A personal note on André-Quillen homology

11 Pages·2006·0.155 MB·English

by  Jinhyun Park

A personal note on Andr´e-Quillen homology
by Jinhyun Park
October 8, 2006
Basic references are the following:
(1) Michel Andr´e, Homologie des alg`ebres commutatives, Grund. Math. Wissen.
206, Springer 1974
(2) Daniel Quillen, On the (co)homology of commutative rings, Proc. Symp. Pure
Math. 17 (1970) 65-87
(3) Jean-Louis Loday, Cyclic homology, 2nd edition, Grund. Math. Wissen. 301,
Springer 1998
(4) Srikanth Iyengar, Andr´e-Quillen homology of commutative algebras, to appear
in Contemp. Math.
(5) Mar´ıa O. Ronco, Smooth Algebras, Appendix to J.-L. Loday’s Cyclic homol-
ogy.
1. Derivations and differentials
A ring is always supposed to be a commutative ring with unity. A homomorphism
of rings is always supposed to preserve the unities. When A is a ring, an A-algebra
B is a ring B with a ring homomorphism A → B.
Definition 1.1. Let A,B be as above and let W be a B-module. An A-derivation
ω : B → W is an element in Hom (B,W) such that for each pair x,y ∈ B, we have
A
the equality
ω(xy) = xω(y)+yω(x).
The A-derivations ω : B → W form a submodule Der (B,W) of the B-module
A
Hom (B,W).
A
An important fact related to the whole discussion is that we have a universal B-
module in a certain sense with respect to derivations, which is written precisely in
the following:
Definition 1.2. Let µ : B ⊗ B → B be the A-algebra homomorphism defined by
A
µ(x⊗y) = xy, called the multiplication. Let I = ker(µ). Consider I/I2. It is also a
B-module (B ’ B ⊗ B/I) because I(I/I2) = 0. We define
A
Ω = I/I2,
B/A
which is the module of B-K¨ahler differentials over A. It is sometimes denoted by
Dif (B).
A
Remark. (1) There is a canonical A-derivation
δ : B → Ω
B/A
x 7→ x⊗1−1⊗x
called the Ka¨her derivation. Exercise: Check that it is indeed an A-derivation,
i.e. δ ∈ Der (B,Ω ).
A B/A
(2) The ideal I is in fact generated by x⊗1−1⊗x (Exercise!). Thus, so is Ω .
B/A
Thus, δ is a surjection.
1
2
(3) We have the following universal property: for any B-module and for any
A-derivation ω : B → W, there exists a unique B-module homomorphism
f : Ω → W such that f ◦δ = ω. (Exercise)
B/A
ω //
B DDDδDDDDD"" yyyyyfyyy<<W
Ω
B/A
In other words, we have a B-module isomorphism
Hom (Ω ,W) ’ Der (B,W).
B B/A A
For the sake of future usages, we define the following:
Definition 1.3. Dif (B,W) = Dif (B)⊗ W = Ω ⊗ W.
A A B B/A B
2. Cotangent complexes
For each m ≥ 0, let A[m] be the free A-algebra in m variables, i.e. it is isomorphic
to the polynomial ring in m-variables with coefficients in A. We now define a set
E (A,B) as follows: an element of E (A,B) is a set of A-algebra homomorphisms of
n n
the following type:
A[i ] →αn A[i ] α→n−1 ··· →α1 A[i ] →α0 B.
n i−1 0
This element will be denoted by (α ,··· ,α ). Note that B is an A[i ]-module via
0 n n
the composition α ◦···◦α : A[i ] → B. Thus, we define
0 n n
Definition 2.1.
T(α ,··· ,α ) := Dif (A[i ],B) = Ω ⊗ B.
0 n A n A[in]/A A[in]
Definition 2.2.
X
T (A,B) := T(α ,··· ,α ).
n 0 n
(α0,···,αn)∈En(A,B)
This is a free B-module. We will use {T (A,B)} to define the cotangent complex.
n
Infact, wegiveitastructureofpresimplicialabeliangroupsfromwhichwewillobtain
its associated complex. In particular, we define homomorphisms
di : T (A,B) → T (A,B), 0 ≤ i ≤ n
n n n−1
with the property
di ◦dj = dj−1 ◦di : T (A,B) → T (A,B), for 0 ≤ i < j ≤ n+1.
n n+1 n n+1 n+1 n−1
This di is defined as follows: for i < n, we send the component
n
T(α ,··· ,α ,α ,··· ,α ) ⊂ T (A,B)
0 i i+1 n n
to the component
T(α ,··· ,α ◦α ,··· ,α ) ⊂ T (A,B)
0 i i+1 n n−1
via the Id. For i = n, we send the component
3
T(α ,··· ,α ,α ) ⊂ T (A,B)
0 n−1 n n
to the component
T(α ,··· ,α ) ⊂ T (A,B)
0 n−1 n−1
via the map Dif (α ,B).
A n
Exercise 2.3. Check that for 0 ≤ i < j ≤ n+1,
di ◦dj = dj−1 ◦di .
n n+1 n n+1
This property shows that
n
X
d := (−1)idi : T (A,B) → T (A,B)
n n n n−1
i=0
has d ◦d = 0, thus, {T (A,B),d } is a complex, called the cotangent complex of
n n+1 ∗ ∗
the A-algebra B. This complex can be augmented as follows: let T (A,B) = Ω .
−1 B/A
For
X
T (A,B) = Ω ⊗ B,
0 A[i0]/A A[i0]
α0:A[i0]→B
d = Dif (α ) : Ω ⊗ B → Ω .
0 A 0 A[i0]/A A[i0] B/A
Remark. In case B is a free A-algebra of finite type, we can give {T (A,B)} a
∗
simplicial structure. In other words, for m ≥ −1, there are homomorphisms s :
m
T (A,B) → T (A,B) such that
m m+1
s ◦d +d ◦s = Id.
n−1 n n+1 n
Indeed, we let B = A[k], a free algebra, and let s : T (A,B) = Ω → T (A,B)
−1 −1 A[k]/A 0
be the identity on Ω ⊂ T (A,B). For m ≥ 0, let s send T(α ,··· ,α ) ⊂
A[k]/A 0 m 0 m
T (A,B) to T(Id,α ,··· ,α ) ⊂ T (A,B) via the Id. The rest is straightforward.
m 0 m m+1
For a ring A and an A-algebra B, B-algebra C, C-algebra W, we define
Definition 2.4. The n-th homology H (A,B,W) = D (B/A,W) as the n-th homol-
n n
ogy module of the complex of C-modules T (A,B,W) = T (A,B)⊗ W. Likewise,
∗ ∗ B
Hn(A,B,W) = Dn(B/A,W) is the n-th cohomology module of the complex of C-
modules T∗(A,B,W) = Hom (T (A,B),W).
B ∗
Remark. In case B is a free A-algebra of finite type, the presence of degenerations
s allows us to have
m
(cid:26)
0 if n 6= 0
D (B/A,W) = .
n Ω ⊗ W if n = 0
B/A B
Remark. In a natural way, when it is making sense, D (B/A,W) is covariant in
n
A,B,W and Dn(B/A,W) is contravariant in A,B and covariant in W. (Exercise.)
We have the following list of basic properties:
4
Proposition 2.5. Let B be an A-algebra, C be a B-algebra, W be a C-module.
Suppose that D (B/A,C) ’ 0 for 1 ≤ i ≤ n. Then, we have natural isomorphisms of
i
C-modules for 0 ≤ j ≤ n
D (B/A,W) ’ TorC(D (B/A,C),W),
j j 0
Dj(B/A,W) ’ Extj (D (B/A,C),W),
C 0
and a surjection
D (B/A,W) → TorC (D (B/A,C),W),
n+1 n+1 0
and an injection
Extn+1(D (B/A,C),W) → Dn+1(B/A,W).
C 0
Proposition 2.6. Let A,B,C,W be as above. Suppose either
(1) W is a flat C-module, or,
(2) D (B/A,·) : (C −mod) → (C −mod) is left-exact.
n−1
Then,
D (B/A,C)⊗ W ’ D (B/A,W).
n C n
Proposition 2.7. Let A,B,C,W be as above. Suppose either
(1) W is an injective C-module, or,
(2) Dn−1(B/A,·) : (C −mod) → (C −mod) is right exact.
Then,
Dn(B/A,W) ’ Hom (Dn(B/A,C),W).
C
Proposition 2.8. Let A,B,C be as above and suppose that we have a short exact
sequence of C-modules
0 → W0 → W → W00 → 0.
Then there exists a natural exact sequence of C-modules in homology
··· → D (B/A,W0) → D (B/A,W) → D (B/A,W00)
n n n
→ D (B/A,W0) → ··· → D (B/A,W00) → 0,
n−1 0
and in cohomology
0 → D0(B/A,W0) → ··· → Dn−1(B/A,W00)
→ Dn(B/A,W0) → Dn(B/A,W) → Dn(B/A,W00) → ··· .
This proposition follows if one proves that
0 → T (A,B,W0) → T (A,B,W) → T (A,B,W00) → 0
∗ ∗ ∗
is exact (and also for T∗). It is however obvious because T (A,B) is free.
n
Regarding limits, we have the following:
Proposition 2.9. Let A,B,C be as above, and we have an inductive system W of
i
C-modules for i ∈ I. Then
limD (B/A,W ) ’ D (B/A,limW ).
n i n i
−→ −→
5
Unfortunately there is not a corresponding one for Dn. In B, we also have the
following:
Proposition 2.10. B is an inductive system of A-algebras, and for a limB -algebra
i i
−→
C and a C-module W, we have a natural isomorphism of C-modules
limD (B /A,W) ’ D (limB /A,W).
n i n i
−→ −→
We have seen that for a free A-algebra of finite type B, D (B/A,W) ’ 0 if n 6= 0
n
and Ω ⊗ W if n = 0. The above proposition now allows us to extend it to any
B/A B
free A-algebra B, not necessarily of finite type.
Proposition 2.11. Let B be a free A-algebra, W be a B-module. Then
(cid:26)
0 if n 6= 0
D (B/A,W) ’
n Ω ⊗ W if n = 0,
B/A B
and,
(cid:26)
0 if n 6= 0
Dn(B/A,W) ’ .
Der (B,W) if n = 0.
A
3. Jacobi-Zariski exact sequence
Let A,B,C be as above and W be a C-module. We state a theorem without a
proof:
Theorem 3.1. There exists a natural exact sequence of C-modules in homology
··· → D (B/A,W) → D (C/A,W) → D (C/B,W)
n n n
→ D (B/A,W) → ··· → D (C/B,W) → 0,
n−1 0
and in cohomology
0 → D0(B/C,W) → ··· → Dn−1(B/A,W)
→ Dn(B/C,W) → Dn(C/A,W) → Dn(C/B,W) → ··· .
Remark. AnA-algebraB isalwaysthequotientofafreeA-algebraF,andD (F/A,W) ’
n
Dn(F/A,W) ’ 0 for n > 0. Thus, the following corollary shows that the calculation
of D (B/A,W) and Dn(B/A,W) reduces to the case when B is a quotient of A:
n
Corollary 3.2. Let B be a free A-algebra, C be a B-module and W be a C-module.
Then there are natural C-module isomorphisms for n ≥ 2
D (C/A,W) ’ D (C/B,W),
n n
Dn(C/A,W) ’ Dn(C/B,W).
Furthermore, when C is a quotient of B, there exists natural exact sequences
0 → D (C/A,W) → D (C/B,W) → D (B/A,W) → D (C/A,W) → 0,
1 1 0 0
0 → D0(C/A,W) → D0(B/A,W) → D1(C/B,W) → D1(C/A,W) → 0.
We have some other results from the Jacobi-Zariski exact sequences:
6
Proposition 3.3. Let B,C be two A-algebras, D be a B ⊗ C-algebra, W be a D-
A
module. Suppose that TorA(B,C) 0 for m > 0. Then we have exact sequences
m
··· → D (D/A,W) → D (D/B,W)⊕D (D/C,W) → D (D/B ⊗ C,W)
n n n n A
→ D (D/A,W) → ··· → D (D/B ⊗ C,W) → 0,
n−1 0 A
and
0 → D0(D/B ⊗ C,W) → ··· → Dn−1(D/A,W)
A
→ Dn(D/B ⊗ C,W) → Dn(D/B,W)⊕Dn(D/C,W) → Dn(D/A,W) → ··· .
A
Proposition 3.4. Let B be an A-algebra, C,D be two B-algebras, W be a C ⊗ D-
B
module. Suppose that TorB(C,D) = 0 for m > 0. Then, we have exact sequences
m
··· → D (B/A,W) → D (C/A,W)⊕D (D/A,W) → D (C ⊗ D/A,W)
n n n n B
→ D (B/A,W) → ··· → D (C ⊗ D/A,W) → 0,
n−1 0 B
and
0 → D0(C ⊗ D/A,W) → ··· → Dn−1(B/A,W)
B
→ Dn(C ⊗ D/A,W) → Dn(C/A,W)⊕Dn(D/A,W) → Dn(B/A,W) → ··· .
B
Corollary 3.5. Let B, C be two A-algebras and W be a B ⊗ C-module. Then,
A
D (B/A,W)⊕D (C/A,W) ’ D (B ⊗ C/A,W),
n n n A
Dn(B/A,W)⊕Dn(C/A,W) ’ Dn(B ⊗ C/A,W).
A
Lastly, we have a result on flat base changes:
Proposition 3.6. Let B be an A-algebra, C be a flat A-algebra. Then,
D (B ⊗ C/C,·) ’ D (B/A,·)
∗ A ∗
4. Another description of D and D∗
∗
Remember that we had used the modules
X
T (A,B) = Ω ⊗ B
n A[in]/A A[in]
(α0,···,αn)∈En(A,B)
to define the Andr´e-Quillen homology and cohomology modules D and D∗. Here is
∗
an another description. Since
Ω ⊕Ω = Ω ,
B1/A B2/A B1⊗AB2/A
we could have in fact taken the infinite tensor product of algebras
O
P = A[i ]
n n
(α0,···,αn)
, and it gives a free resolution of the algebra A. Here the infinite tensor product of
N
algebras is defined carefully as the submodule of A[i ] (tensor product as modules)
n
that is generated by all images of A[i ]. The face maps are defined as follows: the
n
i < n-th face sends the (f ,··· ,f )-th factor to (f ,··· ,f ◦ f ,··· ,f )-th factor
0 n 0 i i+1 n
and the n-th face sends (f ,··· ,f )-th factor to the (f ,··· ,f )-th factor.
0 n 0 n−1
A general discussion on free resolutions of an algebra now shows that the choice of
a free resolution doesn’t affect the D and Dn. Thus, we could have used any free
n
resolution P → B instead and define T (A,B) = Ω ⊗ B.
∗ n Pn/A Pn
7
5. Higher Andre´-Quillen (co)homology
Suppose that q ≥ 1. For an A-algebra B, let P → B be a free resolution. Define
∗
T(q)(A,B) = Ωq ⊗ B and the higher Andr´e-Quillen homology
n Pn/A Pn
D(q)(B/A,W) := H (T(q)(A,B)⊗ W).
n n n B
If W = B, we write this module simply as D(q)(B/A). First of all, we have the
n
following basic result.
Theorem 5.1. Suppose that B is smooth over A. Then
(cid:26)
0 if n > 0,
D(q)(B/A,W) =
n Ωq ⊗ W if n = 0.
B/A B
One very important (at least to me. This is my personal note!! not a paper to be
refereed!!) resultisthefollowingfundamentalspectralsequencefortheAndr´e-Quillen
homology for flat A-algebra B.
Theorem 5.2. Suppose that B is a flat A-algebra. Then, there is a spectral sequence
abutting to the Hochschild homology HH (B/A):
∗
E2 = D(q)(B/A) ⇒ HH (B/A).
pq p p+q
Moreover, if A ⊃ Q, then the spectral sequence degenerates and
M
HH (B/A) = D(q)(B/A).
n p
p+q=n
Proof. Choose a simplicial resolution of B by free A-algebras P → B. Consider a
∗
double complex L :
∗∗
P⊗3 oo P⊗3 oo P⊗3 oo ···
0 1 2
b −b b
(cid:15)(cid:15) (cid:15)(cid:15) (cid:15)(cid:15)
P⊗2 oo P⊗2 oo P⊗2 oo ···
0 1 2
b −b b
(cid:15)(cid:15) (cid:15)(cid:15) (cid:15)(cid:15)
P oo P oo P oo ···
0 1 2
where the bottom is the resolution and B is the Hochschild boundary map. Notice
that once P → B is a resolution, so is P⊗q → B⊗q so that for each row
∗ ∗
(cid:26)
0 n > 0,
0H (P⊗q) =
n ∗ B⊗q n = 0.
On the other hand, for each column, we have
00H (P⊗∗) = HH (P /A).
n p n p
SinceP isasymmetricalgebraoverA, ageneraltheoremontheHochschildhomology
p
of symmetric algebras shows that HH (P /A) ’ Ωn . Thus, 00H (P⊗∗) = Ωn =
n p Pp/A n p Pp/A
T(n)(A,P ), and by definition we have a spectral sequence
p p
E2 = 0H 00H (P⊗∗) = 0H (Ω1 ) = D(q)(B/A) ⇒ HH (B/A).
pq p q ∗ p P∗/A p p+q
8
When k ⊃ Q, the canonical projection from L to Ω∗ :
∗∗ P∗/A
Ω2 oo Ω2 oo Ω2 oo ···
P0/A P1/A P2/A
0 0 0
(cid:15)(cid:15) (cid:15)(cid:15) (cid:15)(cid:15)
Ω1 oo Ω1 oo Ω1 oo ···
P0/A P1/A P2/A
0 0
(cid:15)(cid:15) (cid:15)(cid:15)
P oo P oo P oo ···
0 1 2
induces an isomorphism on homology. Both of them have isomorphic vertical com-
plexes, thus have the same total homology. On the other hand, it apparently degen-
erates so that HH (B/A) = L D(q)(B/A). (cid:3)
n p+q=n p
Corollary 5.3 (Hochschild-Kostant-Rosenberg). When B is smooth over A, we have
HH (B/A) = D(n)(B/A) = Ωn .
n 0 B/A
Exercise 5.4. Let A → B → C be rings and C is flat over B. Then, the Jacobi-
Zariski exact sequence for D generates the spectral sequence (one for each m)
∗
M
E2 = D(i)(B/A,D(j)(C/B)) ⇒ D(m)(C/A).
pq p q p+q
i+j=m
(Hint: If L = A[I ] is a resolution of B over A, then choose a resolution of C over A
∗ n
of the form A[I ][J ].)
n n
Remark. In fact, for the λ-decomposition of HH (B/A)
n
n
M
HH (B/A) = HH(i)(B/A),
n n
i=0
we have HH(i)(B/A) = D(i) (B/A).
n n−i
6. Extensions of fields
The following are some interesting results on D (B/A,W) when A ⊂ B are fields
n
and W is a B-vector space. Firstly, since B is a field, we have
D (B/A,W) ’ D (B/A,B)⊗ W, Dn(B/A,W) ’ Hom (D (B/A,B),B).
n n B B n
Thus, we reduce to the following simple consideration: we choose a subfield Ω ⊂ B
so that the Ω-vector space D (B/A,Ω) determines all B-vector spaces D (B/A,W)
n n
by taking −⊗ W.
Ω
Proposition 6.1. Let A ⊂ B be a monogenic extension of fields. Then,
(1) if the generator is transcendental, then
(cid:26)
0 if n > 0,
rk D (B/A,Ω) =
Ω n 1 if n = 0,
and
9
(2) if the generator γ is algebraic, then

0 if γ is separable,

rk D (B/A,Ω) = 0 if n ≥ 2 and γ is inseparable,
Ω n
 1 if n = 0,1, and γ is inseparable.
Thus, by combining an inductive argument for the finite generation case with the
inductive limit for the general case, we obtain the following result.
Corollary 6.2. When A ⊂ B are fields, and W is a B-vector space, for all n ≥ 2,
D (B/A,W) = 0, Dn(B/A,W) = 0.
n
Corollary 6.3. For extensions of fields A ⊂ B ⊂ C, we have the Jacobi-Zariski exact
sequence
0 → D (B/A,Ω) → D (C/A,Ω) → D (C/B,Ω)
1 1 1
→ D (B/A,Ω) → D (C/A,Ω) → D (C/B,Ω) → 0
0 0 0
An induction also gives the following corollary.
Corollary 6.4. For an extension of fields A ⊂ B of finite type, the vector spaces
D (B/A,B) and D (B/A,B) have finite ranks and the Cartier difference
0 1
rk D (B/A,B)−rk D (B/A,B)
B 0 B 1
is equal to the transcendence degree of the extension A ⊂ B.
Corollary 6.5. A field extension A ⊂ B is separable if and only if
D (B/A,B) = 0.
1
Somecombinationsofseveraltricksgiveamoredetailedcorollarylikethefollowing:
Corollary 6.6 (Mac Lane’s separability criterion). An extension of fields A ⊂ B of
characteristic p > 0 is separable if and only if the ring B ⊗ A1/p is a field.
A
We can have the following generalization. In the following A is a field and B is a
local A-algebra.
Proposition 6.7. Let A ⊂ B be as above and B is noetherian with its residue field
K. Then the following conditions are equivalent.
(1) For all k 6= 0 and a K-module W, D (B/A,W) = 0.
k
(2) For all k 6= 0 and a K-module W, Dk(B/A,W) = 0.
(3) D (B/A,K) = 0.
1
(4) D1(B/A,K) = 0.
7. Extensions of algebras
Let A be a ring, B an A-algebra and W a B-module.
Definition 7.1. Anextension oftheA-algebraB bytheB-moduleW isanA-algebra
X with an exact sequence of A-modules
i p
0 → W → X → B → 0
where
(1) p is an A-algebra homomorphism,
10
(2) x·i(w) = i(p(x)·w) for x ∈ X and w ∈ W.
Lemma 7.2. i(w) is a square zero ideal of X.
Proof. That this is an ideal follows immediately from (2). To show that it is square
free, consider
i(w )·i(w ) = i(p(i(w ))·w ) = i(0·w ) = 0.
1 2 1 2 2
(cid:3)
The converse is also true.
Lemma 7.3. Let B be an A-algebra with a surjective homomorphism of A-algebras
p : X → B with the square-zero kernel I. Then I is in fact a B-module.
Proof. Let b ∈ B and choose any lifting x ∈ X of b so that p(x) = b. For all y ∈ I,
define b · y := x · y. We just need to see that it is well-defined. Suppose that for
two x ,x ∈ X, we have p(x ) = p(x ) = b. Then, x − x ∈ ker(p) = I so that
1 2 1 2 1 2
x y −x y = 0, because I is square-zero. (cid:3)
1 2
We can define the equivalence in the usual way. The trivial extension of B by W is
defined as follows: let X := B⊕W as modules, with the A-algebra product structure
defined by
(b,w)·(b0,w0) := (bb0,bw0 +b0w).
ThesetofequivalenceclassesoftheextensionsofB byW isdenotedbyExalcomm (B,W).
A
The main result its identification with the Andr´e-Quillen cohomology.
Theorem 7.4. For A,B,W as above,
Exalcomm (B,W) ’ D1(B/A,W),
A
and thus consequently the Jacobi-Zariski exact sequence of D∗(B/A,W) has its begin-
ning that looks like
0 → Der (C,W) → Der (C,W) → Der (B,W)
B A A
→ Exalcomm (C,W) → Exalcomm (C < W) → Exalcomm (B,W).
B A A
Thus, D∗(B/A,W) can be seen as the generalization of the Der and Exalcomm
functors.
8. Smooth algebras
The following are all taken from Ronco [5]. The point is that so many different
looking notions of smoothness of algebras are in fact equivalent, and even though the
statementsdonotinvolvetheappearanceoftheAndr´e-Quillenhomologygroups, they
furnish an efficient way of proving the equivalence of the equivalence. The detailed
proofs should be found in [5]. The following are some basic definitions.
Definition 8.1. (1) An ideal J ⊂ A is a local complete intersection if for all
maximal ideal m of A containing J, J ⊂ A is generated by a A -regular
m m m
sequence.
(2) A k-algebra A is unramified if Ω = 0.
A/k
(3) A k-algebra A is essentially of finite type over k if it is a localization of a finite
type k-algebra.