TY - JOUR

T1 - Naïve liftings of DG modules

AU - Nasseh, Saeed

AU - Ono, Maiko

AU - Yoshino, Yuji

N1 - Publisher Copyright:
© 2022, The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature.

PY - 2022/5

Y1 - 2022/5

N2 - Let n be a positive integer, and let A be a strongly commutative differential graded (DG) algebra over a commutative ring R. Assume that (a)B= A[X1, … , Xn] is a polynomial extension of A, where X1, … , Xn are variables of positive degrees; or(b)A is a divided power DG R-algebra and B= A⟨ X1, … , Xn⟩ is a free extension of A obtained by adjunction of variables X1, … , Xn of positive degrees. In this paper, we study naïve liftability of DG modules along the natural injection A→ B using the notions of diagonal ideals and homotopy limits. We prove that if N is a bounded below semifree DG B-module such that ExtBi(N,N)=0 for all i⩾ 1 , then N is naïvely liftable to A. This implies that N is a direct summand of a DG B-module that is liftable to A. Also, the relation between naïve liftability of DG modules and the Auslander-Reiten Conjecture has been described.

AB - Let n be a positive integer, and let A be a strongly commutative differential graded (DG) algebra over a commutative ring R. Assume that (a)B= A[X1, … , Xn] is a polynomial extension of A, where X1, … , Xn are variables of positive degrees; or(b)A is a divided power DG R-algebra and B= A⟨ X1, … , Xn⟩ is a free extension of A obtained by adjunction of variables X1, … , Xn of positive degrees. In this paper, we study naïve liftability of DG modules along the natural injection A→ B using the notions of diagonal ideals and homotopy limits. We prove that if N is a bounded below semifree DG B-module such that ExtBi(N,N)=0 for all i⩾ 1 , then N is naïvely liftable to A. This implies that N is a direct summand of a DG B-module that is liftable to A. Also, the relation between naïve liftability of DG modules and the Auslander-Reiten Conjecture has been described.

KW - DG algebra

KW - DG module

KW - DG quasi-smooth

KW - DG smooth

KW - Free extensions

KW - Lifting

KW - Naïve lifting

KW - Polynomial extensions

KW - Weak lifting

UR - http://www.scopus.com/inward/record.url?scp=85123100335&partnerID=8YFLogxK

U2 - 10.1007/s00209-021-02951-z

DO - 10.1007/s00209-021-02951-z

M3 - Article

AN - SCOPUS:85123100335

SN - 0025-5874

VL - 301

SP - 1191

EP - 1210

JO - Mathematische Zeitschrift

JF - Mathematische Zeitschrift

IS - 1

ER -