
dvigunit module

amoort bijganitemadhye, dwaitkaran module, jyala canonical module dekhil mhanatat, heey eka parivartaneey ringvareel module aahe jeey gulgulit vividhatechya canonical bundleshee samaan aahe. tyacha vapar grauthendic sthanik dwaitamadhye keela jaato.
vyakhya.
notherian ring "r" sathee dwaitkaran module heey eka maryaditpane tayar kelele module "m" aahe, jase qee konatyahi kamal aadarsh "m" sathee, "n" ≤ unchi ("m") asalyas "r"/"m" vector space nahishi hote aanee "n" = unchi ("m") asalyas 1-aayami asate.
duherikaran module adviteey asanyachi garaj nahee kaaran rank 1 projective modules konatyahi duherikaran moduleche tensor utpadan dekhil duherikaran module aahe. tathapi, haah ekmev marg aahe jyamadhye dwaitkaran module adviteey honyas apayashi thartoh konatehi done dwaitkaran module dilyas, eka shreni 1 prakshepak modules dusyachya tensor utpadanasathi samaroop aahe. 
visheshtah ring sthanik asalyas dwaitkaran module samarupateparyant adviteey aahe. 
notherian ringmadhye duherikaran module asane aavashyak nahee. duherikaran module asaleli konatihi ring cohain-mccaulay asane aavashyak aahe. yaulat jarr cohain-mccaulay ring hii gorenstein ringcha bhag asel tarr tyat duherikaran module aahe. visheshtah konatyahi sampoorn sthanik cohain-mccaulay ringmadhye duherikaran module asate. dvigunit moduleshivay ringjasathi paryay mhanoon dvigunit complexes vaparane kadhikadhi shakya aahe.
udaharane.
jarr "r" hii gorenstein ring asel, tarr "r" heey swatahvareel module manale jaanare eka dvigunit module aahe.
jarr "r" hii eka artinian sthanik ring asel tarr "r" chey matallis module (avashesh kshetrache injective huul) heey duherikaran module aahe. 
artinian sthanik ring "r" = "k" ["x", "y"] ("x" 2 ", y" 2 ", xy") madhye eka adviteey dwaitkaran module aahe, parantu teey "r" sathee samaroop nahee.
ring z [] madhye done naun-isomorphic dualizing modules aahet, jeey invertible idealschya done varganshi sambandhit aahet. 
sthanik ring "k" ["x", "y"] ("y" 2 ", xy") cohain-mccaulay nahee mhanoon tyat dvigunit module nahee.