Publicacions
Ma emá iques,
Vol
37
(1993),
285-303
.
Abs ac
MINIMAL
RESOLUTIONS
AND
OTHER
MINIMAL
MODELS
AGUSTÍ
RoIG
In
many
si ua ions,
minimal
models
a e
used
as
ep esen a i es
o
homo opy
ypes
.
In his
pape
we
s a e
his
ac
as
an
equi -
alence
o
ca ego ies
.
This
equi alence
ollows
om
an
axioma ic
de ini ion
o
minimal
objec s
.
We
see
ha
his
de ini ion
includes
examples
such
as
minimal
esolu ions
o
Eilenbe g-Nakayama-
Ta e,
minimal
ibe
spaces
o
Kan
and
A-minimal
A-ex ensions
o
Halpe in
.
Fo
he
i s
one,
his
is
done
by
gene alizing
he
cons uc ion
o
minimal
esolu ions
o
modules
o
complexes
.
The
o he s
ollow
by a
ca ac e iza ion
o
minimal
objec s
in
bi ib ed
ca ego ies
.
In oduc ion
Minimal
models appea
in
di e en
si ua ions
as
ep esen a i e
o
ho-
mo opy
ypes
.
We
claim
ha
his
is
an
in insic
p ope y
o
hose
objec s
called
"minimal",
and
does
no
depend
on
he
pa icula
cons uc ion
employed
in
each
ca ego y
.
To
his
end,
we
p opose
an
axioma ic
de ini-
ion
o
minimali y,
inspi ed
by
he
de ini ion o
minimal
(R,
)-algeb as
o
[H-TI
.
Ou
esul
is
ha
he
homo opy
ca ego y
o
a
model
ca ego y
and
he
ca ego y
o
i s
minimal
objec s
and
homo opy
classes o
maps
a e
equi alen
(Theo em
1
.17)
.
We
also
p o e
ha
ou
axioma ic
de ini ion
p oduces
he
objec s
ha
a e
commonly
known
as
"minimals"
.
In pa icula ,
we
see
ha
he
min-
imal
esolu ions
o
Eilenbe g-Nakayama-Ta e
a e
minimal
objec s
o
a
ca ego y
o
cochain
complexes
.
This
is
done by
gene alizing
he
cons uc-
ion
o
hese
minimal
esolu ions
o
modules
o
complexes
;
Le
.,
g aded
modules
wi h
di e en ial
(Theoem
2
.4
and
2
.5)
.
We
also e i y
ha
minimal
ib e
spaces
o
[Kan]
and
A-minimal
A-ex ensions
o
[Hal]
a e
minimal
objec s
o
bi ib ed
ca ego ies
(see
[Roigl])
in
e ms
o
ou
de -
ini ion
(Theo em
3
.2)
.
O he
examples
may
be
ound
in
[Roig2]
and
[Roig3]
.
286

A
.
Role
1
am
g a e ul
e
F ancisco
Guillén,
Vicen e
Na a o
Azna ,
Pe e
Pascual-Gainza
and
Daniel
Tan é
o
help ul
discussions
and
sugges ions
.
1
am
also
g a e ul
e
he
e e ee
o
his
ema ks
and
c i icisms
.
1
.
Minimal
ob
jec s
and
he
homo opy
ca ego y
Le
Cbe a
ca ego y
and
S
C
mo
C
a
class o
mo phisms
o
C
.
We
will
o en
deno e he
ac
ha
a
mo phism
s
:
a
-->
b
o
C
is
in
S
by
s
:
a
->
b
and
we
will
loosely
say
ha
s
is
a
S-quasi-isomo phism
(quism)
and
a
is
S-quasi-isomo phic
o
b
.
We
will
also
say
ha
a
is
a
S-le
model
o
b
o
ha
b
is
a
S- igh
model
o a
.
De ini ion
1
.1
.
An
objec
m
o
C
is
S-le
minimal
i ,
o
all
s
x
-3
m
E
S,
he e
exis s
a
sec ion
s'
:
m
-
x
;
Le
.,
ss'
=
1
.
A
S-le
minimal
model
o
an
objec
a
E
obj
C
is
a
S-le
model
m
-->
a
wi h
m
a
le
minimal
objec
.
By
in e ing
a ows
in
he
p e ious
de ini ion
we
ha e
he
no ions
o
S- igh
minimal
objec
and
S- igh
minimal
model
(see
[Roig3])
.
F om
now
en
we
suppose
ha
he
class
S
is
ixed
o
e e y
ca ego y
and
con-
side
only
he
"le
pa "
o
he
heo y,
unless
o he wise
s a ed
.
So
we
will
simply
say
minimal
objec ,
model
and minimal model
.
The
ela i o
e sion
o de ini ion 1
.1 is
he
ollowing
(c
.
[B-G],
[Hall
and
[H-T])
:
le
C
be
a
ca ego y,
a an
objec o
C
and
S
C
mo
C
.
Le
a S
be
he
class
o
mo phisms
o
he
ca ego y
o
objec s
unde
a,
a C,
made
up by
he
commu a i e
iangles
o
C
in
which
s
E
S
.
Hypo hesis
1 .3
.
c
a
b
De ini ion
1
.2
.
A
mo phism
a
-->
b
o
C
is
a minimal
mo phism
i i
is
a a S-minimal
objec
o
a C
.
Fo
he
i s
esul s,
i is
necessa y
ha
he
class
S
e i ies
he
(i)
isomo phisms
o
C
a e
in S,
and
(ii) i
in
he
diag am
o C,
x
~
y
~
z,
wo
o
he
mo phisms
{ ,
g,
g
}
a e
in S,
hen
so
is
he
hi d
.
hen
MINIMAL
RESOLUTIONS
AND
OTHER
MINIMAL
MODELS

287
These
hypo hesis
a e
ul illed
i ,
o
ins an e,
C
is
a
model
ca ego y
and
S
=
we
is
he
class
o
i s
weak
equi alen es
o
i
S
is
he
class
o
mo phisms
made
in e ible
by
soma
unc o
H
:
C
-->
D
.
Thenwe
can
easily
p o e
(see
[Roig3])
.
P oposi ion
1 .4
.
I
m
and
n
a e
minimal
objec s
and
s
:
m
-->
n
is
a quism, hen
s
is
an
isomo phism
.
De ini ion
1 .5
.
Le
C
be
a
ca ego y
wi h
an
ini ial
objec
e
.
An
objec
x
o
C
is
acyclic
i i is
quasi-isomo phic
o
e
.
Co olla y
.
An
objec
x
o
C
is
minimal
and
acyclic
i
and
only
i
i
is
ini ial
.
P oo
:
An
ini ial
objec
e
is
always
minimal
and,
since 1,
is
in
S
by
1
.3(i),
i is
also
acyclic
.
On
he
o he
hand
i
x
is
minimal
and
acyclic
he e
is
a
mo phism
in
S
x
-
ewhich
is,
by
1 .4,
an
isomo phism
.
The
ollowing
is
a
use ul
c i e ium
o
minimali y
.
P oposi ion
1 .6
.
I
C
admi s
a
class
o
objec s
M
such
ha
:
(a)
e e y
objec
has a
model
in
M,
and
(b)
e e y
mo phism
o
S
be ween
objec s
o
M
is
an
isomo phism,
(1)
he
objec s
o
M
a e
minimal,
and
(2)
e e y
minimal
objec o
C
is
isomo phic
o
an
objec o
M
.
P oo
.
.-
See
[Roig3]
.
De ini ion
1 .7
.
I
C
has
a
class
o
objec s
M
ha
ul iles
condi ions
(a)
and
(b) o
P oposi ion
1
.6,
we
will
say
ha
C
has
enough
minimals
.
We
will
deno e
by
HoC
he
homo opy
ca ego y
;
Le
.,
he
ca ego y
ob-
ained
om
C by
adjoining
he
in e sas
o
he
mo phisms
in
S
(see
[Qui])
.
I
is
also callad
he
localizad
ca ego y,
CS
([G-Z]),
o
he
de i ed
ca ego y,
in
he
case
whe e C
is
he
ca ego y
o
complexes
o
an
abelian
ca ego y
([Ha ])
.
I
wo
objec s
o
C
a e
isomo phic
in
HoC
hey
a e
said " o
ha e
he
same homo opy
ype"
.
Fo he
momen
we
will
assume
also
Hypo hesis
1 .8
.
S
admi s
a
calculus
o igh
ac ions
.
This
is
he
case
when
D
is
a
model
ca ego y
wi h
only
ib an
objec s,
C
=
7 D
being
he
ca ego y
ob ained
iden i ying
homo opymaps
o
288

A
.
RoiG
D
and
S C
mo
C
being
he
image
o
he
weak
equi alences
o
D
by
he p ojec ion
unc o
D
--
i D
.
We
will
de elop
ou
esul s
unde
hypo hesis
1
.8
and
hen
es a e
hem
"up
o
homo opy"
o a
model
ca ego y
.
Finally, in
ou
pa icula
examples,
all
objec s
will
be
ib an
.
When
S
admi s
a
calculus
o
igh
ac ions,
he
mo phisms
o
HoC
om
a o
b
can
be
ep esen ed
by
sequences
o
mo phisms
o
C
as
a+

1
-
----> b
and
he
isomo phisms
a e
exac ly
hose
sequences
whe e bo h
mo phisms
o
(1)
a e
in
S
.
Then,
wi h
his
hypo hesis,
wo minimal
objec s
o
C
which
a e
isomo phic
in
HoC
a e
necessa ily
isomo phic
in
C
:
I
in
(1)
a and
b
a e
minimal,
we
ha e
a
sec ion
o
a
<-
-
which
is
in
S
by
(i)
and
(ii)
o
1
.3
.
Hence,
i s
composi ion
wi h
-
->
b is
also in
S
.
Thus
we
ha e
a
quism
be ween
minimal
objec s
which,
by
1
.4,
is
an
isomo phism
.
The e o e,
homo opy
ypes
become
isomo phism
classes
when we
use
minimal
objec s
.
We
a e
going
o
s a e
his
ac as
an
equi alence
o
ca ego ies
.
To
begin
wi h,
we
ha e
a
li ing
p ope y
.
P oposi ion
1 .9
.
Le
p
:
a
=> b
E
S
_and
:
m
->
b a
mo p_hism
wi h
m
minimal
.
Then
he e
exis s
a
unique
:
m
->
a
such
ha
p
=
.
P oo
.-
Because
o
he p ope ies
o
he
calculus
o
ac ions,
o
e e y
diag am
in
C,
a
he e
exis s
mo phisms
p'
:
x
=>
m
and
'
:
x
-->
a
such
h
ha
p
'
=
p'
.
T
hn
hen,
as
m
is
minimal,
p'
has
a
sec ion
s
:
m
-~
x
.
Take
=
's
and
so
P
=
P
,
s
=
p
,
s
=
.
_
Secondly,
le
g
:
m
--->
a be
ano he
mo phism
such
ha
pg
=
.
Then
p
=
pg
.
Bu
pE
S,
and
so_,
by
he
calculus
o
ac ions,
he e
exis s
:
z
-->
m
E
S
such
ha
=
g
and
because
m
is
m
_
inima
_
l,
his
mo phism
has
a
sec ion
s
:
m
--> z,
s
=
1

,
.
So
g
=
g s
=
s
=
.
Co olla y
.
Two
minimal
models
o
an
objec a
a e
isomo phic
by a
unique
isomo phism
o
C/a
.
The
nex
p oposi ion
will
allow
us
o
de ine
he
"minimal
model
unc-
o ",
when
we
ha e
enough
minimals
.
MINIMAL
RESOLUTIONS
AND
OTHER
MINIMAL
MODELS

289
P oposi ion
1
.10
.
Le
:
a
-> b
be
a
mo phism
o
C
and
Pa
:
ma
a
and
Pb
:
mb
-
b
wo
minimal
models
.
Then
he e
exis s
a
unique
mo phism
m
:
m
a
->
mb,
ha
ende s
commu a i e
he
diag am
m
aPa
->a
l
b
P oo
.
Applying
he
calculus
o
ac ions
wo
imes,
we
ha e
a
com-
mu a i e
diag am
y
-------
>
x--,mb
Pa

'
P
b

-1
P,,

-IPb
Pa

So,
by
he
minimali y
o
ma,
we
ha e
s
:
m
a
->
y
such
ha
pes
=
1

,,
a
.
We
le
m
=
'p'
s
and
ha e
pbm
=
Pb 'P'
S
=
P6Pá
s
=
PaPb
s
=
Pa-
Le
cp
:
m
a
-`
mb
be
ano he
mo phism
ha
makes
commu a i e
diag am
(2)
:
PbW
=
Pa
=
pbm
.
Then,
because
Pb
E
S,
by
he
calculus
o
ac ions,
he e
exis s
:
z
->
m
a
E
S
such ha
cp
=
m
and,
because
'
m
a
is
minimal,
we
ha e
a
sec ion
s
:
m
a-+
z
o
.

So,
cp
=
cp s
=
m s
=
M
.
Rema k
1
.11
.
m
is
no
necessa ily
he
minimal
model
o
in
he
sense
o
de ini ion
1 .2
.
Fo
ins an e,
unless
a
is
a minimal
objec ,
m
needs
no
o
be an
objec
o
a C
.
We
ha e
no
assumed
ye he exis en e
o a
minimal
model
o
e e y
objec
.
Le
us
suppose
now
ha
C
has
enough
minimals
:
o
e e y
aE
obj
C,
we
choose
a minimal
model
which
will
be
deno ed
by Pa
M(a)
-> a
.
Le
us
also
no e
by
C,
(o
Cmin
when
we
will
alk
abou
bi ib ed
ca ego ies)
he
ull
subca ego y
o
minimal
objec s
.
Thenwe
ha e
Co olla y
.
The
choice
o
a
minimal
model
de ines
a
unc o
M
:C->C
'
,
by
M(a)
=
m
a
and
M( )
=
m
;
and
a
mo phism
o
unc o s
p
:M
-
~
1C

29
0

A
.
ROIG
byp
.
:m
a
~a
.
P oo
.
Al hough
he e
a e
di e en
choices
o
M(a)
and
M(
),
he
unici y
o
P oposi ions
1
.9
and
1 .10
makes
he
co espondence
unc o-
ial
.
FYom
wha
we
ha e
seen,
i is
clea
ha
he
unc o
M
ca ies
mo -
phisms
o
S
in o
isomo phisms
o
C
m
.
In
ac ,
i
in
(2)
E
S,
hen
pa
:
m
a
~
b
is
a minimal
model
o
b
.
Hence,
he e
exis s
an
isomo -
phism
o
C/b,
cp
:
m
a
`
mb,
which,
because
o
he
uniqueness
is
equal
o
m
.
The e o e,
M
ac o izes in a
unique
ashion
h ough
he
localiza ion
unc o
,y
:
C
-->
Cs
.
Tha
is
o
say,
we
ha e
he
commu a i e
diag am
o
unc o s
M
e
s
We
will
no e
also
by
M
his
ac o iza ion
and
call
i
he
minimal
model
unc o
.
Theo em
1
.12
.

The
unc o
M
:
Cs
-->
C
m
is
an
equi alen e
o
ca -
ego ies
.
P oo
.
Le
c
:
Cm
->
CS
be
he
composi ion
o
he
inclusion
C
m
-
C
and
he
localiza ion
C
-->
Cs
.
We
will
show
ha
c is
an
equi alen e,
quasi-in e se
o
M,
by
de ining
isomo phisms
:
lc
m
--->
M
and
s
¿M
--->
lc,
Le
m
be minimal
.
Then,
by
co olla y
o P oposi ion
1
.9,
he e
exis s
an isomo phism
?7
m
:
m
->
M (m)
.
On
he
o he
hand,
de ine
Ea
:
¿M(a)
-->
a
as
he
isomo phism
in
C
s
induced
by
he choice
in
C
o
he
minimal
model
o
a,
M(a)
-->
a
.
I
is
easy o
check
ha
hese
de ini ions
a e
na u al
in
m
and
a,
espec i ely
.
Rema k
1
.13
.
We
can
easily
dualize
he
p e ious
esul s
aking
in o
accoun
ha
Hypo hesis
1
.3
a e
sel -duals
and
eplacing
he
exis en e
o
calculus
o
igh
ac ions
by
ha
o
le
ac ions
.
Aswe
ha e
said,
he exis en e
o a
calculus
o
igh
ac ions
is
ull iled
i
we
a e
wo king
in
a
model
ca ego y
.
So,
om
now
on,
we
will
assume,
ins ead
o
Hypo hesis
1
.3
and
1.8,
Hypo hesis
1
.14
.
C
is
a
model
ca ego y,
S
=
we
is
he
class
o
i s
weak
equi alen es
and
all
he
objec s
o
C
a e
ib an
.
Le
us
ecall
ha , as
a
consequence,
we
ha e
p ope ies
(i)
and
(ii)
o
Hypo hesis
1
.3
and
a
calculus
o
ac ions
in
7 C
.
To
begin
wi h
he
MINIMAL
RESOLUTIONS
AND
OTHER
MINIMAL
MODELS

291
ansla ion
o
p e ious
esul s in
his
se ing,
we
ha e
a
ela i e
e sion
o
1
.9
.
P oposi ion
1
.15
.
Le
be
a
commu a i e
diag am
o
C
in
which
a
is
a
minimal
objec
and
i
is
a
minimal
mo phism
.
Then,
i
p
is
aweak
equi alen e,
he e
exis s
~3
:
b
~
x,
unique
up
o
homo opies,
such
ha
p,C
-
,0
and
bi
-
a
.
P oo
.
Le
ús
ac o ize
p
in
a
co ib a ion
j
:
x
->
z and
a
ib a ion
q
:
z
-->
y,
bo h
i ial
ones
.
Take
he pull-back
o
~3
and
q
:
q~
b
T i ial
ib a ions
a e
s able
unde
pull-backs,
so
q'
is
a
i ial
ib a ion,
which we
can
conside
a
i ial
ib a ion
o
a C om
he
mo phism
in-
duced by ja
and
i,
y
:
a
--->
c,
o
i
.
Bu
i is
a
minimal
mo phism,
so
he e
exis s s
:
b
,
c
such ha
q's
=
16
and
si
=
y
.
On
he
o he
hand,
j
is
a
i ial
co ib a ion
:
so
i
has
a homo opic
in e se
:
z
-->
x
.
Then,
,Q's
is
he
li ing
we
a e
looking
o
:
,Qi
=
lo'si
=
~3'-y
=
ja
-
a
and
pR
=
qj ,C's
-
qO's
=
,Cq's
=
0
.
The
uniqueness
up
o
homo opy
o
/b
is
also
an
easy
e i ica ion
.
Co olla y
1
.
Le
p
:
a
Z
b
E
we
and
_
:
m
-->
b be a
mo phism
wi h
m
aminimal
objec_
.
Then
he e
exis s
:
m
->
a,
unique
up
o
homo opies,
such
ha p
-
.
P oo
..
Take
a
=
e
in
he
p e ious
p oposi ion
.
Co olla y
2
.
Two
minimal
models
o
a
E
C
a e
isomo phic
.

The
isomo phism
is
unique
up
o
homo opies
o
C/a
.
The
es o
he
esul s
admi
analogous
modi ica ions
.
29
2

A
.
RoIG
P oposi ion
1
.16
.
Le
:
a
->
b
be
a
mo phism
o
C
and
pa
:
m
a
-> a
and
pb
:
mb
-> b
wo
minimal
models
.
Then
he e
exis s
a
mo phism
m
:
m
a
`
mb,
unique
up
o
homo opies,
ha
ende s
commu a i e
up
o
homo opy
he
diag am
byp
.
:m
a
-+a
.
ma
Pa
.
a
Le
us
assume
ha
C
has
enough
minimals
.
Ci en
aE
obj
C,
le
us
choose
a minimal
model
:
pa
:
M(a)
-> a
.
Since
p e ious
esul s a e
s a ed
"up
o
homo opy",
he
co espondence
a
H
M(a)
is
no
neces-
sa ily
unc o ial
and
he
weak
equi alen es
Pa
do
no
necessa ely
de ine
a
mo phism
o
unc o s
M
-> lc
.
Ne e heless,
we
ha e
a
well-de ined
unc o
and
a
mo phism
o
unc o s
i
we
ake
as
a ge
he
ca ego y
7 C,,,
which
has
as
objec s
he
minimal
ones
o
C and
as
mo phisms
he
homo opy
classes o
mo phisms
o
C
.
Co olla y
.
The
choice
o a
minimal
model
de ines
a
unc o
M
:C-+7 C,,,
by
M(a)
=
m
a
and
M( )
=
m
,
and
a
mo phism
o unc o s
P
:M--->
l,c
We
also
ha e a
unique
ac o iza ion
o
M
h ough
he
localiza ion
unc o
and
Theo em
1 .12
now
eads
:
Theo em
1
.17
.
The
unc o
M
:
HoC
-->
7 C

,,
is
an
equi alen e
o
ca ego ies
.
Rema ks
1
.18
.
(1)
All
he
abo e
ac s
a e
independen
o
he
classes
o ib a ions
and
co ib a ions
we
choose
.
So,
o
a
gi en
class
o
weak
equi alen es,
all
he
possible
model
s uc u es
sha e
one
class o
co ib an
objec s
in
common,
i
hey
exis
:
he
minimal
ones
.
(2)
The
las
heo em
could
also
be deduced om
he
ac
ha
minimal
mo phisms
a e
necessa ily
co ib a ions
(in
a
closed
model
ca ego y,
a
leas )
and
hen
applying
[Qui,
Theo em
1',
Sec ion
1,
chap e
I],
aking
in o
accoun
ha
we
do
no
need
all
he
co ib an
objec s,
bu
only
one
ep esen a i e
o
each
homo opy
ype
( o
ins an e,
a
minimal
one)
.
(3)
In
o de
o
dualize
p e ious
esul s,
we
need
only o
subs i u e
co ib a ion
o
ib a ion
and
ice- e sa
.
MINIMAL
RESOLUTIONS
AND
OTHER
MINIMAL
MODELS

293
2
.
Minimal
esolu ions
o
complexes
Le
R
be
a
uni a y
commu a i e
ing
.
A
R-dg module
is
a
g aded
module
M
wi h
a
di e en ial
o deg ee
+1
;
Le
.,
a
cochain
complex
.
Le
us
ake
as
S
he
class
o
quasi-isomo phisms
;
ha
is
o
say,
he
mo phisms
which
induce
an
isomo phism
in
cohomology
.
Be o e
we
es ic
ou sel es
o
local
ings,
le
us
examine
some
examples p oduced
by
ou
de ini ion
.
P oposi ion
2
.1
.
Le
M
be
a
R-dg
module
wi h ze o di e en ial
.
Then
M
is
minimal
i
and
only
i
M
i is
a
p ojec i e
module
o
each
i
.
P oo
..
Le
M
be
a
cochain complex,
wi h
a
p ojec i e
module
in
each
deg ee,
ze o
di e en ial
and
X
=>
M
aquism
.
Then,
o
e e y
i,
we
can
choose
a
sec ion
o
Z'X
,
H'X
=
M
i
.
These
gi e
us
a
mo phism
o
R-dg
modules
M
-->
X
which
is
a
sec ion
o he
quism
abo e
.
Recip ocaly,
assume
ha
M
is
minimal
and
le
:
X
-->
M
i
be an
epimo phism
o
R-modules
.
Conside
he
mo phism
o
R-dg
modules
...

>
M
i-2

o

->
Mi-1
®
ke
(o
~
j
)
X

o
-~
Mi+l
es
.
.
.
11

1(1
0)

J-

11
>
M
i-2
0

Mi-1

----
0
----->
Mi

°

-->
Mi+l
(whe e
he
ho izon al
a ows
a e he
di e en ials
and
j
:
ke
y
X
is
he
inclusion)
.
Ob iously,
i is
a
quism
and,
since
M
is
minimal,
i
has
a
sec ion
.
In
pa icula ,
has
a
sec ion
and
so
Mi
is
a
p ojec i e
R-module
.
Co olla y
.
Le
M
be a
R-module,
conside ed
as a
homogeneous
dg
module
wi h ze o
di e en ial
.
Then
M
is
minimal
i
and
only
i i is
p ojec i e
.
P oposi ion
2
.2
.
Le
M
be a
R-dg
module
such
ha
H
i
M
is
a
p o-
jec i e
R-module
o
each
i
.
Then
HM
is
a
minimal
model
o
M
.
P oo
.
Le
Z
i
M
-H
i
M
be
he na u al
p ojec ion
and
s
i
:
H
i
M
->
Z
i
M
a
sec ion
.
Then
s
=
(si)
:
HM
->
M
is
a
quism
o
R-dg
modules
and
so
HM
is
a
model
o
M
.
Because
o
P oposi ion
2
.1
i is
a
minimal
model
.
Co olla y
1
.
I
R
has
ze o global
dimension,
hen
:
(1)
E e y R-dg module
has
a minimal
model
.
(2)
M
is
minimal
i
and
only
i
M
has
ze o
di e en ial
.
300

A
.
Roic
Example
3 .3
.
Le
R
be a
uni a y
commu a i e
ing,
Adgc(R)
he
ca ego y
o
R-dgc
algeb as
and
Adgc(R)
2
he
ca ego y
o
mo phisms
o
R-dgc
algeb as
.
The
objec s
o his
ca ego y
a e
mo phisms
a
:
A
,
B
o
R-dgc
algeb as
and
he
mo phisms
commu a i e
squa es
B
A
C
o
Adgc(R)
.
We
ake
as
P
he
domain
unc o
Adgc(R)
2
-+
Adgc(R)
which
sends
a
o
A
.
Then
he
ibe
ca ego ies
a e
he
ca ego ies
o
A-
dgc
algeb as
:
Adgc(A)
=
A Adgc(R)
.
Fo
each
mo phism
o
R-dgc
algeb as
:
A
-
C
we
ha e
a
ecip ocal
image
unc o
and a
di ec
image
unc o
*
:
Adgc(C)
---->
Adgc(A)

and

*
:
Adgc(A)
--->
Adgc(C)
de ined
by
*
(l0)
=
/3 and
*
(a)
=
_
®1
:
C
`
C
®A
B
.
The
ca esian
mo phism
,(3
and
he
coca esian
mo phism
a
a e
he
commu a i e
squa es
D
1
)
D

B

1®
-
)
C
®A
B
1a =
-(a)
1,3
A
i
C
and
C
[Roigl]
shows
how
o
endow
A
wi h
a
model
s uc u e
om
gi en
s uc u es
en £ and
A
.,
.
Ne e heless,
in
o de
e
alk
o
minimal
objec s
in
A
we
only
need
a
class o
dis inguished
mo phisms,
which
we
de ine as
ollows
:
suppose
we
a e
gi en
classes
S£
C
mo
£
and
S,,
C
mo
A,,
o
e e y
x
E
obj
£
in
such
a
way
ha
hey
a e
compa ible
wi h
ecip ocal
images
;
Le
.,
o
e e y
:
x
-->
y E
mo
£,
one
has
*
(S
9
)
C S
.
.
Then,
pu
S=
{W
:
a
->
b
E
mo
AI Po,
E
SE
and
o,Pa
E
SP
Q
}
Fo
ins ance,
a
ca esian
mo phism
belongs
o
S
i
and
only
i
i s
p o-
jec ion
is
in
SE,
and
a
ib e-mo phism
co
E
mo
A
x
belongs
o
S
i
and
only
i i is
in
S,,
.
I
is
clea
ha
his
choice
o
S
ag ees
wi h
ha
o
weak
equi alences
made
in
[Roigl,
Theo em
5
.1],
conside ing
SE
and
S,,
he
weak
equi alences
o
he
model
ca ego ies
£
and
A
.,,
espec i ely
.
In
pa icula ,
i
o
he
ca ego y
Adgc(R)
2
we
ake
SAdgc(R)
and
SA
he
classes o
quism
o
Adgc(R)
and
Adgc(A),
espec i ely,
he
mo phism
(1)
is
in
S
i
and
only
i
and
g a e
quism
.

MINIMAL
RESOLUTIONS
AND
OTHER
MINIMAL
MODELS

301
Theo em
3 .4
.
Le
m
E
obj
A
.
Then,
m
is
a
minimal
objec o
A
i
and
only
i
m
E
obj
Apm
and
Pm
E
£
a e
minimal
objec s
.
P oo
.
Le
us assume
ha
m
is
a minimal
objec
o
A
.
Le
us
show
ha
i is
so
in
Apm
.
Le
a
:
a
->
m
E
Spm
.
Then
u
E
S
and,
as
m
is
minimal
in
A,
he e
exis s
u'
:
m
--->
a
E
mo
A
such
ha
uu'
=
l
m
.
The
only
hing
we
ha e
o
p o e
is
ha
a'
E
mo
Apm
and
his
ollows
by
e alua ing
P
on bo h
sides
o
he
las
equali y,
which
gi es
us
Po,'
=
lpm
.
Le
us
see
ha
Pm
E
obj
£
is
also
a minimal
objec
.

Le
s
:
x
-->
Pm
E
S
£
.
Conside
he
ca esian
mo phism
ms
:
s*m
->
m
.
I
belongs
o
S
because
P(ms)
=
s
.
Then,
since
m
is
minimal
in
A,
he e
exis s
p
:
m
-->
s*m
such
ha
m
s
p
=
l
m
.
Hence
s(Pp)
=
lpm
.
Con e sely,
le
m
be
a
minimal
objec
o
Apm
,
Pm
a
minimal
objec
o
£ and
u
:
a
->
m
a
mo phism
o
S
.
Then
s
=
Pa
:
Pa
->
Pm
ESE
and,
as
Pm
is
minimal,
he e
exis s
:
Pm
-->
Pa
such ha
s
=
lpm
.
Le
us
conside
he
sou ce
ac o iza ion
o
a
:
o,
=
m'u'
.
Then
s
E
SPa
and
so
*
(m')
:
*a
--->
*s*m
=
m
is
in
Sp
m
.
Because
o
he
minimali y
o
m
in
Apm
,
his
mo phism
has
a
sec ion cp
:
m
-->
*
a
and
a
(p
:
m
->
a
is
a
sec ion
o
a
.
Co olla y
1
.
I
o
all
x
E
obj
£,
A
x
and £
ha e
enough
minimals,
hen
A
has
enough
minimals
.
P oo -
Le
us
build
a minimal
model
o
a
E
obj
A
:
ake
a minimal
model
o
Pa, which
exis s
by
hypo hesis
:
s
:
m'
->
Pa
.
Take
a
minimal
model
o
s*a
in
.
.4

,,
:
p
:
m
--->
s*a
.
Then
a
s
p
:
m
-->
a
is
a minimal
model
o
a
in
A
.
Co olla y
2
.
Le
A
be
endowed
wi h
a
model
ca ego y
s uc u e
such
as in
[Roig1,
Theo em
5
.1]
.
Assume
also
ha ,
o
e e y
x
E
obj
£,
A,
and
£
ha e
enough
minimals
.
Then
he
inclusion
Ami,,
--->
A
induces
an
equi alen e
o
ca ego ies
7 Amin
=
Ho
A
The
ollowing
esul
could
also
be
p o ed
o
any
sui able
ca ego y
o
mo phisms
di ec ly
om
he
de ini ions
:
Co olla y
3
.
A
mo phism
o
R-dgc
algeb as
:
A
->
B
is
a
minimal
objec
o
Adgc(R)
2
i
and
only
i
A
is
a minimal R-dgc
algeb a
and
is
a
minimal
mo phism
.
Adgc(R)
2
has
enough
minimals,
o
ins an e,
i
we
ake
R
a
ze o
global
dimension
ing
and
es ic
ou sel es
o
non-nega i e
homologi-
cally
connec ed
R-dgc
algeb as
(see
[Hal,
Chap e
9])
.
A
minimal
model
302

A
.
RoIG
o
:
A
->
B
in
he
ca ego y
o
mo phisms
is
cons uc ed
as in
Co ol-
la y
1
:
we
ake
a
minimal
model
o
A
in
Adgc(R),
p
:
MA
-
A,
and
aminimal model
o
p
:
MA
->
B
in
Adgc(MA)
;
ha
is
o
say,
a
commu a i e
iangle o
Adgc(R),
Re e en es
B
whe e
o, is
a quism
o
R-dgc
algeb as
.
In [Hal],
MA
-~
M
pis
called
a
A-minimal
A-ex ension
.
Dualizing
he
p e ious
esul s o igh
minimal
objec s
and
igh
mod-
els,
we
see
ha
in
he
ca ego y
0°Se
o
simplicial
se s,
aking
S
o
be
he
mo phisms
which
induces
isomo phisms
in
all
he
homo opy
g oups,
minimal
Kan
complexes
a e
he
minimal
objec s
in
e ms
o
( he
dual
o )
ou
De ini ion
1
.1
.
This
ollows
om
( he
dual
o )
ou
P oposi ion
1
.6
and
[May,
9
.5
and
9
.7]
.
The
same
p oposi ion
and
[May,
10
.6,
10
.7
and
10
.13],
aking
in o
accoun
ha
we
do
no
peed
he
de o ma ion
e ac s o
be
s ong,
show
ha
he
minimal
Kan
ib a ions
wi h base
B
a e
he
minimal
objec s
o
AlSe /B
.
These
esul s
and
he
dual
o
ou
Theo em
3
.2,
gi e
us
Co olla y
4
.
Minimal
ib e
spaces
cons i u e
he
minimal
objec s o
(AOSe )
2
.
He e,
(AOSe )
Z is
he
ca ego y
o
mo phisms
o
simplicial
se s,
bi ib ed
wi h
he
codomain
unc o
and minimal
ib e
space
means
aminimal
Kan
ib a ion
p
:
E
-->
B
wi h
B
a
minimal
Kan
complex
.
This
las
co olla y
could
also
be deduced om
he
dual
o
1 .6
and
[May,
10
.11
and
10
.16]
.
[B-G]
BOUSFIELD,
A
.
K
.
AND
GUGENHEIM,
V
.
K
.
A
.
M
.,
"On
PL
De
Rham
heo y
and
a ional
homo opy
ype,"
Mem
.
AMS
179, 1976
.
[Ei]
EILENBERG,
S
.,
Homological
dimensions
and
syzygies,
Ann
.
o
Ma hs
.
64
(1956),
328-336
.
[G-Z]
GABRIEL,
P
.
AND
ZISMAN,
M
.,
"Calculas
o
ac ions
and
ho-
mo opy
heo y,"
Sp inge ,
1967
.
[H-T]
HALPERIN,
S
.
ANDTANRÉ
D
.,
Homo opie
il ée
e
ib és
C°°,
Ill
.
J
.
o
Ma hs
.
34
(1990),
284-324
.
MINIMAL
RESOLUTIONS
AND
OTHER
MINIMAL
MODELS

303
[Ha ]
HARTSHORNE,
R
.,
"Residues
and
duali y,"
Sp inge
LNM
20,
1966
.
[Hal]
HALPERIN,
S
.,
"Lec u es
on
minimal
models,"
Mem
.
SMF,
nou-
elle
sé ie,
9/10,
1983
.
[Kan]
KAN,
D
.,
Minimal
ee
CSS
g oups,
Ill
.
J
.
o
Ma hs
.
2
(1958),
449-476
.
[Ma ]
MATSUMURA,
H
.,
"Commu a i e
ing
heo y,"
Camb idge
Uni-
e si y
P ess,
1986
.
[May]
MAY,
J
.
P
.,
"Simplicial
objec s
in
algeb aic
opology,"
Van
Nos-
and,
1967
.
[Qui]
QUILLEN,
D
.
G
.,
"Homo opical
Ageb a,"
Sp inge
LNM
47,
1967
.
[Roigl]
RoIG
A
.,
Model
ca ego y
s uc u es
in
bi ib ed
ca ego ies, o
appea
in
J
.
o
Pu e
and
Applied Algeb a
.
[Roig2]
RoIG,
A
.,
Fo malizabili y
o
DG
modules and
mo phisms
o
DGC
algeb as,
o
appea
in
Ill
.
J
.
o
Ma hs
. .
[Roig3]
RoIG,
A
.,
Modéles
minimaux
e
onc eu s
dé i és, o
appea
in
J
.
o
Pu e
and
Applied
Algeb a
.
[SGA1]
GROTHENDIECK,
A
.,
"Séminai e
de
Géome ie
Algéb ique
1,"
Sp inge
LNM
224,
1971
.
[Ta e]
TATE,
J
.,
Homology
o
noe he ian
ings
and
local ings,
Ill
.
J
.
o
Ma hs
.
1
(1957),
14--27
.
Depa amen
de
Ma emá ica
Aplicada
I,
ETSEIB
Uni e si a Poli écnica
de
Ca alunya
A
.
Diagonal
647
08028
Ba celona
SPAIN
P ime a
e sió
ebuda
el
29 de
Se emb e
de
1992,
da e a
e si6
ebuda
el
8
de
Gene de
1993