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If A B are commutative R algebras then Aa zB is the categoval sum f Ad B i e tlom.p.algfAQBD tlomar.ayCA.dk can HompalglB c com Mone generally if A.Bnotnesc.com motte Homer.az aBiC SfiA C giB oe flagCbl gCblfCa aHaeA.bo B pitsaw qiA B e A AapB 40C are a I AaB se Bu A e a b HalgCb e fog D
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then Aa zB - GitHub Pages · nm 10h N Saen E Is L. Is our commotterys R nano s s miff MasN in R mad T if R is commutate M au R med can define an R R b mad stneeheanM by m r nm r m

Jul 10, 2020

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Page 1: then Aa zB - GitHub Pages · nm 10h N Saen E Is L. Is our commotterys R nano s s miff MasN in R mad T if R is commutate M au R med can define an R R b mad stneeheanM by m r nm r m

If A B are commutative R algebras

then Aa zB is the categoval sumf Ad B

i e tlom.p.algfAQBD tlomar.ayCA.dk

can HompalglB ccom

Monegenerally if A.Bnotnesc.commotte

Homer.az aBiC SfiA C giB oe

flagCbl gCblfCaaHaeA.boB

pitsawqiA B e A AapB 40C

are a I

AaB seBu A

e

a b HalgCb e fog D

Page 2: then Aa zB - GitHub Pages · nm 10h N Saen E Is L. Is our commotterys R nano s s miff MasN in R mad T if R is commutate M au R med can define an R R b mad stneeheanM by m r nm r m

Adjointness of a sinestratnfsaksR 45 N an R module

L an S module

Dd yl L thought I as an R maddens'sX

Hom z N.ph HomslSqzN.L

HompCN xL s HomsCSaN L

N Sqn Ln 7 Ian

Thinkgren g N L III Saf L

suchthat diagram comatesnm 10h

N Saen EIs L

Page 3: then Aa zB - GitHub Pages · nm 10h N Saen E Is L. Is our commotterys R nano s s miff MasN in R mad T if R is commutate M au R med can define an R R b mad stneeheanM by m r nm r m

Is our commotterys

R nanos s

miff

MasNin R mad T

if R iscommutate M au R med

can define an R R bmad stneeheanM

by m r nm r m r m

rLmr r r'm r nm

Crm r

M N RMads can console hath as1212 binds

d argues another R R bmed

R mad

r man I r m an m ran man

manor

Page 4: then Aa zB - GitHub Pages · nm 10h N Saen E Is L. Is our commotterys R nano s s miff MasN in R mad T if R is commutate M au R med can define an R R b mad stneeheanM by m r nm r m

Digression

runletif he afull ofcharatiste 2

Let ULFbe avectorspace and consider

Boku vxv b.FI b isbdrm11

Vat a q get vav F

Ceb sq raw

Claim if be Billu Z b b suchthat

b is skew symmetricb is symmetric

b b th

let on BillvBike defedby

orb yw bCwi

Define a hem fangs FEDEnd BillD

F algebrasx or

noti E I O sa getR F EuffBiIN

Page 5: then Aa zB - GitHub Pages · nm 10h N Saen E Is L. Is our commotterys R nano s s miff MasN in R mad T if R is commutate M au R med can define an R R b mad stneeheanM by m r nm r m

BillDhasthestuuteRanRmadule.xbeobaixiJb Eq.o'M

R FGY zF

exit

e FEY 11 4

1

GHFaf

e s bo fc slo.IT

etf thx 1

eke fkfefefe.io

Xe Cl 1 yo 1,0 e

Xf lb DcoD fo D f

inprtla.goeub.lb tetflbebi.tt

Page 6: then Aa zB - GitHub Pages · nm 10h N Saen E Is L. Is our commotterys R nano s s miff MasN in R mad T if R is commutate M au R med can define an R R b mad stneeheanM by m r nm r m

olds Ceb Gelb eb eb symm

offb xlfh kf b fb fb skew

b ebtfb

uniquenessnote

b b'tb D to b d syn

b d d bb c slaw

skewsyn

if d is slow dsymmor'd D 20 0

Io d o

Page 7: then Aa zB - GitHub Pages · nm 10h N Saen E Is L. Is our commotterys R nano s s miff MasN in R mad T if R is commutate M au R med can define an R R b mad stneeheanM by m r nm r m

vedwspacescc.hn

DeIAuedv space a module on a field

skeg field

Def If U a uspa elfdivisionring

and Sou p a shut we say SB

independent if Egaisio si

ai O alle

Det S spans V is all reV can hewritten es

Sais v ai et pies

Det s is a basis if it is an independent

spanmyset

Prope If S is a spanningsetsuchthat no

proper shutspansthenS is a basis

dconwsly

Page 8: then Aa zB - GitHub Pages · nm 10h N Saen E Is L. Is our commotterys R nano s s miff MasN in R mad T if R is commutate M au R med can define an R R b mad stneeheanM by m r nm r m

PI Basis min l spayif 5 Es spans

s Eaisi sits

Ea si lose 0

He indeg

indep since ifGai Si 0 9 O

s ai E9isii l

s Stai'aisi

Can any spayatcont.us ales

notanimal

Proveit yourself orwilldiscuss later


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