Introduction This tex t constitute s a recor d o f th e cours e i n homologica l algebr a give n a t th e Virginia Polytechni c Institut e i n July , 1970 , unde r th e auspice s o f th e Nationa l Science Foundation' s Regiona l Conference s project . Th e natur e o f th e audienc e re - quired tha t th e cours e begi n wit h a n introductio n t o th e notio n o f module s ove r a uni - tary ring , bu t permitte d rapi d developmen t o f th e theor y fro m tha t startin g point . Th e first thre e chapter s ma y b e regarde d a s containin g materia l essentia l t o an y intro - ductory cours e i n homologica l algebra , whil e th e late r chapter s reflec t th e choice s actually mad e b y th e audienc e amon g man y possibl e specia l topic s accessibl e t o thos e who ha d mastere d th e earl y material . Thu s i t ma y b e claime d tha t th e cours e achieve d depth o f penetratio n o n a narro w front whil e i t i s admitte d tha t breadt h o f coverag e o f the entir e domai n o f homologica l algebr a wa s neithe r attempte d no r achieved . In on e importan t respec t neithe r th e cours e no r thes e note s ar e self-contained throughout, familiarit y wit h th e basi c concept s o f categor y theor y ha s bee n assumed . As basi c concept s w e refe r t o the notion s o f category , functor , duality , natura l trans - formation, natura l equivalence , monomorphism , epimorphism , abelia n category , adjoin t functors. Thi s doe s no t reflec t th e vie w tha t al l thes e notion s should , i n an y well - structured program , b e availabl e t o thos e embarkin g o n a cours e i n homologica l alge - bra rather , i t wa s a simpl e matte r o f choosin g t o utiliz e al l th e lectur e tim e availabl e to discus s homologica l algebr a an d dependin g o n th e audienc e t o acquir e th e neces - sary categorica l concept s b y thei r ow n efforts , a s an d whe n thes e concept s wer e needed. Exceptionally , w e di d tak e time , a t th e outse t o f Chapte r 4 , t o discus s push - outs an d pull-back s wit h som e care , sinc e thei r propertie s ar e centra l t o th e result s o f that chapter . A further wor d shoul d b e sai d abou t abelia n categories . Th e firs t thre e chapter s (in particular , o f course , th e first ) ar e explicitl y abou t categorie s o f modules , bu t w e remark periodicall y tha t th e argument s ar e vali d in—o r ma y b e adapte d to—an y abelia n category. Th e motivatio n fo r muc h o f th e materia l o f Chapte r 4 , however , reside s i n generalizing Ext , an d it s properties , t o abelia n categorie s no t possessin g sufficien t projectives o r injectives . Th e materia l o f Chapter s 5 an d 6 remain s highl y significan t if on e restrict s attentio n t o categorie s o f module s (especiall y i f on e permit s graded , o r bigraded, modules) however , w e continu e t o emplo y categorica l language , s o tha t our descriptions ar e expresse d i n genera l terms , whil e ou r arguments , wher e convenient , vii
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