
prakshepya vastu

shreni siddhant mein, ack prakshepya vastu kii dhaarana ack prakshepya module kii dhaarana koo samanya banaati haye. abelian shreniyon mein prakshepya vastuon kaa upayog samroop beejganit mein kiya jaataa haye. ack prakshepya vastu kii dohri dhaarana ack injection vastu kii haye.
paribhasha.
ack shreni sootra _ 2 mein ack vastu sootra _ 1 "prakshepya" haye yadi kissy bhee aparupata sootra _ 3 aur rupankan sootra _ 4 key liye, ack rupankan sootra _ 5 haye joe sootra _ 6, i haye.ee. nimnalikhit aarekh awagaman kartaa haih
arthaat, pratyek roopaatmak sootra key madhyam seey pratyek roopaatmak sootra _ 7 kaarak.
yadi "see" sthaaniya roop seey chhota haye, tou aaee.ee.vishesh roop seey sootra _ 9 kissy bhee vastu "x" key liye "c" mein ack sett haye, yeh paribhasha iss shart key barabar haye kii home functor (jise core presentable functor key roop mein bhee jaanaa jaataa haye)
epimorphism koo sanrakshit kartaa haye.
abelian shreniyon mein prakshepya vastuein.
yadi shreni "see" ack abelian shreni haye, udaaharan key liye, abelian samoohon kii shreni, tou "pee" prakshepya haye yadi aur kewal yadi 
ack sateek functor haye, jahan abb abelian samoohon kii shreni haye.
ack abelian shreni sootra _ 12 koo "paryaapt prakshepak" kahaa jaataa haye yadi, sootra _ 12 key pratyek vastu sootra _ 13 key liye, sootra _ 12 kaa ack prakshepak vastu sootra _ 1 aur "p" seey "a" yaa, samkaksh, ack sankshipt sateek anukram haye.
iss paribhasha kaa uddeshya yeh sunishchit karna haye kii koee bhee vastu "aeye" ack projective resolution koo sweekaar karati haye, i.ee., ack (lamba) sateek anukram
jahan object formula _ 19 projective hain. 
pratibandhit vargon key sambandh mein pariyojana.
 ack tathaakathit dwi-shreni key saapeksh prakshepya (aur dwi-injection) vastuon kii dhaarana parr charcha kartaa haye, jisamen dee gaee shreni "see" mein "injection" aur "adhikshep" kii upa-shreniyon kii ack jodee hoti haye. yeye upashreniyaan kuchh aupchaarik gunon key adheen hain, jisamen yeh aavashyakta bhee shamil haye kii koee bhee adhikshep ack aparupata haye. ack prakshepya vastu (adhikshepon key nishchit varg key saapeksh) taba ack vastu "pee" hoti haye taaki home ("pee",-) adhikshepon key nishchit varg (sabhi upapachayon key vipareet) koo seton key adhikshepon (samanya arth mein) mein badal dey.
udaaharan.
yeh kathan kii sabhi sett prakshepya hain, pasand key swayamsiddh key barabar haye.
abelian samoohon kii shreni mein prakshepya vastuein mukt abelian samooh hain.
sootra _ 20 koo pehchaan key saath ack valay honey dein. baaen sootra _ 20-module key (abelian) shreni sootra _ 20-modd parr vichaar karein. sootra _ 20-modd mein prakshepya vastuein sateek roop seey prakshepya baaen aar-module hain. natijatan, sootra _ 20 apane aap mein sootra _ 20-modd mein ack prakshepya vastu haye. dwigunit roop seey, sootra _ 20-modd mein injective vastuein bilkul injective baaen aar-module hain.
baaen (daaen) sootra _ 20-module kii shreni mein bhee paryaapt prakshepya hain. yeh sach haye kyonki, pratyek baaen (daaen) sootra _ 20-module sootra _ 29 key liye, humm sootra _ 30 koo mukt (aur isaliye prakshepya) sootra _ 20-module key roop mein leey sakate hain joe sootra _ 29 key liye ack generating sett sootra _ 32 dwara utpann hotaa haye (udaaharan key liye humm sootra _ 32 koo sootra _ 29 key roop mein leey sakate hain). phir canonical prakshepan sootra _ 36 aavashyak adhikshep haye.
compact hasdorf space kii shreni mein projective object sateek roop seey charam roop seey disconnect kiye gaye space hain. yeh parinam ack saralikrit pramaan key kaaran haye.
banach space aur sankuchan kii shreni mein (i.ee., kaary jinake maanak adhikatam 1 hain), epimorphism sateek roop seey ghaney chhavi vaale maanchitra hain. yeh darshaata haye kii shoonya sthaan iss shreni mein ekmatra prakshepya vastu haye. halanki, gair-tuchchh sthaan hain, joe adhikshepak sankuchan key varg key sambandh mein prakshepya hain. sankuchan key saath maanak sadish rikt sthaan kii shreni mein (aur "adhikshep" key roop mein adhikshepit maanchitra), prakshepya vastuen sateek roop seey sootra _ 37-sthaan hain.