Almost smooth algebras

(1)

C AHIERS DE

TOPOLOGIE ET GÉOMÉTRIE DIFFÉRENTIELLE CATÉGORIQUES

A LFREDO R. G RANDJEAN

M ARIA J. V ALE Almost smooth algebras

Cahiers de topologie et géométrie différentielle catégoriques, tome 32, n^o2 (1991), p. 131-138

<http://www.numdam.org/item?id=CTGDC_1991__32_2_131_0>

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ALMOST SMOOTH ALGEBRAS

by

^Alfredo^R.GRANDJEAN ^{and Maria}J. ^VALE

CAHIERS DE TOPOLOGIE

F’T GÉOMÉTRIE DIFFÉRENTIELLE CA TÉGORIQUES

VOL. XXXII-2 (1991)

RÉSUMÉ. Le but de cet article est d’ introduire et de caractériser les

alg6bres "presque

lisses". Une

A-alg6-

bre B est presque lisse si ^et seulement

si,

pour ^tout

homomorphisme surjectif

^de

A-alg6bres

commutatives de but B la seconde suite exacte fondamentale associ6e est une suite exacte courte.

INTRODUCTION.

We introduce and characterize "almost smooth

algebras",

which

generalize

^the

formally

^smooth

algebras

^of Grothendieck

[1].

^There ^are "almost smooth

algebras"

^that ^are ^not ^for-

mally

^smooth.

It is well known that for every

epimorphism

of groups there is a short exact sequence of modules

[2]. Also,

^for

every

surjective homomorphism

of commutative

algebras

^there

is a

right

^exact sequence

(the

^second fundamental exact

sequence

[5]),

whose lack of exactness is measured

by

^the

cotangent

^functors

[3],

^which

play

^an

important

^{role in}

alge-

braic deformation

theory.

"Almost smooth

algebras"

^are ^char-

acterized

by

^the ^short ^exactness property of the second fundamental exact sequence.

DEFINITION.

An

A-algebra

^B^is

almost2

smooth if for any

A-algebra C ,

any ideal ^I of C

satisfying ^1=0,

and any

A-algebra

^homo-

morphism

g : B ->

C/I

such that I is an

injective

^B-module

via g , there exists ^a

lifting

f : B -> C of g that is ^an

A-algebra homomorphism.

(3)

THEOREM. Let B be an

A-algebra.

^{Then the}

following

^condi-

tions are

equivalent:

(1)

^{B is}^analmost smooth

A-algebra;

(2)

T1(BBA,I =

⁰

for every injective

^B-madule

I ,

^where

T1

is Lichtenbaum and

Schlessinger’s first

upper

cotangent fun-

ctor

[3];

(3) T1 (BBA,B) = 0 ,

^where

T 1

^is ^the

first

^lower cotangent

functor [3];

(4) for

every

surjective homomorphism of A-algebras

g : ^D⁾⁾

B ,

^the

imperfection

^module

of

^the

D-algebra

^B

relytive

^to

^{A ,}

^eBBDBA

^([1], ^p.136),

^is

isomorphic

^to

J/J ,

^{where J}⁼

Kerg ;

(5) for

every

surjecfive homomorphism of A-algebras

g : ^D->

B ,

the second

fundamental

^exact sequence

of

^B-

modules

is short _exact,where J ⁼

Kerg ;

(6)

^{there is} ^a

polynomial ring

^R ôver Â ândâ

surjective homomorphism of A-algebras

^R-> B such that

is short exact, where J ⁼

Kerg ;

(7)

^{there is} ^an

isomorphism Ext1B(R DBA,M = T1(BBA,M) , for

every B-module M .

(8)

^a

singular

A-extension

of

^B

by

^an

injective

^B-module ^I

is A-trivial

if

^its

image by

^a

surjective homomorphism of A-algebras

^{u :}^{D ---+7}^{B is}

^A-trivial;

(9) if

⁰-> M -> E -> B -> 0 is a

singular

A-extension

of

^B

by

^the ^B-module

M ,

^then ^the ^second

fundamental

^exact

sequence

of

^B-modules

is short exact;

(10)

every

singular

A-extension

of B ,

(4)

of

^B^with

McN , EcH ,

such that the

diagram

is commutative.

PROOF.

(1)

^-

(2). Let 0 -> + I -> + E P-> B -> + 0 be a _singular

A-extension of B

by

^an

injective

^B-module ^{I .} ^Since ^B

is an almost smooth

A-algebra,

^there ^is ^an

A-algebra

^section

s of p . Thus 0 -> I -> E -> + B -> + 0 is

split.

(2)

==>

(1).

^Let^{C be}^an

^A-algebra

^{and I} ^anideal of C satis-

fying

^{I2 = 0 .} ^Let p : C ->

GI

be the

projection

^and

g : B ->

GI

^an

A-algebra homomorphism

such that I is an in-

jective

B-module via _{g .} Consider the commutative

diagram

where

Since

T’(BBA,I)

⁼

^{0 ,}

is a

split

A-extension. For the section s of q , hs is

a

lifting

of g .

(2)

^4===t

(3).

Let P be a

polynomial ring

^over^A

mapping

^onto

B with kernel J . The sequence

is exact. If I is an

injective ^B-module,

^thesequence

(5)

is also exact. Thus

if

Conversely,

^let

T1(BBA,I)

^{= 0} ^for every

injective

^B-module

I . Choose I to contain

T1(BW,B) ;

^it follows that

T1 (BBA,B)

⁼^{0 .}

(3)

=>

(4)

^Consider ^the

surjective homomorphism

^of ^A-

algebras

^D->> B and the associated exact sequences

Since

be a free extension of B over A

[3] and

^let ^J be the kernel of R -- B . From the exact sequence

we obtain the exact sequence

(6)

module. Since

T1(BBA,I)= EXt1B (QBBA,I) I

⁼

^{0 ,}

^u* ^is

injec-

tive.

(8) =>(9).

^Let 0 -> M -> E -> B -> 0 be a

singular

^A-

extension and M -> CE’A + E B -> nBV" - 0 the second fundamental sequence associated

with

^the

A-algebra homomorphism

E ->> B . Since p :

T(BBA,I)

^-

T (EBA,I)

^is

injective

for every

injective

^B-module

^{I ,}

the sequence

is exact for any

injective

B-module I . Thus M - QEBA OE B is

injective.

(9)

=>

(3).

^Let^P^be^a

polynomial ring

^over^A

mapping

^onto^B

and with kernel J . Consider the

diagram

If

J/J2 ->QEBA +E B-> QBBA -> 0

^{is the} second fundamental

exact sequence associated with the

A-algebra homomorphism

E -H

B ,

^then

by

^the

naturality

^of ^the Jacobi-Zariski sequence, the

following diagram

is commutative:

Since

(2) ==* (10).

^Let ⁰-> M -> E -> B -> 0 be a

singular

^A-

extension of

B ,

and choose an

injective

^B-module ^I ^contain-

ing

^{M .} Consider the

diagram

(7)

where E’ =

(1 9 E)/H ,

^{1 9 E}^is ^an

A-algebra

^with

multiplica-

tion

and H ⁼

{(m,-im)| m

^e

^{M} .}

^The

singular

A-extension 0 -> 1 -> E’ -> B -> 0 is

split,

^because ^T

(BBA,I)

⁼^{0 .}

(10) => (2).

^Let 0 -> I -> E -> B -> 0 be ^a

singular

^A-

extension of B

by

^an

injective

^B-module

^{I ,}

^and

0-> N-> D-> B-> 0 ^a

split embedding.

^Let

j : I

-> N be inclusion and 03A8 : N - I ^a B-module

homomorphism

^such

that

03C8j

⁼

1, .

Consider the commutative

diagram

where

D03C8 _ ^{(I e} D)/{(03C8n,-in) n

^E

^{N) .}

^Since ^the

^singular

extension 0 -> I ->

D-> B-> 0 ^splits,

^so ^too ^does

0-> I -> E-> B-> 0.

NOTE.

Every formally

^smooth

algebra

is almost

smooth,

^but

there are almost smooth

algebras

^that ^are ^not

formally

(8)

and I the ideal of S

generated by

Then B is an almost smooth

A-algebra

^but ^is ^not

formally

smooth,

^because ^the

homological

dimension of CB’A ^{is infin-} ite

[4].

(9)

REFERENCES.

1. A.

GROTHENDIECK,

Eléments de

géometrie algébrique ^IV,

Première

Partie,

Publ. Math.

I.H.E.S.,

1964.

2. P.J. HILTON & U.

STAMMBACH, A

^courseⁱⁿ

homological algebra, Springer,

^1971.

3. S. LICHTENBAUM & S.

SCHLESSINGER,

^The

cotangent complex

of a

morphism,

Trans. Amer. Math. Soc. 128

(1967),

^41-

70.

4. T.

MATSUOKA,

On almost

complete intersections,

^Manuscr.

Math. 21

(1977),

^329-340.

5. H.

MATSUMURA,

Commutative

algebra, Benjamin,

^New

^York,

1970.

6. P.

SEIBT,

Infinitesimal extensions of commutative

alge- bras,

^{J. Pure}

Appl. Algebra

¹⁶

(1980),

^197-206.

Departamento

^de

Algebra

Facultad de Matemiticas

Universidad de

Santiago

^de

Compostela

ESPAGNE

Almost smooth algebras

C AHIERS DE

TOPOLOGIE ET GÉOMÉTRIE DIFFÉRENTIELLE CATÉGORIQUES

A LFREDO R. G RANDJEAN

M ARIA J. V ALE Almost smooth algebras

by

alg6bres "presque

A-alg6-

si,

homomorphisme surjectif

A-alg6bres

algebras",

generalize

formally

algebras

[1].

algebras"

mally

epimorphism

[2]. Also,

surjective homomorphism

algebras

right

(the

[5]),

by

cotangent

[3],

play

important

alge-

theory.

algebras"

by

A-algebra

almost2

A-algebra C ,

satisfying 1=0,

A-algebra

morphism

C/I

injective

lifting

A-algebra homomorphism.

A-algebra.

following

equivalent:

(1)

A-algebra;

T1(BBA,I =

for every injective

I ,

T1

Schlessinger’s first

cotangent fun-

[3];

(3) T1 (BBA,B) = 0 ,

T 1

first

functor [3];

(4) for

surjective homomorphism of A-algebras

B ,

imperfection

of

D-algebra

relytive

A ,

([1], p.136),

isomorphic

J/J ,

Kerg ;

(5) for

surjecfive homomorphism of A-algebras

B ,

fundamental

of

Kerg ;

(6)

polynomial ring

satisfying ^1=0,

^{A ,}

^([1], ^p.136),

^A-trivial;

(2). Let 0 -> + I -> + E P-> B -> + 0 be a _singular

^A-algebra

^{0 ,}

injective ^B-module,

^{0 ,}

(8) =>(9).