Download Koszul resolutions September 27, 2012

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