Pure derived categories and weak balanced big CohenMacaulay modules

Tsutomu Nakamura University of Verona 27 8 2019

The 8th ChinaJapan Korea International Symposium on Ring Theory

Setup R comm noeth ring d dimR は

Cot R イMEModRI E怵 F M 0 た 0 FEFlatR l

FICot R Flat RACER

1 Wi は ESpecRI dimRI i

W ニ Wiki inidModR が恣 はRd パ 無に䘏ば

X E ModR

八 人 ii

Ofたがが i が がい っがが が心 o

XX tot fOfたがX i がべ っ 一 パパ がN o

八 八 八

quasi isom

XE FlatR X がX N Yoshino2018

Nik FlatR 氺 Flat R

rK FKot R

Th me

K FlatR t NFKotR Kerが二 kpa FlatRinc

が inc adjoint pair

ModR が pure acyclic島 XRM acyclic for TIEModR

or K Flat R kpa FlatR EK FlCot RUU

X l s が

ヨ

Remark Flat R が く に は な パ 銀Enochs 1984 LESpecR

Airing D FlatA K FlatA kpa FHAな袋品を悊

MartelSalarian 2011

K Pro A ED FlatA 三 K FlCotANeeman2008 Gillespie2004

5tovicek2014

Notations

Kta Prog R イX E Ka Proj R l Hom X P acyclic た ProR

Kita Flat R YXEKadFlatRIE p.X i acyclic VEE Inj R

UlKita FKot R

11 fact

Kt.clFlCotR 1 X E Ka FKot R l HomR X F acyclic た Flat R

GProj R 州 留 改Ekta Proj R st M Kerdi

GFlatR の 留ヨに Kid FlatR st M Kerdi

GHCotR GFlatR ACot Rs Mii 文Ekta FKot R st M Kerdi

Known GPri RE Ke Pro Risked Flat REEFKot Rmoduloprojective moduloflat cotorsions

Thin Kid Proj R で Kid FICot Rが

Cor If Rm has an isolated singularity then

GProjR た GFK七 R

2

Def Holm2017 Rm local

An R module is weak balanced big CohenMacaulay

wbbCM if any system of parameters of m is a weak

regular sequence

Rem A wbbCM module M with MlmM FO has been

traditionally called a balanced big CM module

Henceforth R will be assumed to be a CM local ring

Fact Holm Flat RE wbbc.HR and the equalityholds if and only if R is regular

Def wbbCMC R wbbc.HR A Cot R

FI Cot Rs wbbCMC R s wbbCMC Rmodulo flat cotorsions

Fact R is regular い wbbCMCR イo

RemWhen R is Gorenstein wbb MR GFlat R Holm

and so wbbCMCR GFKot R

Moreover KidProj R KadProj R and this is compactly

generated Jorgensen2005 so we can talk about

purity in the triangulated category Krause 2000

where GProj RE Kac ProR E Ka FKotR E G FICot R

Thm R Gorenstein

ME GFICot R is pure injective in ModR if and only if

M is pure injective in G FICot R

Rem A similar result does not holds for Gorenstein projectives

Ex に kには1作 char k キ2 k I

The list of indec.wbt.CM pure inj R module isx お In 20 U 脈 U Rt な持 N Ii theintegralclosure

Puninski 2018 of R in Rgg x

KdFKot R EG FlCot R wbbCMC RU U

t Rt RtY 1 E