Lecture 16 10/28
- Morita context
• Time shift from the next week.
• Next Homework is due Nov. 9 .( Monday )
Morita equivalenceS @ QQ ④ = F$2324 )
R - mod -→ S - modc-
P ④ C-D= GR#Q
is c- s.
Got = ④ Q) C - )sme
R - bimood.
Go Ft idr - mod <⇒ X :P ④sQ # Ras 42,12) -binned
FOG = ids - mod ⇐ M : Q ,xqP ISas CSS) -binned .
+ conditions: Op QP P
Q P Q Q
existence of X and M ⇒ ¥9 is progenerator .
SQ - a-
Ps - Ii-
Q - n-
R
Morita equiv Cian be summarized as
( Rg S,P,Q, I , tf) .
Morita context : not requiring X, In to be =.
t PQ P and QPQ conditions.
⇒ ( Ra SP ) is a ring -
Denied Morita context :
start from ITR.PT/ P: left R - mod .
-
S : = Endp( p)
TP ( RC Pf s )
Q : = Hom,fP, E) E SE Q GR )
X -- ev : P q Q = Pg Hom! P, R)→ R
p ⑤ q 1- qlp ) .q : p→ R .
M : Q P = Him,fP, R) q P → End,Cp)
of ④ p → ( B -4 R'''
*)
check : PQP , QPQ ✓
EI,
R - mod.
P=f¥. ( not nee. generator)-
→ ( R, P, S -- Enuff PIP, Q, - - - )
R - mod --⇒ S - mailHom ( P
,-)
R
Bre : idempotent in R
.( e?- e)
P# Rie, S ⇐ e Re
( unit of S is e ).
Hoyle ,M) = EM
-
rRe → M
.
⑧e t x = ex
.
$
if e M ⇐ M,
M t M.
it em = o.
M tr o.
This is a
quotient functor .
R%/ . . . . > Is S -mod
.
1¥-
Serre quotientclosed under
M sit . eM=o .
taking extensionsand
<Hom,{ P, - )
R - mod→ S - mod
P④skf¥oh£gM|--
Pref;
Q = Ho mpf B R )
(A) Hompl P , - ) = Q ④pC - )
( write P as summand of Rt"
)Homrlp, Pg K- s ) = Q ④qP④s C - )
§ " % e -P.
want µ : Qxopp → S is E.
A
lit ) : Hom,{ P, P) = Q ,⑤zP .
Conversely ,
if
( R , P)→ ( R , S, P, Q, d , µ ) any Morita cntixtThen TFA E :
④ Q④rP -5 s.
µ
D Q ④ P -⇒ S.
* MR -
3) P is a f.g . projective module
If,
2) ⇒ 3).
X gives Q→ Hom,fp, R )µ can be written as composition .
Q④ P→Horn,! P , R) ④ P
Es s.
'@soinngjdfntfetie.s)s s
⇒ Home ( P , R ) go P X-D S
I t
⇐④Ti-F ( E9i④Pi) = I pic 9£ C- Enid ( P)
( oh , . . - ion)⑦n
(%÷ )P→ R -3 P
id⇒ P is a direct summand of Rt
"
.
-
Any Morita context ( R , S , BQ, X . M.
Qxop # s is an =⇐⇒ suj .R2
r
Pg Q ×→ R is = ⇒ surj .
Categorical properties) invariants of a ring--
I.
injective module vprojective module . v
free module. X k - us
.
→ Much) - mod .
f. g -
fk '→④ not free
.
finite presentation . Rt "-⇒ Rtb→ M→ o.( ⇒ ltomp.CM , - ) comms with fi n
.
S M is f. g . ⇒ any { Ma} getsnbjobj-ofMS.tt. ① Ma -37 M .
then ⇒ finite I' c I.
¥17- ⇒ M.
2- ( R) categorical invariant .
"End ( idk - mod ) .
Dually .( cocenter)
.
[ R , R) CR(Z - span of ab - ba . a. b GR .
Not an ideal in general .④arcb.ba#TrlR):=R/CR,R]
abelian gp .
( not a ring in general)
Mn ( h ) . k=MnlkY[ Much), Mach) ] .Tr Lab) =Tr( ba )
.
e-
R → RI? R,R]I !¥
A = ab. gp .
C- ( ab) = tcba ) t a,b ER
Ed.
R=kCx, y ) .#kEx,RKBR] : Ky = ya
xyxy f x' y'
xyxy = yxyx
[ xyx , y ]x' y'
= x y-x = y
' x' = y x'y
MCR, R ) = Span ( words in x, yl up to cyclic permutation ) .
Prof .
Counter is a categorical int -
If.
R N SMorita
( R, s, P, Q, d, tu ) . Morita context
realizing equiv .X : P q Q IR
$
In : Q q P IS
Define R → S/o, s] .
X (Epi ④ 9 :) 1-7 MCE Ei ④ Pi)only well- defined mod (SS]
.
Npi ④ S9 i) → µ ( soli ④ Pi ) = Sm (Ei④ Pi)"
X ( Pis ④ 9 i ) 1-3 µ ( Pf , ④ Pis. ) = µ ( Ei ④ Pi ) s
Factors through RICR, RT → Sl SI
M ④ N.
R
If
R-bimvd.mn: 14¥77
k
it Holla,Mt = /M/(rmFmmry,rERT#M = R
.
.
(Rg? = Rdr,RI
-
P g Q T R. i§og¥ !
c :*.QQ P Q
P Q/(#p④gf, Cpxoapr) #
[email protected]:DFun.co/R.Omt.Rj.mff-Tcs.Rj.biimgd .
TnTght exact , armmto)composition c- p④
$22
get ④ -if P QQP.in#s....Q=/T:.s:J7-
-
