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Koszul resolutions September 27, 2012 History Theorem (H.Hopf, H.Samelson 1941) Let G be a compact connected Lie group and PG = {α ∈ H ≥1 (G; R)|∆(α) = 1 ⊗ α + α ⊗ 1} be the space of primitive elements (here, ∆ is induced on V H ∗ (G; R) by the multiplication on G). Then H ∗ (G; R) ' ∗ PG . j π Consider a principle G-bundle G ,→ E → M . Let {xi } be a basis of PG . There are G-invariant differential forms ξi ∈ A∗ (E) on E such that j ∗ (ξi ) represent xi dξi = π ∗ (ci ) for some closed ci ∈ A∗ (M ). V One equips the graded algebra ∗ PG ⊗ A∗ (M ) with the differential d defined by d(xi ⊗ 1) = 1 ⊗ ci , d(1 ⊗ b) = 1 ⊗ db. Koszul resolutions History Theorem (H.Hopf, H.Samelson 1941) Let G be a compact connected Lie group and PG = {α ∈ H ≥1 (G; R)|∆(α) = 1 ⊗ α + α ⊗ 1} be the space of primitive elements (here, ∆ is induced on V H ∗ (G; R) by the multiplication on G). Then H ∗ (G; R) ' ∗ PG . j π Consider a principle G-bundle G ,→ E → M . Let {xi } be a basis of PG . There are G-invariant differential forms ξi ∈ A∗ (E) on E such that j ∗ (ξi ) represent xi dξi = π ∗ (ci ) for some closed ci ∈ A∗ (M ). V One equips the graded algebra ∗ PG ⊗ A∗ (M ) with the differential d defined by d(xi ⊗ 1) = 1 ⊗ ci , d(1 ⊗ b) = 1 ⊗ db. Koszul resolutions History Theorem (H.Hopf, H.Samelson 1941) Let G be a compact connected Lie group and PG = {α ∈ H ≥1 (G; R)|∆(α) = 1 ⊗ α + α ⊗ 1} be the space of primitive elements (here, ∆ is induced on V H ∗ (G; R) by the multiplication on G). Then H ∗ (G; R) ' ∗ PG . j π Consider a principle G-bundle G ,→ E → M . Let {xi } be a basis of PG . There are G-invariant differential forms ξi ∈ A∗ (E) on E such that j ∗ (ξi ) represent xi dξi = π ∗ (ci ) for some closed ci ∈ A∗ (M ). V One equips the graded algebra ∗ PG ⊗ A∗ (M ) with the differential d defined by d(xi ⊗ 1) = 1 ⊗ ci , d(1 ⊗ b) = 1 ⊗ db. Koszul resolutions History Theorem (J.L. Koszul 1950) V The natural map ψ : ∗ PG ⊗ A∗ (M ) → A∗ (E) defined by ψ(xi ⊗ 1) = ξi , ψ(1 ⊗ b) = π ∗ (b) is a quasi-isomorphism. In particular, if M is formal (say, it is a compact Kähler manifold [DGMS75]) then V the de Rham cohomology of E is computed by the complex ∗ PG ⊗ H ∗ (M ). Koszul resolutions History Theorem (J.L. Koszul 1950) V The natural map ψ : ∗ PG ⊗ A∗ (M ) → A∗ (E) defined by ψ(xi ⊗ 1) = ξi , ψ(1 ⊗ b) = π ∗ (b) is a quasi-isomorphism. In particular, if M is formal (say, it is a compact Kähler manifold [DGMS75]) then V the de Rham cohomology of E is computed by the complex ∗ PG ⊗ H ∗ (M ). Koszul resolutions History Let g be a Lie algebra over k, A be a left g-module and C be a right g-module Definition U (g) H∗ (g, A) = Tor∗ (A, k), H ∗ (g, A) = Ext∗U (g) (k, C), where U (g) is the universal enveloping algebra of g. Tor and Ext can be computed using the standard bar-complex, but there is a smaller V complex that does the job. We equip Vp (g) = U (g) ⊗k p g with the differential d(u ⊗ x1 ∧ . . . xp ) = θ1 + θ2 , where θ1 = p X (−1)i+1 uxi ⊗ x1 . . . x̂i . . . xp i=1 θ2 = X (−1)i+j u ⊗ [xi , xj ]x1 . . . x̂i . . . xˆj . . . xp . i<j Koszul resolutions History Let g be a Lie algebra over k, A be a left g-module and C be a right g-module Definition U (g) H∗ (g, A) = Tor∗ (A, k), H ∗ (g, A) = Ext∗U (g) (k, C), where U (g) is the universal enveloping algebra of g. Tor and Ext can be computed using the standard bar-complex, but there is a smaller V complex that does the job. We equip Vp (g) = U (g) ⊗k p g with the differential d(u ⊗ x1 ∧ . . . xp ) = θ1 + θ2 , where θ1 = p X (−1)i+1 uxi ⊗ x1 . . . x̂i . . . xp i=1 θ2 = X (−1)i+j u ⊗ [xi , xj ]x1 . . . x̂i . . . xˆj . . . xp . i<j Koszul resolutions History Let g be a Lie algebra over k, A be a left g-module and C be a right g-module Definition U (g) H∗ (g, A) = Tor∗ (A, k), H ∗ (g, A) = Ext∗U (g) (k, C), where U (g) is the universal enveloping algebra of g. Tor and Ext can be computed using the standard bar-complex, but there is a smaller V complex that does the job. We equip Vp (g) = U (g) ⊗k p g with the differential d(u ⊗ x1 ∧ . . . xp ) = θ1 + θ2 , where θ1 = p X (−1)i+1 uxi ⊗ x1 . . . x̂i . . . xp i=1 θ2 = X (−1)i+j u ⊗ [xi , xj ]x1 . . . x̂i . . . xˆj . . . xp . i<j Koszul resolutions History Theorem The complex V∗ (g) (known as the Chevalley-Eilenberg complex) is an acyclic subcomplex of the bar-resolution. Thus, H∗ (g, A) (resp. H ∗ (g, A)) can be computed as the homology of the complex A ⊗U (g) V (g) (resp. Hom(V (g), C)). Koszul resolutions Minimal free resolutions Let A be a (non-commutative) non-negatively graded associative augmented k-algebra such that dim(Ai ) < ∞ and A0 = k. One often needs to have a free resolution of the ground field k. Moreover, for practical purposes such a resolution should be ”small“. Definition A bounded above complex of free graded A-modules d d · · · → P2 → P1 → P0 → 0 is called minimal provided all the induced maps k ⊗A Pi+1 → k ⊗A Pi vanish. In other words, d(Pi ) ⊂ A+ Pi−1 . Lemma A free resolution is minimal iff for each i, a basis of Pi−1 maps onto a minimal set of generators of coker(di ). Proof. Use Nakayama’s lemma. Koszul resolutions Minimal free resolutions Let A be a (non-commutative) non-negatively graded associative augmented k-algebra such that dim(Ai ) < ∞ and A0 = k. One often needs to have a free resolution of the ground field k. Moreover, for practical purposes such a resolution should be ”small“. Definition A bounded above complex of free graded A-modules d d · · · → P2 → P1 → P0 → 0 is called minimal provided all the induced maps k ⊗A Pi+1 → k ⊗A Pi vanish. In other words, d(Pi ) ⊂ A+ Pi−1 . Lemma A free resolution is minimal iff for each i, a basis of Pi−1 maps onto a minimal set of generators of coker(di ). Proof. Use Nakayama’s lemma. Koszul resolutions Minimal free resolutions Let A be a (non-commutative) non-negatively graded associative augmented k-algebra such that dim(Ai ) < ∞ and A0 = k. One often needs to have a free resolution of the ground field k. Moreover, for practical purposes such a resolution should be ”small“. Definition A bounded above complex of free graded A-modules d d · · · → P2 → P1 → P0 → 0 is called minimal provided all the induced maps k ⊗A Pi+1 → k ⊗A Pi vanish. In other words, d(Pi ) ⊂ A+ Pi−1 . Lemma A free resolution is minimal iff for each i, a basis of Pi−1 maps onto a minimal set of generators of coker(di ). Proof. Use Nakayama’s lemma. Koszul resolutions Minimal free resolutions Let A be a (non-commutative) non-negatively graded associative augmented k-algebra such that dim(Ai ) < ∞ and A0 = k. One often needs to have a free resolution of the ground field k. Moreover, for practical purposes such a resolution should be ”small“. Definition A bounded above complex of free graded A-modules d d · · · → P2 → P1 → P0 → 0 is called minimal provided all the induced maps k ⊗A Pi+1 → k ⊗A Pi vanish. In other words, d(Pi ) ⊂ A+ Pi−1 . Lemma A free resolution is minimal iff for each i, a basis of Pi−1 maps onto a minimal set of generators of coker(di ). Proof. Use Nakayama’s lemma. Koszul resolutions Minimal free resolutions Definition A resolution d d · · · → P2 → P1 → P0 → M → 0 of a graded A-module by free graded A-modules is called a linear free resolution if each Pi is generated in degree i. Lemma Any linear free resolution is minimal. Proof. Since d is assumed to be homogeneous of degree zero, then the image d(Pi ) ⊂ Pi−1 is sitting in degrees ≥ i, that is, d(Pi ) ⊂ A+ Pi−1 . Koszul resolutions Minimal free resolutions Definition A resolution d d · · · → P2 → P1 → P0 → M → 0 of a graded A-module by free graded A-modules is called a linear free resolution if each Pi is generated in degree i. Lemma Any linear free resolution is minimal. Proof. Since d is assumed to be homogeneous of degree zero, then the image d(Pi ) ⊂ Pi−1 is sitting in degrees ≥ i, that is, d(Pi ) ⊂ A+ Pi−1 . Koszul resolutions Koszul algebras Definition A k-algebra A is called Koszul iff k admits a linear free resolution. Example The symmetric algebra S(V ) over a f.d. k-vector space is Koszul. The linear free resolution of k as a trivial S(V )-module is given by the so-called standard (or tautological) Koszul complex: 3 2 ^ ^ ∗ · · · (V ) ⊗ S(V ) → (V ∗ ) ⊗ S(V ) → V ∗ ⊗ S(V ) → S(V ) → k where the differential is a∗ ⊗ a 7→ n P (a∗ ∧ e∗i ) ⊗ (ei a). i=1 Koszul resolutions Quadratic algebras Definition An associative k-algebra A is called quadratic if A ' T (V )/(R), where T (V ) is the tensor algebra over a k-vector space V and R ⊂ V ⊗ V is a subspace. Example 1 The tensor algebra T (V ) is quadratic (R = 0). 2 The symmetric algebra S(V ) is quadratic (R = hei ⊗ ej − ej ⊗ ei i). V The exterior algebra (V ) is quadratic (R = hei ⊗ ej + ej ⊗ ei i). 3 4 The quantum plane is the quadratic algebra generated by V = hx, yi and R = hx ⊗ y − qy ⊗ xi). 5 Steenrod algebra Ap (Priddy’s work). Koszul resolutions Quadratic algebras Definition An associative k-algebra A is called quadratic if A ' T (V )/(R), where T (V ) is the tensor algebra over a k-vector space V and R ⊂ V ⊗ V is a subspace. Example 1 The tensor algebra T (V ) is quadratic (R = 0). 2 The symmetric algebra S(V ) is quadratic (R = hei ⊗ ej − ej ⊗ ei i). V The exterior algebra (V ) is quadratic (R = hei ⊗ ej + ej ⊗ ei i). 3 4 The quantum plane is the quadratic algebra generated by V = hx, yi and R = hx ⊗ y − qy ⊗ xi). 5 Steenrod algebra Ap (Priddy’s work). Koszul resolutions Quadratic algebras Definition An associative k-algebra A is called quadratic if A ' T (V )/(R), where T (V ) is the tensor algebra over a k-vector space V and R ⊂ V ⊗ V is a subspace. Example 1 The tensor algebra T (V ) is quadratic (R = 0). 2 The symmetric algebra S(V ) is quadratic (R = hei ⊗ ej − ej ⊗ ei i). V The exterior algebra (V ) is quadratic (R = hei ⊗ ej + ej ⊗ ei i). 3 4 The quantum plane is the quadratic algebra generated by V = hx, yi and R = hx ⊗ y − qy ⊗ xi). 5 Steenrod algebra Ap (Priddy’s work). Koszul resolutions Koszul algebras Theorem A Koszul algebra A is quadratic. Idea of the proof. Invstigate the first three terms of the linear free resolution of k: . . . P2 = Ab2 → P1 = Ab1 → P0 = A → k → 0 Koszul resolutions