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equivalence - MIT Mathematics

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Lecture 16 10/28 - Morita context Time shift from the next week . Next Homework is due Nov . 9 . ( Monday ) Morita equivalence S @ QQ = F $2324 ) R - mod - S - mod c- P C- D= G R # Q is c- s . Got = Q ) C - ) s me R - bimood . Go Ft idr - mod < X :P sQ # R as 42,12 ) - binned FOG = ids - mod M : Q ,xqP IS as CSS ) - binned . + conditions : Op QP P Q P Q Q
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Page 1: equivalence - MIT Mathematics

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

Page 2: equivalence - MIT Mathematics

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 .

Page 3: equivalence - MIT Mathematics

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.

Page 4: equivalence - MIT Mathematics

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.

Page 5: equivalence - MIT Mathematics

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)

Page 6: equivalence - MIT Mathematics

( 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.

Page 7: equivalence - MIT Mathematics

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'

Page 8: equivalence - MIT Mathematics

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

Page 9: equivalence - MIT Mathematics

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) #

Page 10: equivalence - MIT Mathematics

[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-

-


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