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Naïve liftings of DG modules

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Abstract

Let n be a positive integer, and let A be a strongly commutative differential graded (DG) algebra over a commutative ring R. Assume that

  1. (a)

    \(B=A[X_1,\ldots ,X_n]\) is a polynomial extension of A, where \(X_1,\ldots ,X_n\) are variables of positive degrees; or

  2. (b)

    A is a divided power DG R-algebra and \(B=A \langle X_1,\ldots ,X_n \rangle \) is a free extension of A obtained by adjunction of variables \(X_1,\ldots ,X_n\) of positive degrees.

In this paper, we study naïve liftability of DG modules along the natural injection \(A\rightarrow B\) using the notions of diagonal ideals and homotopy limits. We prove that if N is a bounded below semifree DG B-module such that \({\text {Ext}}_B ^i (N, N)=0\) for all \(i\geqslant 1\), 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.

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Notes

  1. Some authors use the cohomological notation for DG algebras. In such a case, A is described as \(A = \bigoplus _{n \leqslant 0} A ^n\), where \(A^{n} = A_{-n}\) and A is called non-positively graded.

  2. “\(\mathrm {Mon}\)” is chosen for “monomial.”

  3. Keller [19] calls these “DG modules that have property (P).”

  4. The definition of diagonal ideals originates in scheme theory. In fact, if \(A \rightarrow B\) is a homomorphism of commutative rings, then the kernel of the natural mapping \(B \otimes _AB \rightarrow B\) is the defining ideal of the diagonal set in the Cartesian product \({\text {Spec}}B \times _{{\text {Spec}}A} {\text {Spec}}B\).

  5. \(J^{[\ell ]}\) is just a notation for the \(\ell \)-th DG \(B^e\)-submodule of J in the sequence. It is not an \(\ell \)-th power of any kind.

  6. In case that \(A \rightarrow B\) is a homomorphism of commutative rings, B is projective over A, and \(J/J^2\) is projective over B, then B is smooth over A in the sense of scheme theory. In this case, \(J/J^2 \cong \Omega _{B/A}\) is the module of Kähler differentials.

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Acknowledgements

We thank the referee for reading the paper and for helpful suggestions.

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Authors and Affiliations

  1. Department of Mathematical Sciences, Georgia Southern University, Statesboro, GA, 30460, USA

    Saeed Nasseh

  2. Institute for the Advancement of Higher Education, Okayama University of Science, Ridaicho, Kitaku, Okayama, 700-0005, Japan

    Maiko Ono

  3. Graduate School of Natural Science and Technology, Okayama University, Okayama, 700-8530, Japan

    Yuji Yoshino

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  1. Saeed Nasseh
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  2. Maiko Ono
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  3. Yuji Yoshino
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Correspondence to Saeed Nasseh.

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Yuji Yoshino was supported by JSPS Kakenhi Grant 19K03448.

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Nasseh, S., Ono, M. & Yoshino, Y. Naïve liftings of DG modules. Math. Z. 301, 1191–1210 (2022). https://doi.org/10.1007/s00209-021-02951-z

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