Smoothness, Regularity and Complete Intersection pdf

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C
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372 Moonshine: The first quarter century and beyond, J. LEPOWSKY, J. MCKAY & M.P. TUITE (eds)
373 Smoothness, regularity and complete intersection, J. MAJADAS & A.G. RODICIO
374 Geometric analysis of hyperbolic differential equations, S. ALINHAC
LONDON MATHEMATICAL SOCIETY LECTURE NOTE SERIES: 373
Smoothness, Regularity and
Complete Intersection
JAVIER MAJADAS
ANTONIO G. RODICIO
Universidad de Santiago
de Compostela, Spain
cambridge university press
Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore,
São Paulo, Delhi, Dubai, Tokyo
Cambridge University Press
The Edinburgh Building, Cambridge CB2 8RU, UK
Published in the United States of America by Cambridge University Press, New York
www.cambridge.org

Information on this title: www.cambridge.org/9780521125727
© J. Majadas and A. G. Rodicio 2010
This publication is in copyright. Subject to statutory exception
and to the provisions of relevant collective licensing agreements,
no reproduction of any part may take place without the written
permission of Cambridge University Press.
First published 2010
Printed in the United Kingdom at the University Press, Cambridge
A catalogue record for this publication is available from the British Library
ISBN 978-0-521-12572-7 Paperback
Cambridge University Press has no responsibility for the persistence or
accuracy of URLs for external or third-party internet websites referred to
in this publication, and does not guarantee that any content on such
websites is, or will remain, accurate or appropriate.
Contents
Introduction page 1
1 Definition and first properties of (co-)homology
modules 4
1.1 First definition 4
1.2 Differential graded algebras 5
1.3 Second definition 11
1.4 Main properties 17
2 Formally smooth homomorphisms 22
2.1 Infinitesimal extensions 23
2.2 Formally smooth algebras 26
2.3 Jacobian criteria 29
2.4 Field extensions 34
2.5 Geometric regularity 39
2.6 Formally smooth local homomorphisms of
noetherian rings 43

2.7 Appendix: The Mac Lane separability criterion 46
3 Structure of complete noetherian local rings 47
3.1 Cohen rings 47
3.2 Cohen’s structure theorems 52
4 Complete intersections 55
4.1 Minimal DG resolutions 56
4.2 The main lemma 60
4.3 Complete intersections 62
4.4 Appendix: Kunz’s theorem on regular local rings
in characteristic p 64
v
vi Contents
5 Regular homomorphisms: Popescu’s theorem 67
5.1 The Jacobian ideal 68
5.2 The main lemmas 74
5.3 Statement of the theorem 83
5.4 The separable case 87
5.5 Positive characteristic 91
5.6 The module of differentials of a regular
homomorphism 108
6 Localization of formal smoothness 109
6.1 Preliminary reductions 109
6.2 Some results on vanishing of homology 115
6.3 Noetherian property of the relative Frobenius 117
6.4 End of the proof of localization of formal
smoothness 120
6.5 Appendix: Power series 121
Appendix: Some exact sequences 126
Bibliography 130
Index 134

Introduction
This book proves a number of important theorems that are commonly
given in advanced books on Commutative Algebra without proof, owing
to the difficulty of the existing proofs. In short, we give homological
proofs of these results, but instead of the original ones involving simpli-
cial methods, we modify these to use only lower dimensional homology
modules, that we can introduce in an ad hoc way, thus avoiding sim-
plicial theory. This allows us to give complete and comparatively short
proofs of the important results we state below. We hope these notes can
serve as a complement to the existing literature.
These are some of the main results we prove in this book:
Theorem (I) Let (A, m,K) → (B, n,L) be a local homomorphism of
noetherian local rings. Then the following conditions are equivalent:
a) B is a formally smooth A-algebra for the n-adic topology
b) B is a flat A-module and the K-algebra B ⊗
A
K is geometrically
regular.
This result is due to Grothendieck [EGA 0
IV
, (19.7.1)]. His proof is
long, though it provides a lot of additional information. He uses this
result in proving Cohen’s theorems on the structure of complete noethe-
rian local rings. An alternative proof of (I) was given by M. Andr´e [An1],
based on Andr´e–Quillen homology theory; it thus uses simplicial meth-
ods, that are not necessarily familiar to all commutative algebraists. A
third proof was given by N. Radu [Ra2], making use of Cohen’s theorems
on complete noetherian local rings.
Theorem (II) Let A be a complete intersection ring and p a prime
ideal of A. Then the localization A

p
is a complete intersection.
1
2 Introduction
This result is due to L.L. Avramov [Av1]. Its proof uses differential
graded algebras as well as Andr´e–Quillen homology modules in dimen-
sions 3 and 4, the vanishing of which characterizes complete intersec-
tions.
Our proofs of these two results follow Andr´e and Avramov’s arguments
[An1], [Av1, Av2] respectively, but we make appropriate changes so as
to involve Andr´e–Quillen homology modules only in dimensions ≤ 2: up
to dimension 2 these homology modules are easy to construct following
Lichtenbaum and Schlessinger [LS].
Theorem (III) A regular homomorphism is a direct limit of smooth
homomorphisms of finite type (D. Popescu [Po1]–[Po3]).
We give here Popescu’s proof [Po1]–[Po3], [Sw]. An alternative proof
is due to Spivakovsky [Sp].
Theorem (IV) The module of differentials of a regular homomorphism
is flat.
This result follows immediately from (III). However, for many years
up to the appearance of Popescu’s result, the only known proof was that
by Andr´e, making essential use of Andr´e–Quillen homology modules in
all dimensions.
Theorem (V) If f :(A, m,K) → (B,n,L) is a local formally smooth
homomorphism of noetherian local rings and A is quasiexcellent, then f
is regular.
This result is due to Andr´e [An2]; we give here a proof more in the
style of the methods of this book, mainly following some papers of Andr´e,
A. Brezuleanu and N. Radu.
We now describe the contents of this book in brief. Chapter 1 intro-

duces homology modules in dimensions 0, 1 and 2. First, in Section 1.1
we give the definition of Lichtenbaum and Schlessinger [LS], which is
very concise, at least if we omit the proof that it is well defined. The
reader willing to take this on trust and to accept its properties (1.4) can
omit Sections (1.2–1.3) on first reading; there, instead of following [LS],
we construct the homology modules using differential graded resolutions.
This makes the definition somewhat longer, but simplifies the proof of
some properties. Moreover, differential graded resolutions are used in
an essential way in Chapter 4.
Introduction 3
Chapter 2 studies formally smooth homomorphisms, and in partic-
ular proves Theorem (I). We follow mainly [An1], making appropriate
changes to avoid using homology modules in dimensions > 2. This part
was already written (in Spanish) in 1988.
Chapter 3 uses the results of Chapter 2 to deduce Cohen’s theorems
on complete noetherian local rings. We follow mainly [EGA 0
IV
] and
Bourbaki [Bo, Chapter 9].
In Chapter 4, we prove Theorem (II). After giving Gulliksen’s result
[GL] on the existence of minimal differential graded resolutions, we fol-
low Avramov [Av1] and [Av2], taking care to avoid homology modules
in dimension 3 and 4. As a by-product, we also give a proof of Kunz’s re-
sult characterizing regular local rings in positive characteristic in terms
of the Frobenius homomorphism.
Finally, Chapters 5 and 6 study regular homomorphisms, giving in
particular proofs of Theorems (III), (IV) and (V).
The prerequisites for reading this book are a basic course in com-
mutative algebra (Matsumura [Mt, Chapters 1–9] should be more than
sufficient) and the first definitions in homological algebra. Though in

places we use certain exact sequences deduced from spectral sequences,
we give direct proofs of these in the Appendix, thus avoiding the use of
spectral sequences.
Finally, we make the obvious remark that this book is not in any
way intended as a substitute for Andr´e’s simplicial homological methods
[An1] or the proofs given in [EGA 0
IV
], since either of these treatments
is more complete than ours. Rather, we hope that our book can serve as
an introduction and motivation to study these sources. We would also
like to mention that we have profited from reading the interesting book
by Brezuleanu, Dumitrescu and Radu [BDR] on topics similar to ours,
although they do not use homological methods.
We are grateful to T. S´anchet Giralda for interesting suggestions and
to the editor for contributing to improve the presentation of these notes.
Conventions. All rings are commutative, except that graded rings are
sometimes (strictly) anticommutative; the context should make it clear
in each case which is intended.
1
Definition and first properties of
(co-)homology modules
In this chapter we define the Lichtenbaum–Schlessinger (co-)homology
modules H
n
(A, B, M) and H
n
(A, B, M), for n =0, 1, 2, associated to
a (commutative) algebra A → B and a B-module M, and we prove
their main properties [LS]. In Section 1.1 we give a simple definition
of H

n
(A, B, M) and H
n
(A, B, M), but without justifying that they are
in fact well defined. To justify this definition, in Section 1.3 we give
another (now complete) definition, and prove that it agrees with that
of 1.1. We use differential graded algebras, introduced in Section 1.2. In
[LS] they are not used. However we prefer this (equivalent) approach,
since we also use differential graded algebras later in studying complete
intersections. More precisely, we use Gulliksen’s Theorem 4.1.7 on the
existence of minimal differential graded algebra resolutions in order to
prove Avramov’s Lemma 4.2.1. Section 1.4 establishes the main prop-
erties of these homology modules.
Note that these (co-)homology modules (defined only for n =0, 1, 2)
agree with those defined by Andr´e and Quillen using simplicial methods
[An1, 15.12, 15.13].
1.1 First definition
Definition 1.1.1 Let A be a ring and B an A-algebra. Let e
0
: R → B
be a surjective homomorphism of A-algebras, where R is a polynomial
A-algebra. Let I =kere
0
and
0 → U → F
j
−−→ I → 0
an exact sequence of R-modules with F free. Let φ:

2

F → F be the R-
module homomorphism defined by φ(x∧y)=j(x)y −j(y)x, where

2
F
4
1.2 Differential graded algebras 5
is the second exterior power of the R-module F . Let U
0
= im(φ) ⊂ U .
We have IU ⊂ U
0
, and so U/U
0
is a B-module. We have a complex of
B-modules
U/U
0
→ F/U
0
⊗
R
B = F/IF → Ω
R|A
⊗
R
B
(concentrated in degrees 2, 1 and 0), where the first homomorphism
is induced by the injection U → F , and the second is the composite
F/IF → I/I

2
→ Ω
R|A
⊗
R
B, where the first map is induced by j, and
the second by the canonical derivation d: R → Ω
R|A
(here Ω
R|A
is the
module of K¨ahler differentials). We denote any such complex by L
B|A
,
and define for a B-module M
H
n
(A, B, M)=H
n
(L
B|A
⊗
B
M) for n =0, 1, 2,
H
n
(A, B, M)=H
n
(Hom
B

(L
B|A
,M)) for n =0, 1, 2.
In Section 1.3 we show that this definition does not depend on the
choices of R and F .
1.2 Differential graded algebras
Definition 1.2.1 Let A be a ring. A differential graded A-algebra (R, d)
(DG A-algebra in what follows) is an (associative) graded A-algebra with
unit R =

n≥0
R
n
, strictly anticommutative, i.e., satisfying
xy =(−1)
pq
yx for x ∈ R
p
,y ∈ R
q
and x
2
= 0 for x ∈ R
2n+1
,
and having a differential d =(d
n
: R
n
→ R

n−1
) of degree −1; that is, d
is R
0
-linear, d
2
= 0 and d(xy)=d(x)y +(−1)
p
xd(y) for x ∈ R
p
, y ∈ R.
Clearly, (R, d)isaDGR
0
-algebra. We can view any A-algebra B as a
DG A-algebra concentrated in degree 0.
A homomorphism f :(R,d
R
) → (S, d
S
)ofDGA-algebras is an A-
algebra homomorphism that preserves degrees (f(R
n
) ⊂ S
n
) such that
d
S
f = fd
R
.

If (R, d
R
), (S, d
S
) are DG A-algebras, we define their tensor product
R ⊗
A
S to be the DG A-algebra having
a) underlying A-module the usual tensor product R⊗
A
S of modules,
with grading given by
R ⊗
A
S =

n≥0


p+q=n
R
p
⊗
A
S
q

6 Definition and first properties of (co-)homology modules
b) product induced by (x⊗y)(x


⊗y

)=(−1)
pq
(xx

⊗yy

) for y ∈ S
p
,
x

∈ R
q
c) differential induced by d(x ⊗ y)=d
R
(x) ⊗ y +(−1)
q
x ⊗ d
S
(y) for
x ∈ R
q
, y ∈ S.
Let {(R
i
,d
i
)}

i∈I
be a family of DG A-algebras. For each finite subset
J ⊂ I, we extend the above definition to

i∈J
A
R
i
; for finite subsets
J ⊂ J

of I, we have a canonical homomorphism

i∈J
A
R
i
→

i∈J

A
R
i
.
We thus have a direct system of homomorphisms of DG A-algebras. We
say that the direct limit is the tensor product of the family of DG A-
algebras {(R
i
,d

i
)}
i∈I
. ItisaDGA-algebra, that we denote by

i∈I
A
R
i
(and is not to be confused with the tensor product of the underlying
family of A-algebras R
i
).
ADGideal I ofaDGA-algebra (R, d) is a homogeneous ideal of the
graded A-algebra R that is stable under the differential, i.e., d(I) ⊂ I.
Then R/I is canonically a DG A-algebra and the canonical map R →
R/I is a homomorphism of DG A-algebras.
An augmented DG A-algebra is a DG A-algebra together with a sur-
jective (augmentation) homomorphism of DG A-algebras p: R → R

,
where R

isaDGA-algebra concentrated in degree 0; its augmentation
ideal is the DG ideal ker p of R.
ADGsubalgebra S ofaDGA-algebra (R, d) is a graded A-subalgebra
S of R such that d(S) ⊂ S. Let (R, d)beaDGA-algebra. Then
Z(R):=kerd is a graded A-subalgebra of R with grading Z(R)=

n≥0


Z(R) ∩R
n

, and B(R):=im(d) is a homogeneous ideal of Z(R).
Therefore the homology of R
H(R)=Z(R)/B(R)
is a graded A-algebra.
Example 1.2.2 Let R
0
be an A-algebra and X a variable of degree
n>0. Let R = R
0
X be the following graded A-algebra:
a) If n is odd, R
0
X is the exterior R
0
-algebra on the variable X,
i.e., R
0
X = R
0
1 ⊕ R
0
X, concentrated in degrees 0 and n.
b) If n is even, R
0
X is the quotient of the polynomial R
0

-algebra
on variables X
(1)
,X
(2)
, , by the ideal generated by the ele-
ments
X
(i)
X
(j)
−
(i + j)!
i!j!
X
(i+j)
for i, j ≥ 1.
1.2 Differential graded algebras 7
The grading is defined by deg X
(m)
= nm for m>0. We set X
(0)
=1,
X = X
(1)
and say that X
(i)
is the ith divided power of X. Observe that
i!X
(i)

= X
i
.
Now let R beaDGA-algebra, x a homogeneous cycle of R of degree
n − 1 ≥ 0, i.e., x ∈ Z
n−1
(R). Let X be a variable of degree n, and
R X = R ⊗
R
0
R
0
X. We define a differential in R X as the unique
differential d for which R → R X isaDGA-algebra homomorphism
with d(X)=x for n odd, respectively d(X
(m)
)=xX
(m−1)
for n even.
We denote this DG A-algebra by R X; dX = x.
Note that an augmentation p: R → R

satisfying p(x) = 0 extends in
a unique way to an augmentation p : R X; dX = x→R

by setting
p(X)=0.
Lemma 1.2.3 Let R beaDGA-algebra and c ∈ H
n−1
(R) for some

n ≥ 1.Letx ∈ Z
n−1
(R) be a cycle whose homology class is c. Set
S = R X; dX = x and let f : R → S be the canonical homomorphism.
Then:
a) f induces isomorphisms H
q
(R)=H
q
(S) for all q<n− 1;
b) f induces an isomorphism H
n−1
(R)/ c
R
0
= H
n−1
(S).
Proof a) is clear, since R
q
= S
q
for q<n,
b) Z
n−1
(R)=Z
n−1
(S) and B
n−1
(R)+xR

0
= B
n−1
(S). 
Definition 1.2.4 If {X
i
}
i∈I
is a family of variables of degree > 0, we
define R
0
{X
i
}
i∈I
 :=

i∈I
R
0
R
0
X
i
 as the tensor product of the DG
R
0
-algebras R
0
X

i
 for i ∈ I (as in Definition 1.2.1). If R isaDG
A-algebra, we say that a DG A-algebra S is free over R if the underlying
graded A-algebra is of the form S = R ⊗
R
0
S
0
{X
i
}
i∈I
 where S
0
is a
polynomial R
0
-algebra and {X
i
}
i∈I
a family of variables of degree > 0,
and the differential of S extends that of R. (Caution: it is not necessarily
a free object in the category of DG A-algebras.)
If R isaDGA-algebra and {x
i
}
i∈I
a set of homogeneous cycles of R,
we define R {X

i
}
i∈I
; dX
i
= x
i
 to be the DG A-algebra
R ⊗
R
0
(

i∈I
R
0
R
0
X
i
; dX
i
= x
i
),
which is free over R.
Lemma 1.2.5 Let R beaDGA-algebra, n − 1 ≥ 0, {c
i
}
i∈I

a set
8 Definition and first properties of (co-)homology modules
of elements of H
n−1
(R) and {x
i
}
i∈I
a set of homogeneous cycles with
classes {c
i
}
i∈I
. Set S = R {X
i
}
i∈I
; dX
i
= x
i
, and let f : R → S be the
canonical homomorphism. Then:
a) f induces isomorphisms H
q
(R)=H
q
(S) for all q<n− 1;
b) f induces an isomorphism H
n−1

(R)/ {c
i
}
i∈I

R
0
= H
n−1
(S).
Proof Similar to the proof of Lemma 1.2.3, bearing in mind that direct
limits are exact. 
Theorem 1.2.6 Let p: R → R

be an augmented DG A-algebra. Then
there exists an augmented DG A-algebra p
S
: S → R

, free over R with
S
0
= R
0
, such that the augmentation p
S
extends p and gives an iso-
morphism in homology
H(S)=H(R


)=

R

if n =0,
0 if n>0.
If R
0
is a noetherian ring and R
i
an R
0
-module of finite type for all
i, then we can choose S such that S
i
is an S
0
-module of finite type for
all i.
Proof Let S
0
= R. Assume that we have constructed an augmented
DG A-algebra S
n−1
that is free over R, such that S
n−1
0
= R
0
and the

augmentation S
n−1
→ R

induces isomorphisms H
q
(S
n−1
)=H
q
(R

) for
q<n− 1. Let {c
i
}
i∈I
be a set of generators of the R
0
-module
ker

H
n−1
(S
n−1
) → H
n−1
(R


)

(equal to H
n−1
(S
n−1
) for n>1), and {x
i
}
i∈I
a set of homogeneous
cycles with classes {c
i
}
i∈I
. Let S
n
= S
n−1
{X
i
}
i∈I
; dX
i
= x
i
. Then
S
n

isaDGA-algebra free over R with S
n
0
= R
0
and such that the
augmentation p
S
n
: S
n
→ R

extending p
S
n−1
defined by p
S
n
(X
i
)=0
induces isomorphisms H
q
(S
n
)=H
q
(R


) for q<n(Lemma 1.2.5).
We define S := lim
−→
S
n
.
If R
0
is a noetherian ring and R
i
an R
0
-module of finite type for all
i, then by induction we can choose S
n
with S
n
i
an S
n
0
= R
0
-module of
finite type for all i, since if S
n−1
i
is an S
n−1
0

-module of finite type for all
i, then H
i
(S
n−1
)isanS
n−1
0
-module of finite type for all i. 
Definition 1.2.7 Let A → B be a ring homomorphism. Let R beaDG
A-algebra that is free over A with a surjective homomorphism of DG
1.2 Differential graded algebras 9
A-algebras R → B inducing an isomorphism in homology. Then we say
that R is a free DG resolution of the A-algebra B.
Corollary 1.2.8 Let A → B be a ring homomorphism. Then a free DG
resolution R of the A-algebra B exists. If A is noetherian and B an A-
algebra of finite type, then we can choose R such that R
0
is a polynomial
A-algebra of finite type and R
i
an R
0
-module of finite type for all i.
Proof Let R
0
be a polynomial A-algebra such that there exists a sur-
jective homomorphism of A-algebras R
0
→ B. (If A is noetherian and

B an A-algebra of finite type, then we can choose R
0
a polynomial A-
algebra of finite type.) Now apply Theorem 1.2.6 to R
0
→ B. 
Definition 1.2.9 Let R beaDGA-algebra that is free over R
0
, i.e.,
R = R
0
{X
i
}
i∈I
.Forn ≥ 0, we define the n-skeleton of R to be the
DG R
0
-subalgebra generated by the variables X
i
of degree ≤ n and their
divided powers (for variables of even degree > 0). We denote it by R(n).
Thus R(0) = R
0
, and if A → B is a surjective ring homomorphism with
kernel I and R a free DG resolution of the A-algebra B with R
0
= A,
then R(1) is the Koszul complex associated to a set of generators of I.
Lemma 1.2.10 Let A be a ring and B an A-algebra. Let

A → S


✒


R → B
be a commutative diagram of DG A-algebra homomorphisms, where S
is a free DG resolution of the S
0
-algebra B and R isaDGA-algebra
that is free over A. Then there exists a DG A-algebra homomorphism
R → S that makes the whole diagram commute.
Proof Let R(n)bethen-skeleton of R. Assume by induction that we
have defined a homomorphism of DG A-algebras R(n − 1) → S so that
the associated diagram commutes. We extend it to a DG A-algebra
homomorphism R(n) → S keeping the commutativity of the diagram.
a) If n =0,R(0) = R
0
and R
0
→ S
0
exists because R
0
is a polyno-
mial A-algebra.
10 Definition and first properties of (co-)homology modules
b) If n is odd, let R(n)=R(n−1) {T
i

}
i∈I
. We have a commutative
diagram
R(n − 1)
n
⊕

i∈I
R
0
T
i
R(n − 1)
n−1
R(n − 1)
n−2

R(n)
n
−−−−→ R(n)
n−1
−−−−→ R(n)
n−2
R(n − 1)
n

✒
❅
❅❘







S
n
−−−−→ S
n−1
−−−−→ S
n−2
and therefore a homomorphism R(n)
n
→ ker(S
n−1
→ S
n−2
)=
im(S
n
→ S
n−1
), and so there exist an R
0
-module homomorphism
R(n)
n
→ S
n

extending R(n − 1)
n
→ S
n
. By multiplicativity
using the map R(n)
n
→ S
n
, we extend R(n − 1) → S to a homo-
morphism of DG A-algebras R(n) → S.
c) For even n ≥ 2, suppose that R(n)=R(n − 1) {X
i
}
i∈I
.As
above, we define R(n)
n
→ S
n
and then extend it to R(n) → S by
multiplicativity using divided power rules based on the binomial
and multinomial theorems.
In more detail, suppose the map R(n)
n
→ S
n
is defined by
X
i

→
v

t=1
a
t
Y
(r
t,1
)
1
···Y
(r
t,m
)
m
∈ S
n
,
where the a
t
are coefficients in S
0
, the Y
i
are variables with
deg Y
i
> 0, and the divided powers Y
(r

t,j
)
j
have integer exponents
r
t,j
≥ 0. (Of course, for deg Y
j
odd and r>1, we understand
Y
(r)
j
= 0.) Then for l>0, the image of X
(l)
i
is determined by the
familiar divided power rules†
(a) (Y
1
+ ···+ Y
v
)
(l)
=

α
1
+···+α
v
=l

α
1
, ,α
v
≥0
Y
(α
1
)
1
···Y
(α
v
)
v
; and
(b) (Y
1
Y
2
)
(l)
= Y
l
1
Y
(l)
2
(if deg Y
1

and deg Y
2
≥ 2 are even).
Thus R(n) → S
n
is given by
X
(l)
i
→

α
1
+···+α
v
=l
α
1
, ,α
v
≥0

v

t=1
a
α
t
t
(Y

(r
t,1
)
1
)
α
t
···(Y
(r
t,m
)
m
)
α
t
α
t
!

,
† Both are justified by observing that the two sides agree on multiplying by l!.
1.3 Second definition 11
where the monomial
(Y
(r
t,1
)
1
)
α

t
···(Y
(r
t,m
)
m
)
α
t
α
t
!
equals
• 1ifα
t
=0;
• 0ifα
t
≥ 2 and r
t,j
= 0 for every j with deg Y
j
even positive;
•
(r
t,j
α
t
)!
α

t
!(r
t,j
!)
α
t
× (Y
(r
t,1
)
1
)
α
t
···Y
(r
t,j
α
t
)
j
···(Y
(r
t,m
)
m
)
α
t
if α

t
=1,or
if for some j deg Y
j
is even and positive and r
t,j
α
t
≥ 1;
note that the coefficient
(r
t,j
α
t
)!
α
t
!(r
t,j
!)
α
t
is an integer.
Using the formula Y
(p)
i
Y
(q)
i
=

(p+q!)
p!q!
Y
(p+q)
i
, we see that (Y
(r
t,i
)
i
)
α
t
=
(r
t,i
α
t
)!
(r
t,i
!)
α
t
Y
(r
t,i
α
t
)

i
, and so this definition does not depend on the chosen j.
A straightforward computation (easier if we multiply “formally” by
p!q!), shows that under this map, X
(p)
i
X
(q)
i
and
(p+q)!
p!q!
X
(p+q)
i
have the
same image. 
Remarks
i) The assumption that S is free over S
0
is only used to avoid defin-
ing divided powers structure.
ii) For the definition of H
n
(A, B, M), for n =0, 1, 2, we use free DG
resolutions only up to degree 3, and so we could have used sym-
metric powers resolutions instead of divided powers resolutions
(since they agree in degrees ≤3). However, in Chapter 4 we use
minimal resolutions and there we need divided powers.
1.3 Second definition

Definition 1.3.1 Let A → B be a ring homomorphism. Let e : R → B
be a free DG resolution of the A-algebra B. Let J = ker(R ⊗
A
B → B,
x ⊗ b → e(x)b). Let J
(2)
be the graded R
0
⊗
A
B-submodule of R ⊗
A
B
generated by the products of the elements of J and the divided powers
X
(m)
, m>1 of variables of J of even degree ≥ 2. Note that J
(2)
is a
subcomplex of R
0
⊗
A
B-modules of J. We define the complex
Ω
R|A
⊗
R
B := J/J
(2)

,
which is in fact a complex of B-modules.
In degree 0 it is isomorphic to Ω
R
0
|A
⊗
R
0
B, where Ω
R
0
|A
is the usual
12 Definition and first properties of (co-)homology modules
R
0
-module of differentials of the A-algebra R
0
. For, we have an exact
sequence of R
0
-modules defined by the multiplication of R
0
(considering
R
0
⊗
A
R

0
as an R
0
-module multiplying in the right factor)
0 → I → R
0
⊗
A
R
0
→ R
0
→ 0,
which splits, and so applying −⊗
R
0
B we obtain an exact sequence
0 → I ⊗
R
0
B → R
0
⊗
A
B → B → 0,
showing that I ⊗
R
0
B = J
0

. On the other hand, the exact sequence of
R
0
-modules
0 → I
2
→ I → Ω
R
0
|A
→ 0
gives an exact sequence
I
2
⊗
R
0
B =(I ⊗
R
0
B)
2
= J
2
0
→ I ⊗
R
0
B = J
0

→ Ω
R
0
|A
⊗
R
0
B → 0,
and therefore J
0
/J
(2)
0
= J
0
/J
2
0
=Ω
R
0
|A
⊗
R
0
B.
In degree 1, (J/J
(2)
)
1

= J
1
/J
0
J
1
=(R
1
⊗
A
B)/J
0
(R
1
⊗
A
B)=
(R
1
⊗
A
B) ⊗
R
0
⊗
A
B
B = R
1
⊗

R
0
B is the free B-module obtained by base
extension of the free R
0
-module R
1
.
Similarly, in degree 2, (J/J
(2)
)
2
= J
2
/(J
0
J
2
+ J
2
1
)=(R
2
/R
2
1
) ⊗
R
0
B.

In general, for n>0, (Ω
R|A
⊗
R
B)
n
=(R(n)/R(n − 1))
n
⊗
R
0
B.
Definition 1.3.2 We say that an A-algebra P has property (L) if for any
A-algebra Q,anyQ-module M,anyQ-module homomorphism u: M →
Q such that u(x)y = u(y)x for all x, y ∈ M, and for any pair of A-algebra
homomorphisms f,g: P → Q such that im(f − g) ⊂ im(u), there exists
a biderivation λ: P → M such that uλ = f − g
P

✠
λf






g
M
u

−−→ Q.
Here we say that λ is a biderivation to mean that λ is A-linear and
λ(xy)=f(x)λ(y)+g(y)λ(x).
Lemma 1.3.3 Let A be a ring, P an A-algebra.
a) If P is a polynomial A-algebra, then P has property (L).
b) If P has property (L) and S is a multiplicative subset of P, then
S
−1
P has property (L).
1.3 Second definition 13
Proof a) Let Q, M, u, f, g be as in (1.3.2). Let P = A[{X
i
}
i∈I
]. Since
im(f − g) ⊂ im(u), there exist elements Y
i
in M such that u(Y
i
)=
f(X
i
) − g(X
i
). Define λ on monomials X
i
1
···X
i
n

by
λ(X
i
1
···X
i
n
)=
n

j=1
f(X
i
1
···X
i
j−1
)Y
i
j
g(X
i
j+1
···X
i
n
).
To see that λ is well defined, we show that the right-hand side of this
formula remains invariant under transpositions (i
p

,i
p+1
) of the indices:
λ(X
i
1
···X
i
p+1
X
i
p
···X
i
n
) − λ(X
i
1
···X
i
p
X
i
p+1
···X
i
n
)
= f(X
i

1
···X
i
p−1
)Y
i
p+1
g(X
i
p
X
i
p+2
···X
i
n
)
+ f (X
i
1
···X
i
p−1
X
i
p+1
)Y
i
p
g(X

i
p+2
···X
i
n
)
− f (X
i
1
···X
i
p−1
)Y
i
p
g(X
i
p+1
···X
i
n
)
− f (X
i
1
···X
i
p
)Y
i

p+1
g(X
i
p+2
···X
i
n
)
= f(X
i
1
···X
i
p−1
)g(X
i
p+2
···X
i
n
)(u(x)y − u(y)x)=0,
where x = Y
i
p+1
and y = Y
i
p
. Extend λ to P by linearity. It follows
easily that λ is a biderivation. Since f − g is also a biderivation and
f(X

i
) − g(X
i
)=u(Y
i
)=u(λ(X
i
)), we have uλ = f − g.
b) Let Q be an A-algebra and M a Q-module and u: M → Q a
Q-module homomorphism such that u(x)y = u(y)x for all x, y ∈ M;
suppose that f

,g

: S
−1
P → Q are two A-algebra homomorphisms such
that im(f

− g

) ⊂ im(u). Let f,g: P → Q be the respective composites
of f

,g

with the canonical map P → S
−1
P . Since P has property (L),
there exists a biderivation λ: P → M such that uλ = f − g.

P



✁
✁
✁
✁
✁☛
λ
S
−1
P
f







g

M
u
−−→ Q.
Let p/s ∈ S
−1
P with p ∈ P and s ∈ S. Solving the equation λ(p)=
λ(s · p/s)=f


(s)λ

(p/s)+g

(p/s)λ(s) for λ

gives
λ

(p/s)=(g

(s)λ(p) − g

(p)λ(s))/(f

(s)g

(s)).
Using the relation u(x)y = u(y)x for x, y ∈ M, one can show by a
14 Definition and first properties of (co-)homology modules
tedious but straightforward calculation that this formula actually defines
a biderivation λ

: S
−1
P → M extending λ. Since f

−g


is a biderivation
extending f − g, it is clear that uλ

= f

− g

. 
Lemma 1.3.4 Let A → B be a ring homomorphism and p: R → B and
q : S → B two free DG resolutions of the A-algebra B.Letf,g: R → S
be two homomorphisms of augmented DG A-algebras, i.e., p = qf, p =
qg. Then there exist B-module homomorphisms
α
i
:(Ω
R|A
⊗
R
B)
i
→ (Ω
S|A
⊗
S
B)
i+1
for i =0, 1, 2
such that
d
S

1
α
0
= f
0
− g
0
, and d
S
i+1
α
i
+ α
i−1
d
R
i
= f
i
− g
i
, for i =1, 2,
where d
R
, d
S
are the differentials of R and S respectively, and denotes
the induced map in the following diagram
(Ω
R|A

⊗
R
B)
3
d
R
3
−−→ (Ω
R|A
⊗
R
B)
2
d
R
2
−−→ (Ω
R|A
⊗
R
B)
1
d
R
1
−−→ (Ω
R|A
⊗
R
B)

0
f
3






g
3
α
2
 f
2






g
2
α
1
 f
1







g
1
α
0
 f
0






g
0
(Ω
S|A
⊗
S
B)
3
d
S
3
−−→ (Ω
S|A
⊗
S

B)
2
d
S
2
−−→ (Ω
S|A
⊗
S
B)
1
d
S
1
−−→ (Ω
S|A
⊗
S
B)
0
.
Proof Let I = ker(R ⊗
A
B → B), J = ker(S ⊗
A
B → B), so that
Ω
R|A
⊗
R

B = I/I
(2)
,Ω
S|A
⊗
S
B = J/J
(2)
. We have f
i
(I
i
) ⊂ J
i
, g
i
(I
i
) ⊂
J
i
, for all i. We begin by defining maps α
i
R
3
d
R
3
−−→ R
2

d
R
2
−−→ R
1
d
R
1
−−→ R
0

f
3






g
3
α
2


f
2







g
2
α
1


f
1






g
1
α
0


f
0







g
0
S
3

d
S
3
−−→ S
2
/S
2
1

d
S
2
−−→ S
1
/d
S
2
(S
2
1
)

d
S

1
−−→ S
0
and the α
i
are the maps induced by the α
i
. Note that the lower row is
still exact.
Definition of α
0
: by Lemma 1.3.3, there exists a biderivation α
0
: R
0
→
S
1
/d
S
2
(S
2
1
) such that

d
S
1
α

0
=

f
0
− g
0
. The A-linear map α
0
induces a
map R
0
⊗
A
B → S
1
/d
S
2
(S
2
1
) ⊗
A
B → S
1
/d
S
2
(S

2
1
) ⊗
S
0
B = S
1
⊗
S
0
B =
(Ω
S|A
⊗
S
B)
1
, and this composite factors through a map α
0
:(Ω
R|A
⊗
R
B)
0
→ (Ω
S|A
⊗
S
B)

1
, since f
0
(I
0
) ⊂ J
0
, g
0
(I
0
) ⊂ J
0
, α
0
is a biderivation
and then α
0
⊗
A
B takes I
2
0
into the image of J
0
J
1
.
Definition of α
1

: let R
1
=

i
R
0
T
i
. For each i, let y
i
∈ S
2
/S
2
1
be
1.3 Second definition 15
such that

d
S
2
(y
i
)=

f
1
(T

i
) − g
1
(T
i
) − α
0
d
R
1
(T
i
); such a y
i
exists, since

d
S
1
(

f
1
(T
i
) − g
1
(T
i
) − α

0
d
R
1
(T
i
)) =

f
0
d
R
1
(T
i
) − g
0
d
R
1
(T
i
) − (

f
0
− g
0
)(d
R

1
(T
i
))=0.
Define α
1
(T
i
)=y
i
and extend it to R
1
by R
0
-linearity (regarding S
2
as
an R
0
-module via f
0
). It is clear that α
1
induces α
1
:(Ω
R|A
⊗
R
B)

1
→
(Ω
S|A
⊗
S
B)
2
, since (

f
0
⊗
A
B)(I
0
) ⊂ J
0
.ByA-linearity, to see that

d
S
2
α
1
+ α
0
d
R
1

=

f
1
− g
1
it is enough to see that

d
S
2
α
1
(aT
i
)+α
0
d
R
1
(aT
i
)=

f
1
(aT
i
) − g
1

(aT
i
)
with a ∈ R
0
. But

d
S
2
α
1
(aT
i
)+α
0
d
R
1
(aT
i
) −


f
1
(aT
i
) − g
1

(aT
i
)

=

f
0
(a)

d
S
2
α
1
(T
i
)+


f
0
(a)α
0
d
R
1
(T
i
)+g

0
(d
R
1
(T
i
))α
0
(a)

−


f
0
(a)

f
1
(T
i
) − g
0
(a)g
1
(T
i
)

=


f
0
(a)


d
S
2
α
1
(T
i
)+α
0
d
R
1
(T
i
) −


f
1
(T
i
) − g
1
(T

i
)


+ g
0
(d
R
1
(T
i
))α
0
(a) −


f
0
(a) − g
0
(a))(g
1
(T
i
)

=

f
0

(a)


d
S
2
α
1
(T
i
)+α
0
d
R
1
(T
i
) −


f
1
(T
i
) − g
1
(T
i
)



+

d
S
1
(g
1
(T
i
))α
0
(a)) −

d
S
1
(α
0
(a))g
1
(T
i
)
=

f
0
(a)



d
S
2
α
1
(T
i
)+α
0
d
R
1
(T
i
) −


f
1
(T
i
) − g
1
(T
i
)


+


d
S
2
(g
1
(T
i
)α
0
(a)),
where the products g
1
(T
i
)α
0
(a) have the obvious meaning. Now the two
summands are zero, the first by definition of α
1
and the second because
it is in d
S
2
(S
2
1
).
Definition of α
2

: let R
2
=(

k
R
0
U
k
) ⊕ (

i=j
R
0
T
i
T
j
). By the
equation

d
S
2
α
1
+ α
0
d
R

1
=

f
1
− g
1
, there exist z
k
∈ S
3
such that

d
S
3
(z
k
)=

f
2
(U
k
) − g
2
(U
k
) − α
1

d
R
2
(U
k
). Define α
2
(U
k
)=z
k
. Define α
2
(T
i
T
j
) ∈ S
3
such that its image in (Ω
S|A
⊗
S
B)
3
is 0, e.g., define α
2
(T
i
T

j
)=0or
α
2
(T
i
T
j
)=

f
1
(T
i
)α
1
(T
j
) −

f
1
(T
j
)α
1
(T
i
) (this second definition has the
following advantage, here unnecessary since we do not need α

2
, but only
α
2
:(

d
S
3
α
2
+ α
1
d
R
2
)(T
i
T
j
)=0=

f
2
(T
i
T
j
) − g
2

(T
i
T
j
)). Extend α
2
to
R
2
by R
0
-linearity (again regarding S
3
as an R
0
-module via f
0
). As for
16 Definition and first properties of (co-)homology modules
α
1
, it is clear that α
2
induces α
2
:(Ω
R|A
⊗
R
B)

2
→ (Ω
S|A
⊗
S
B)
3
. The
equation
d
S
3
α
2
+ α
1
d
R
2
= f
2
− g
2
follows from

d
S
3
α
2

(U
k
)=

d
S
3
(z
k
)=

f
2
(U
k
) − g
2
(U
k
) − α
1
d
R
2
(U
k
), and
from the R
0
-linearity of


d
S
3
, d
R
2
, α
2
, α
1
and of f
2
− g
2
. 
Remark 1.3.5 Again, the assumption that S is free is not necessary.
Definition 1.3.6 Let A → B be a ring homomorphism and M a B-
module. Let R be a free DG resolution of the A-algebra B.Forn =
0, 1, 2, define the (co-)homology B-modules by
H
n
(A, B, M)=H
n
(Ω
R|A
⊗
R
M):=H
n

((Ω
R|A
⊗
R
B) ⊗
B
M)
H
n
(A, B, M)=H
n
(Hom
B
(Ω
R|A
⊗
R
B,M)).
In view of Lemma 1.2.10 and Lemma 1.3.4, the definition does not de-
pend on the choice of R, and is natural in A, B and M.
Proposition 1.3.7 The (co-)homology modules of Definition 1.3.6 agree
with those of Definition 1.1.1, which are therefore well defined.
Proof Let A be a ring and B an A-algebra. Let e
0
: R → B be a
surjective homomorphism of A-algebras, where R is a polynomial A-
algebra. Let I = ker(e
0
) and
0 → U → F

j
−→ I → 0
an exact sequence of R-modules with F free. Let φ:

2
F → F be as
in Definition 1.1.1, and U
0
= im(φ).
Let S be a free DG resolution of the A-algebra B with S
0
= R (see
the proof of corollary (1.2.8)) and with S
1
= F as S
0
= R-module (see
the proof of Theorem 1.2.6).
We prove that there is an isomorphism between the complex
U/U
0
→ F/IF → Ω
R|A
⊗
R
B
of Definition 1.1.1, and
(Ω
S|A
⊗

S
B)
2
/d
3
(Ω
S|A
⊗
S
B)
3
→ (Ω
S|A
⊗
S
B)
1
→ (Ω
S|A
⊗
S
B)
0
.
1.4 Main properties 17
Since H
n
(Ω
S|A
⊗

S
B) for n =0, 1, 2, do not depend on the choice of
S, this proves the proposition (note also that
(Ω
S|A
⊗
S
B)
2
/d
3
(Ω
S|A
⊗
S
B)
3
⊗
B
M =(Ω
S|A
⊗
S
M)
2
/d
3
(Ω
S|A
⊗

S
M)
3
,
Hom
B
((Ω
S|A
⊗
S
B)
2
/d
3
(Ω
S|A
⊗
S
B)
3
,M)=
ker(Hom
B
((Ω
S|A
⊗
S
B)
2
,M) → Hom

B
((Ω
S|A
⊗
S
B)
3
,M)), for any B-
module M).
It is clear that (Ω
S|A
⊗
S
B)
0
=Ω
R|A
⊗
R
B,(Ω
S|A
⊗
S
B)
1
= S
1
⊗
S
0

B =
F ⊗
R
B = F/IF. To see that U/U
0
=(Ω
S|A
⊗
S
B)
2
/d
3
(Ω
S|A
⊗
S
B)
3
,
first note that U/U
0
is the first Koszul homology module of the ideal I
of R, that is, U/U
0
= H
1
(S(1)), where S(n)isthen-skeleton of S.
Associated to the exact sequence
0 → S(1) → S(2) → S(2)/S(1) → 0

we have an exact sequence
H
2
(S(2))
β
2
−→ H
2
(S(2)/S(1))
δ
2
−→ H
1
(S(1))
α
1
−→ H
1
(S(2)) = 0
and so U/U
0
= H
1
(S(1)) = H
2
(S(2)/S(1))/ im(β
2
).
We have isomorphisms H
n

(S(n)/S(n−1))=(S(n)/S(n−1))
n
⊗
S
0
B =
(Ω
S|A
⊗
S
B)
n
for n>1, and with this identification, the diagram
H
3
(S(3)/S(2))
δ
3
−→ H
2
(S(2))
β
2
−→ H
2
(S(2)/S(1))

(Ω
S|A
⊗

S
B)
3
d
3
−→ (Ω
S|A
⊗
S
B)
2
is commutative. On the other hand, H
2
(S(3)) = 0, so that α
2
=
0 and then δ
3
is surjective. Thus (Ω
S|A
⊗
S
B)
2
/d
3
(Ω
S|A
⊗
S

B)
3
=
H
2
(S(2)/S(1))/ im(β
2
)=U/U
0
. So we have vertical isomorphisms in
the diagram
U/U
0
−−−−→ F/IF −−−−→ Ω
R|A
⊗
R
B

(Ω
S|A
⊗
S
B)
2
/d
3
(Ω
S|A
⊗

S
B)
3
→ (Ω
S|A
⊗
S
B)
1
→ (Ω
S|A
⊗
S
B)
0
and it is an easy computation to see that the diagram commutes. 
1.4 Main properties
Proposition 1.4.1 Let A be a ring, B an A-algebra, M a B-module.
We have