F, F )) where the first equality is just by definition, the second isomorphism is by theinductive assumption and Lemma 23, the third is the quasi-isomorphism given inLemma 24 and the final equality is just the definition. (cid:3) A chain of isomorphisms.
Let us recapitulate what we have done so fartowards our goal of computing the extension algebra y q of a block of GL ( F ) with p q simples modules. We have the following sequence of isomorphisms y q ≅ HO F, EO q c , t ( F, F ) Theorem 29 ≅ HO F, EO q c , ˜ t ( F, F ) Lemma 30 ≅ HO F, O q c ! , t ! E ( F, F ) Corollary 28 ≅ HO F, O q c ! , t ! ( F, F ) E ( F, F ) = ( F, F ) ≅ HO F, O q Ω , Ψ ( F, F ) Corollary 31 ≅ H O F O q T Ω ( Ψ ) ( F [ z ]) Corollary 21 ≅ O H F O q HT Ω ( Ψ ) H ( F [ z ]) Lemma 18 ≅ O F O q HT Ω ( Ψ ) ( F [ z ]) H F = F, H ( F [ z ]) = F [ z ] . Recalling our definition of λ q = O F O q Λ ( F [ z ]) and our goal of proving an isomor-phism of y q and λ q , we therefore focus our attention on on proving the followingresult: Theorem 32.
We have an isomorphism of ijk -trigraded algebras H ( T Ω ( Ψ )) ≅ Λ . This will take up most of the remainder of this article, and is largely independentof the preceding subsections, whose aim it was to establish the above sequence ofisomorphisms. 8.
An explicit dg algebra.
The homology H ( Ψ ) . This subsection is concerned with verifying the state-ment of Theorem 32 for i -degree one.In the following, we will write differential k -graded modules as complexes by spec-ifying the 0-th homological position and interpreting ⋯ → M s → M s + → . . . as ⊕ M s [ − s ] . Warning:
The modules M s may carry their own k -grading, and will generallydo so whenever we look at modules over Ψ, which, unlike c , is not an algebraconcentrated in k -degree zero. In order to simplify our computations and to use results from the theory of Koszulduality in their original setting, we will therefore initially work in the (bounded)derived category of j -graded Ω-modules (e.g. in Lemma 33, Lemma 34, Lemma35, Lemma 37(i),(ii)), ignoring any internal k -grading of the algebra and thenestablishing the k -grading on the j -graded bimodules subsequently.In order to compute the homology of Ψ, which we need for our computation of H ( T Ω ( Ψ )) , we first compute the functor t − ⊗ − on various c -modules.The algebra c has irreducible modules L ( l ) indexed by integers 1 ≤ l ≤ p . Thealgebra has standard modules ∆ ( l ) which have top L ( l ) and socle L ( l − ) in case2 ≤ l ≤ p , and ∆ ( ) = L ( ) . The algebra has costandard modules ∇ ( l ) which havesocle L ( l ) and top L ( l − ) in case 2 ≤ l ≤ p , and ∇ ( ) = L ( ) . We define L ( l ) , ∆ ( l ) and ∇ ( l ) as j -graded modules by insisting their tops are concentrated in degree 0. Lemma 33.
We have exact triangles in the derived category of j -graded c -modulesas follows: L ( p − l ) → t − ⊗ c L ( l ) → L ( p )⟨ − l ⟩[ − l ] ↝ , for ≤ i < p , and → t − ⊗ c L ( p ) → L ( p )⟨ − p ⟩[ − p ] ↝ . Proof.
We can identify t − ⊗ c − with Hom c ( t , − ) , which by Ringel duality takescostandard modules ∇ to standard modules ∆. Easy computations establish thefollowing formulae: t − ⊗ c ∇ ( l ) = ∆ ( p − l + ) , for 2 ≤ l ≤ p and t − ⊗ c ∇ ( ) = ∆ ( p )⟨ − ⟩ . Since we have a quasi-isomorphism L ( l ) ≅ ( ∇ ( l )⟨ − ⟩ → ∇ ( l − )⟨ − ⟩ → ⋅ ⋅ ⋅ → ∇ ( )⟨ − ( l − )⟩ → ∇ ( )⟨ − ( l − )⟩ we obtain t − ⊗ c L ( l ) = ( ∆ ( p − l + )⟨ − ⟩ → ∆ ( p − l + )⟨ − ⟩ → ... → ∆ ( p )⟨ − l ⟩) for 1 ≤ l ≤ p , where in both complexes the leftmost terms lie in homological position0. For 1 ≤ l ≤ p −
1, this is exact except in homological position 0, where the kernelis L ( p − l ) and k -degree l −
1, where the cokernel is L ( p )⟨ − l ⟩ . For l = p , this is exactexcept in homological position p −
1, where the cokernel is L ( p )⟨ − p ⟩ . Noting thata simple L in homological position t is the same as L [ − t ] , we obtain the statementof the lemma. (cid:3) We now compute a one-sided tilting complex which is quasi-isomorphic to Ψ. Theadvantage of this complex over Ψ is that its structure is extremely explicit, makingcomputations possible.Consider the differential jk -bigraded Ω-module which, written as a two-term com-plex W = ( W d W → W ) , is given byΩ e p ⊗ F e p F Qe → Ω e OSZUL DUAL 2-FUNCTORS AND EXTENSION ALGEBRAS OF SIMPLE MODULES FOR GL with Ω e in homological position zero, where Q is the type A p subquiver of the quiverof Ω generated by arrows x , where F Q is the corresponding hereditary subalgebraof Ω, where e = ∑ ≤ l ≤ p − e l is the sum over the idempotents e l at l and where thedifferential on the complex is given by the algebra product. In other words, W isgiven by p − ⊕ l = ( Ω e p ⟨ l ⟩ → Ω e p − l ) , where the maps are given by right multiplication with x l . Consider also the dif-ferential jk -bigraded Ω-module which, written as a two term complex, is givenby Ω e p ⟨ p ⟩ → X denote the sum of these two-term complexes. Lemma 34.
We have a quasi-isomorphism between complexes of j -graded Ω -modules X and Ψ .Proof. Under Koszul duality an irreducible c -module corresponds to a projective c ! -module, hence a projective Ω-module. A shift ⟨ j ⟩ for c -modules corresponds toa shift ⟨ j ⟩[ − j ] for c ! -modules, therefore a shift ⟨ − j ⟩[ − j ] for Ω-modules (again thehomological grading in the derived category of Ω-modules does not coincide withthe k -grading since in the first Ω Ω is concentrated in a single degree whilst in thesecond Ω Ω is concentrated in many degrees). Lemma 33 therefore gives us exacttriangles in the derived category of j -graded Ω-modules as follows:Ω e p − l → Ψ ⊗ Ω Ω e l → Ω e p ⟨ l ⟩[ ] ↝ , for 1 ≤ l < p , and 0 → Ψ ⊗ Ω Ω e p → Ω e p ⟨ p ⟩[ ] ↝ . Here Ω e l denotes the projective indecomposable Ω-module indexed by l . We there-fore have exact triangles in the derived category of j -graded Ω-modulesΩ e p ⟨ l ⟩ → Ω e p − l → Ψ ⊗ Ω Ω e l ↝ , for 1 ≤ l < p . Morphisms in this derived category between projective objects liftto morphisms of chain complexes. The object Ψ ⊗ Ω Ω e l is indecomposable. Up toa scalar, there is a unique graded homomorphism φ from Ω e p ⟨ l ⟩ to Ω e p − l . Thisimplies we have an isomorphism in the derived category of j -graded Ω-modulesΨ ⊗ Ω Ω e l ≅ ( Ω e p ⟨ l ⟩ φ → Ω e p − l ) , or to put it another way, an isomorphismΨ ⊗ Ω Ω e l ≅ ( Ω e p ⊗ F e p F Qe p − l → Ω e p − l ) , where the term Ω e p − l is concentrated in homological degree 0, and the differentialis given by multiplication. We thus find X is quasi-isomorphic to Ψ as a complexof j -graded Ω-modules as required. (cid:3) Lemma 35.
We have an exact sequence of j -graded Ω -modules, → Ω e l ⟨ p ⟩ → Ω e p ⟨ l ⟩ → Ω e p − l → Θ e p − l → , for ≤ l ≤ p − . Proof.
This is a direct computation visible from the Loewy structure of these smallmodules, as described in Section 1. (cid:3)
Taking the first two terms in the sum from 1 to p − W ⟨ p ⟩ and W , respectively. We define the sum of the mapsbetween these terms to be δ ∶ W ⟨ p ⟩ → W . Let γ denote the automorphism of Ω that sends a homogeneous element ω to ( − ) ∣ ω ∣ k ω . The following is an easy direct computation. Lemma 36. γ ↺ Ω is an inner automorphism that sends ω to gωg − , where g = ∑ ( − ) l e l . The next lemma establishes the aim of this subsection, by showing that the algebrasΛ and H ( Ψ ) are isomorphic in i -degree one. Lemma 37. (i) X is a j -graded tilting complex for Ω .(ii) We have an isomorphism of j -graded algebras H End Ω ( X ) ≅ Ω . (iii) As a jk -graded left Ω -module, the kernel of the differential on X is isomorphicto Ω ⟨ p ⟩[ − p ] , and the cokernel of the differential on X is isomorphic to Θ σγ .(iv) We have isomorphisms of jk -bigraded Ω - Ω -bimodules with zero differential Λ ◇● ≅ Ω ⟨ p ⟩[ − p ] ⊕ Θ σ ≅ Ω ⟨ p ⟩[ − p ] ⊕ ( Θ σ ) γ ≅ H ( X ) ≅ H ( Ψ ) . Before proving the lemma, let us pause a moment on the statement of part (iv).Implicit is the existence of a Ω-Ω-bimodule structure on H ( X ) . The reason forthe existence of such a structure is the action of H End Ω ( X ) on H ( X ) , and theisomorphism H End Ω ( X ) ≅ Ω of part (ii).
Proof. (i) and (ii). As the image of the autoequivalence t ⊗ c − of D b ( c ) under Koszulduality (and grading reversal), the functor Ψ ⊗ Ω − induces an autoequivalence of D b ( Ω ) . The image of the regular module Ω Ω under a derived auto equivalence ↻ D b ( Ω ) is necessarily a tilting complex whose endomorphism ring is isomorphicto Ω.(iii) The isomorphism as j -graded left Ω-modules is immediate from the definitionof X and Lemma 35. The k -grading is forced upon us by the requirement thatthe differential has k -degree one, the existing k -grading on Ω and the fact thatthe term Ω e p − l in homological position zero does indeed appear without any shiftsas the image of the (unshifted) simple c -module L ( p − l ) under Koszul duality.Indeed, since x p − l has k -degree p − l , but the map given by right multiplication hasto have k -degree one, the top of Ω e p in the summand Ω e p → Ω e l of X has to beconcentrated in k -degree p − l −
1, meaning that (since our k -grading conventionfollows that of the derived category, see Section 3.1) the corresponding k -graded OSZUL DUAL 2-FUNCTORS AND EXTENSION ALGEBRAS OF SIMPLE MODULES FOR GL version of the summand is Ω e p [ + l − p ] → Ω e l . The top of the homology givenby the kernel of these maps (in every summand) is concentrated in k -degree p − H ( X ) is shifted by [ − p ] .(iv) The first isomorphism follows from definition of Λ. Indeed Λ ◇● = Ω ⊗ ζ ⊕ Θ σ ⊗ ζ is a formal variable placed in jk -degree ( p, p − ) (noting that the degree of the top of Ω ⟨ p ⟩[ − p ] is concentratedin j -degree p and k -degree p −
1, see Section 3.1). The second isomorphism holdsby Lemma 36.The remaining isomorphisms as left jk -bigraded Ω-modules follow from (iii) andLemma 34. To establish these isomorphisms of Ω-Ω-bimodules, we need to examinethe right action of Ω on X .There are two obvious endomorphisms ˜ y and ˜ x of X given by the commutativediagrams(4) Ω e p ⟨ p − l ⟩ ⋅ x p − l / / (cid:15) (cid:15) Ω e l ⋅ x (cid:15) (cid:15) Ω e p ⟨ p − l + ⟩ ⋅ x p − l + / / Ω e l − (5) Ω e p ⟨ p − ⟩ ⋅ x p − / / (cid:15) (cid:15) Ω e (cid:15) (cid:15) Ω e p ⟨ p ⟩ / / e p ⟨ p − l ⟩ ⋅ x p − l / / ⋅ xy (cid:15) (cid:15) Ω e l ⋅ y (cid:15) (cid:15) Ω e p ⟨ p − l − ⟩ ⋅ x p − l − / / Ω e l + (7) Ω e p ⟨ p ⟩ / / ⋅ xy (cid:15) (cid:15) (cid:15) (cid:15) Ω e p ⟨ p − ⟩ ⋅ x p − / / Ω e respectively. On the kernel of the differential which is as a left module isomorphicto Ω, it is easy to see that these endomorphisms satisfy the relations of Ω. Indeed,in (4) the kernel of the first row is Ω e p − l and the kernel of the second row is Ω e p − l + and the induced action on there is right multiplication by y . Similary, the inducedaction on homology of ˜ x is multiplication by x on the kernel of the differential. To obtain our twist by γ , i.e. our claim on signs of this action, we phrase thispurely in terms of dg bimodules. As a dg-bimodule, the first row in (4) is equal toΩ e p ⟨ p − l ⟩[ + l − p ] ⊕ Ω e l with differential given byΩ e p ⟨ p − l ⟩[ + l − p ] ⊕ Ω e l ⎛⎝ ⋅ x p − l [ p − l − ] ⎞⎠ / / Ω e p ⟨ p − l ⟩[ + l − p ] ⊕ Ω e l where the shifts in k -grading are again forced on us taking into account that x has k -degree 1 and the differential needs to have k -degree 1. So (4) translates toΩ e p ⟨ p − l ⟩[ + l − p ] ⊕ Ω e l ⎛⎝ [ − ]
00 0 ⎞⎠ (cid:15) (cid:15) ⎛⎝ ⋅ x p − l [ p − l − ] ⎞⎠ / / Ω e p ⟨ p − l ⟩[ + l − p ] ⊕ Ω e l ⎛⎝ ⋅ x ⎞⎠ (cid:15) (cid:15) Ω e p ⟨ p − l + ⟩[ l − p ] ⊕ Ω e l ⎛⎝ ⋅ x p − l + [ p − l ] ⎞⎠ / / Ω e p ⟨ p − l + ⟩[ l − p ] ⊕ Ω e l In order to make the dg-formalism ( a.m ) d = a. ( m ) d, ( m.a ) d = ( − ) ∣ a ∣ k m ( d ) .a described in Section 3.1 work, we see that for our generator for our generator˜ y ≅ End D b ( Ω ) ( X ) , we need (setting m = e p and a = ˜ y ), that(8) e p ˜ y ⋅ x p − l [ p − l − ] = − e p ⋅ x p − l + [ p − l ] ⋅ ˜ y, and similarly for the action of ˜ x . Therefore, choosing the isomorphism End D b ( Ω ) ( X ) → Ω to send ˜ x and ˜ y to x and y respectively, we need to twist the action on Θ σ by γ on the right, i.e. ˜ y will correspond to right multiplication by − x on Θ, which then,picking up the sign from (8), will make our diagrams commute. (cid:3) The homology H ( Ψ ⊗ Ω i ) . In order to prove that Λ is isomorphic to H ( T Ω ( Ψ )) ,we now need to consider higher i -degrees, in other words we need to compute thehomology of Ψ ⊗ Ω i .Consider the complex X i of Ω-modules, obtained by splicing together n copies of X . To be more precise, let X i denote the complex X ⟨( i − ) p ⟩ → W ⟨( i − ) p ⟩ → ... → W ⟨ p ⟩ → W → W whose differentials are obtained via the composition W ⟨ p ⟩ / / d W ⟨ p ⟩ $ $ ■■■■■■■■■ W W ⟨ p ⟩ . δ ; ; ①①①①①①①①① In other words, X i is the direct sum ofΩ e p ⟨ ip ⟩ → → ... → OSZUL DUAL 2-FUNCTORS AND EXTENSION ALGEBRAS OF SIMPLE MODULES FOR GL where the nonzero term is in the homological position − i , and p − ⊕ l = ( Ω e p ⟨ l + ( i − ) p ⟩ → Ω e p ⟨ p − l + ( i − ) p ⟩ → ... → Ω e p ⟨( p − l ) + p ⟩ → Ω e p ⟨ l ⟩ → Ω e p − l ) if i is odd or p − ⊕ l = ( Ω e p ⟨ l + ( i − ) p ⟩ → Ω e p ⟨ p − l + ( i − ) p ⟩ → ... → Ω e p ⟨ l + p ⟩ → Ω e p ⟨ p − l ⟩ → Ω e l ) if i is even. Here again the maps right multiplication by x l respectively x p − l for therightmost arrow and right multiplication by the appropriate power of xy dictatedby the grading for the remaining arrows.We define Ψ i to be Ψ ⊗ Ω i . Lemma 38. (i) We have a quasi-isomorphism of differential bigraded Ω -modules X i ≅ Ψ i . (ii) We have isomorphisms of jk -bigraded Ω - Ω -bimodules with zero differential Λ i ◇● ≅ Ω ⟨ ip ⟩[ i ( − p )] ⊕ Θ σ ⟨( i − ) p ⟩[( i − )( − p )] ⊕ ... ⊕ Θ σ i ⟨ ⟩[ ] ≅ Ω ⟨ ip ⟩[ i ( − p )] ⊕ Θ σγ ⟨( i − ) p ⟩[( i − )( − p )] ⊕ ... ⊕ Θ ( σγ ) i ⟨ ⟩[ ] ≅ H ( X i ) ≅ H ( Ψ i ) . (iii) For a suitable choice of such isomorphisms, the multiplication map Λ i ⊗ Λ i ′ → Λ i + i ′ corresponds to H applied to the multiplication map Ψ i ⊗ Ψ i ′ → Ψ i + i ′ . Proof.
The of the functor Ψ ⊗ Ω − (as an autoequivalence of the bounded derivedcategory of Ω-modules) is to shift the projective Ω e p by 1, whilst taking Ω e l to acomplex Ω e p → Ω e l concentrated in degrees − e p → Ω e l . If we iterate this construction i times in a jk -graded setting, we obtain precisely X i . This establishes (i).(ii) The first isomorphism comes from the definition of Λ. Indeed, we haveΛ i ◇● = Ω ⊗ ζ i ⊕ Θ σ ⊗ ζ i − ⊕ ... ⊕ Θ σ i ⊗ ζ is placed in jk -degree ( p, p − ) and that by our conventions in Section 3.1 these are indeed the degrees in whichthe components of homology are concentrated, we obatin the claim. The secondisomorphism again follows from Lemma 36.For the third isomorphism, consider the homology concentrated in the middle termof W ⟨( l + ) p ⟩ / / d W ⟨( l + ) p ⟩ ' ' PPPPPPPPPPPP W ⟨ lp ⟩ / / d W ⟨ lp ⟩ $ $ ❏❏❏❏❏❏❏❏❏ W ⟨( l − ) p ⟩ W ⟨( l + ) p ⟩ δ ⟨ lp ⟩ qqqqqqqqqq W ⟨ lp ⟩ δ ⟨( l − ) p ⟩ qqqqqqqqqq which is given by Q = Ker d W ⟨ lp ⟩ Im ( δ ⟨ lp ⟩ ○ d W ⟨( l + ) p ⟩) . Note that the subsequence W ⟨( l + ) p ⟩ → W ⟨ lp ⟩ → W ⟨ lp ⟩ which has summands of the formΩ e l ⟨( l + ) p ⟩ → Ω e p ⟨ lp + l ⟩ → Ω e p − l ⟨ lp ⟩ is exact on the left and in the middle by Lemma 35, so in fact Q ≅ Coker d W ⟨( l + ) p ⟩ . This was computed in Lemma 35 as a left Ω-module.The claim on the k -grading follows in the same way as in Lemma 37(iv), whosecomputations we also follow to establish a bimodule structure. Again, we knowthat we have an isomorphism End D b ( Ω ) ( X i ) ≅ Ω, and we have obvious generators(in the example where i is odd and 1 ≤ l ≤ p − e p ⟨ l + ( i − ) p ⟩ / / ⋅ (cid:15) (cid:15) Ω e p ⟨ p − l + ( i − ) p ⟩ ⋅ xy (cid:15) (cid:15) ⋯ Ω e p ⟨ l ⟩ ⋅ (cid:15) (cid:15) / / Ω e p − l ⋅ x (cid:15) (cid:15) Ω e p ⟨ l + + ( i − ) p ⟩ / / Ω e p ⟨ p − l − + ( i − ) p ⟩ ⋯ Ω e p ⟨ l + ⟩ / / Ω e p − l − and(10) Ω e p ⟨ l + ( i − ) p ⟩ / / ⋅ xy (cid:15) (cid:15) Ω e p ⟨ p − l + ( i − ) p ⟩ ⋅ (cid:15) (cid:15) ⋯ Ω e p ⟨ l ⟩ ⋅ xy (cid:15) (cid:15) / / Ω e p − l ⋅ y (cid:15) (cid:15) Ω e p ⟨ l − + ( i − ) p ⟩ / / Ω e p ⟨ p − l + + ( i − ) p ⟩ ⋯ Ω e p ⟨ l − ⟩ / / Ω e p − l + Again choosing the identification with Ω in such a way that it coincides with naturalright multiplication on the component in homology isomorphic to Ω appearing inthe left-most homological position and obeying the sign rule we pick up a twist by σγ in every step.The last isomorphism follows from (i).(iii) Here we write Λ i = ⊕ j,k ∈ Z Λ ijk . By its definition as a tensor algebra, the algebraΛ is quadratic with respect to the i -grading, with generators Λ = Ω in degree 0,generators Λ = Ω ζ ⊕ Θ σ in degree 1, and relations R ⊂ ( ζ Ω ⊗ Ω Θ σ ) ⊕ ( Θ σ ⊗ Ω ζ Ω ) ⊂ Λ ⊗ Λ Λ given by the image of the compositionΘ σ a ↦( a, − a ) / / Θ σ ⊕ Θ σ ( a,b )↦( ζ ⊗ a,b ⊗ ζ ) / / ( ζ Ω ⊗ Ω Θ σ ) ⊕ ( Θ σ ⊗ Ω ζ Ω ) , OSZUL DUAL 2-FUNCTORS AND EXTENSION ALGEBRAS OF SIMPLE MODULES FOR GL which simply says that the formal variable ζ and the bimodule Θ σ commute.It follows that to prove (iii), it is sufficient to check that the hexagonΘ σ ∼ v v ♠♠♠♠♠♠♠♠♠♠♠♠♠♠ ∼ ( ( ◗◗◗◗◗◗◗◗◗◗◗◗◗◗ Θ σ ⊗ Ω Ω ≀ (cid:15) (cid:15) Ω ⊗ Ω Θ σ ≀ (cid:15) (cid:15) H ( Ψ ) ⊗ Ω H ( Ψ ) µ ( ( PPPPPPPPPPPP H ( Ψ ) ⊗ Ω H ( Ψ ) µ v v ♥♥♥♥♥♥♥♥♥♥♥♥ H ( Ψ ) commutes, where µ denotes the multiplication map in HT Ω ( Ψ ) and H q denotesthe homology in the q th position of Ψ i regarded as a complex of Ω-Ω-bimodulesin D b ( Ω ) (which is not the same as taking the q th homology H q with respect tothe k -grading (cf. Section 3.1) as again Ω is not concentrated in k -degree zero, sowe have two different decompositions H = ⊕ q H q and H = ⊕ q H q ). Hence we havebimodule isomorphisms H ( Ψ ) = H ( Ψ ) ⊕ H ( Ψ ) ≅ Ω ⟨ p ⟩[ − p ] ⊕ Θ σγ H ( Ψ ) = H ( Ψ ) ⊕ H ( Ψ ) ⊕ H ( Ψ ) ≅ Ω ⟨ p ⟩[ − p ] ⊕ Θ σγ ⟨ p ⟩[ − p ] ⊕ Θ . (11)Our explicit description of the action of Ω on X i also gives fixed bimodule isomor-phisms H ( X ) = H ( X ) ⊕ H ( X ) ≅ Ω ⟨ p ⟩[ − p ] ⊕ Θ σγ H ( X ) = H ( X ) ⊕ H ( X ) ⊕ H ( X ) ≅ Ω ⟨ p ⟩[ − p ] ⊕ Θ σγ ⟨ p ⟩[ − p ] ⊕ Θ . (12)Let us now explain how once we have fixed our quasi-isomorphism φ between X and Ψ, we can define an associated fixed isomorphism H ( X ) ≅ H ( Ψ ) hence fixing bimodule isomorphisms in (11) as induced by those in (12).Indeed we have an isomorphism H ( Ψ ⊗ φ ) ∶ H ( Ψ ⊗ Ω X ) → H ( Ψ ) ;and the isomorphisms H ( Ψ ⊗ Ω X ) = H ( Ψ ⊗ Ω (( Ω e p ⊗ F e p F Qe → Ω e ) ⊕ ( Ω e p ⟨ p ⟩ → ))) , H ( φe l ) ∶ H ( Xe l ) ≅ H ( Ψ ⊗ Ω Ω e l ) imply an isomorphism H (( Xe p ⊗ F e p F Qe → Xe ) ⊕ ( Xe p ⟨ p ⟩ → )) ≅ H ( Ψ ⊗ Ω X ) , Our explicit description of X , of X , and of the right action of Ω on X , induces anisomorphism H ( X ) → H (( Xe p ⊗ F e p F Qe → Xe ) ⊕ ( Xe p ⟨ p ⟩ → )) . Composing these isomorphisms gives us our fixed isomorphism H ( X ) ≅ H ( Ψ ) .To establish our commuting hexagon we now proceed to examine in detail theeffect of right multiplication by H ( Ψ ) upon fixing the above identifications, and bycompatibility of our morphisms we can do this on the level of X , with our veryexplicit description. We have an explicit isomorphism H ( X ) ≅ H ( X ⊗ Ω (( Ω e p ⊗ F e p F Qe → Ω e ) ⊕ ( Ω e p ⟨ p ⟩ → ))) and in this formulation, given our analysis in Lemma 37(iv), we can see that anelement ω ∈ H ( X ) ≅ Ω ⟨ p ⟩[ − p ] in the second tensor factor (which is isomorphic toΩ), will act on the right of an element θ ∈ H ( X ) ≅ Θ σγ by the formula θ ⋅ ω = θγσ ( ω ) .Similarly and element θ ′ ∈ H ( X ) ≅ Θ σγ in the second tensor factor will act on theright on and element ω ′ ∈ H ( X ) ≅ Ω ⟨ p ⟩[ − p ] simply by the formula ω ′ ⋅ θ ′ = ωθ .Note that the canonical bimodule isomorphism Θ σγ → Θ σ implied by Lemma 36 isgiven by the map θ ↦ θg , where g was the self-inverse element inducing the innerautomorphism γ .This implies that the hexagon of fixed bimodule isomorphisms (where we have usedour identifications H ( X ) ⊗ Ω H ( X ) ≅ Θ σγ ⊗ Ω Ω, H ( X ) ⊗ Ω H ( X ) ≅ Ω ⊗ Ω Θ σγ and H ( X ) ≅ Θ σ ) θ ∈ ✲ v v ♠♠♠♠♠♠♠♠♠♠♠♠♠ Θ σ ∼ v v ♠♠♠♠♠♠♠♠♠♠♠♠♠ ∼ ( ( ◗◗◗◗◗◗◗◗◗◗◗◗◗ ⋅ g (cid:15) (cid:15) ∋ θ ✑ ( ( ◗◗◗◗◗◗◗◗◗◗◗◗◗ θ ⊗ ∈ ❴ (cid:15) (cid:15) Θ σ ⊗ Ω Ω ≀ (cid:15) (cid:15) Ω ⊗ Ω Θ σ ≀ (cid:15) (cid:15) ∋ ⊗ θ ❴ (cid:15) (cid:15) θg ⊗ ∈ ✏ ' ' PPPPPPPPPPPP H ( X ) ⊗ Ω H ( X ) ' ' PPPPPPPPPPPP H ( X ) ⊗ Ω H ( X ) w w ♥♥♥♥♥♥♥♥♥♥♥♥ ∋ ⊗ θg ✳ w w ♥♥♥♥♥♥♥♥♥♥♥♥♥ θgσγ ( ) = θg ∈ H ( X ) ∋ θg commutes, which completes the proof of the Lemma. (cid:3) Proof of Theorem 32.
The isomorphism H ( T Ω ( Ψ )) ≅ Λ as an Ω-Ω-bimodulecomes from Lemma 38(ii). It is an algebra isomorphism, thanks to Lemma 38(iii).Recall the gradings on both algebras:The i -grading on the tensor algebra T Ω ( Ψ ) has Ω in i -degree 0 and Ψ in i -degree1, while the j -grading is inherited by regarding Ψ as a differential jk -bigradedΩ-module, where the j - and k -gradings on Ω are both by path length.We have Λ = T Ω ( Θ σ ) ⊗ F [ ζ ] ; we let Ω have i -degree 0 and let ζ and Θ σ have i -degree 1; the j -grading on Λ is obtained by grading Ω and Θ σ by path length andplacing ζ in degree p ; finally the k -grading is obtained by grading Ω and Θ σ bypath length and placing ζ in degree p − OSZUL DUAL 2-FUNCTORS AND EXTENSION ALGEBRAS OF SIMPLE MODULES FOR GL These gradings match up under the isomorphism, hence Λ ≅ HT Ω ( Ψ ) as ijk -trigraded algebras. (cid:3) Computing y . Here we apply the theory of the preceding sections to compute y . Proof of Theorem 1 . We have algebra isomorphisms y q ≅ HO F, EO q c , t ( F, F ) Theorem 29 ≅ HO F, EO q c , ˜ t ( F, F ) Lemma 30 ≅ HO F, O q c ! , t ! E ( F, F ) Corollary 28 ≅ HO F, O q c ! , t ! ( F, F ) E ( F, F ) = ( F, F ) ≅ HO F, O q Ω , Ψ ( F, F ) Corollary 31 ≅ H O F O q T Ω ( Ψ ) ( F [ z ]) Corollary 21 ≅ O H F O q HT Ω ( Ψ ) H ( F [ z ]) Lemma 18 ≅ O F O q Λ ( F [ z ]) Theorem 32 ≅ λ q definition of λ q Sending q to ∞ , we obtain y ≅ λ as required. References [1] A. Beilinson, V. Ginzburg, W. Soergel,
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Vanessa Miemietz, Will Turner
School of Mathematics, University of East Anglia, Norwich, NR4 7TJ, UK, [email protected]
Department of Mathematics, University of Aberdeen, Fraser Noble Building, King’sCollege, Aberdeen AB24 3UE, UK, [email protected]@abdn.ac.uk