WHEN ARE n-SYZYGY MODULES n-TORSIONFREE?
HIROKI MATSUI, RYO TAKAHASHI, AND YOSHINAO TSUCHIYAAbstract. Let R be a commutative noetherian ring. We consider the question of whenn-syzygy modules over R are n-torsionfree in the sense of Auslander and Bridger. Our
tools include Serre’s condition and certain conditions on the local Gorenstein property ofR. Our main result implies the converse of a celebrated theorem of Evans and Griﬃth.

1. Introduction
The notion of n-torsionfree modules was introduced by Auslander and Bridger [1] totreat the theory of torsionfree modules over integral domains in general situations. They
proved that all n-torsionfree modules are n-syzygy. It is thus natural to ask when theconverse holds, that is:
Question 1.1. When are n-syzygy modules n-torsionfree?
This question has been investigated by several authors so far. Among other things, thefollowing results of Evans and Griﬃth and of Auslander and Bridger are celebrated; see[2, Theorem 3.8], [5, Lemma 1.3] and [1, Theorem (2.17) and Proposition (4.21)].
Theorem 1.2 (Evans–Griﬃth). Let R be a commutative noetherian ring. Let n ≥ 0 be aninteger. Suppose that R satisfies (Gn−1 ) and (Sn ). Then for a finitely generated R-module
M one has:
M is n-syzygy ⇐⇒ M is n-torsionfree ⇐⇒ M satisfies (Sn ).
Theorem 1.3 (Auslander–Bridger). Let R be a commutative noetherian ring. Let n ≥ 0be an integer. Then the following are equivalent.
e n−1 ).
(1) R satisfies (G(2) For a finitely generated R-module M one has:
M is (n + 1)-syzygy ⇐⇒ M is (n + 1)-torsionfree.
e n−1 ) are certain conditions on theHere, (Sn ) is Serre’s condition, while (Gn−1 ) and (Glocal Gorenstein property of a commutative noetherian ring. See Deﬁnition 2.1 in the next
section for the deﬁnitions of the notions appearing in the theorem. Recently, Goto andTakahashi [4] have proved that the converse of Theorem 1.2 holds under the additional
assumption that R is local.
The purpose of this paper is to proceed with the study of Question 1.1. Our main result
given in the next section includes the following.
2010 Mathematics Subject Classification. 13D02, 13C60, 13H10.
Key words and phrases. n-torsionfree module, n-syzygy module, Serre’s condition, Gorenstein ring.
Hiroki Matsui was partly supported by JSPS Grant-in-Aid for JSPS Fellows 16J01067. Ryo Takahashi
was partly supported by JSPS Grants-in-Aid for Scientiﬁc Research 25400038 and 16K05098.
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HIROKI MATSUI, RYO TAKAHASHI, AND YOSHINAO TSUCHIYA

Theorem 1.4. Let R be a commutative noetherian ring. Let n ≥ 0 be an integer. Thenthe following three conditions are equivalent.
(1) R satisfies (Gn−1 ) and (Sn ).
e n−1 ).
(2) R satisfies (G(3) For a finitely generated R-module M one has:
M is n-syzygy ⇐⇒ M is n-torsionfree ⇐⇒ M satisfies (Sn ).
(4) For a finitely generated R-module M one has:
M is (n + 1)-syzygy ⇐⇒ M is (n + 1)-torsionfree.
It looks surprising that the conditions (3) and (4) are equivalent, both in the sense that(4) does not contain any requirements on Serre’s condition but (3) does, and in the sensethat (3) is on the number n while (4) is on n + 1. Furthermore, this theorem implies thatthe converse of Theorem 1.2 due to Evans and Griﬃth holds true even if R is non-local,
which is not covered by [4].
In the next section, we will ﬁrst recall the precise deﬁnitions of the notions appearing
above, and then state our main result and prove it.
2. The main result
Throughout this section, let R be a commutative noetherian ring. Denote by mod R
the category of ﬁnitely generated R-modules. We begin with recalling several deﬁnitions.
Definition 2.1. (1) A full subcategory X of mod R is said to be closed under extensions
provided that for each exact sequence 0 → L → M → N → 0 in mod R, if L and N
are in X , then so is M .
f(2) Let P1 −→ P0 → M → 0 be an exact sequence in mod R with P0 , P1 projective. The(Auslander) transpose of M is deﬁned as the cokernel of the homomorphism
HomR (f, R) : HomR (P0 , R) → HomR (P1 , R)and denoted by Tr M .
(3) Let n ≥ 0 be an integer. Let M be a ﬁnitely generated R-module. We say that(a) M is n-syzygy if there exists an exact sequence 0 → M → Pn−1 → Pn−2 → · · · →P1 → P0 in mod R with each Pi projective.
(b) M is n-torsionfree if ExtiR (Tr M, R) = 0 for all 1 ≤ i ≤ n.
(c) M satisﬁes Serre’s condition (Sn ) if the inequality depth Mp ≥ min{n, ht p} holds
for all prime ideals p of R.
e n )) if(4) For an integer n ≥ −1 the ring R is said to satisfy the condition (Gn ) (resp. (Gthe local ring Rp is Gorenstein for every prime ideal p of R with ht p (resp. depth Rp )is at most n.
We denote by Syzn (R), TFn (R) and Sn (R) the full subcategories of mod R consisting ofn-syzygy modules, n-torsionfree modules and modules satisfying (Sn ), respectively. Hereare some remarks and basic properties which we will use.
Proposition 2.2. (1) The transpose is uniquely determined up to projective summands.
(2) Every R-module is both 0-syzygy and 0-torsionfree, and satisfies (S0 ).

WHEN ARE n-SYZYGY MODULES n-TORSIONFREE?

3e −1 ).
(3) The ring R always satisfies (G−1 ) and (G(4) The finitely generated projective R-modules are both n-syzygy and n-torsionfree for all
integers n ≥ 0.
(5) Let n be a non-negative integer. The following assertions hold.
(a) If R satisfies (Sn ), then so do all the n-syzygy modules.
(b) The full subcategory Sn (R) of mod R is closed under extensions.
Proof. The statement (1) is follows from [1, Proposition (2.6)(c)], while (2) and (3) are=evident. For a ﬁnitely generated projective R-module P , the sequence 0 → P −→P →0 → · · · → 0 is exact, and Tr P = 0. This shows (4). The two assertions of (5) can beproved by using the depth lemma.
■The following theorem is the main result of this paper, including Theorem 1.4, whence
including the converse of Theorem 1.2. (Note that Theorem 1.4 asserts the equivalences(2) ⇔ (4) ⇔ (6).)
Theorem 2.3. Let R be a commutative noetherian ring. Let n be a non-negative integer.
Consider the following eight conditions.
e n−1 ).
(1) R satisfies (G(2) R satisfies (Gn−1 ) and (Sn ).
(3) TFn (R) = Sn (R).
(4) TFn (R) = Syzn (R) = Sn (R).
(5) TFn (R) = Syzn (R), and Syzn (R) is closed under extensions.
(6) TFn+1 (R) = Syzn+1 (R).
(7) Syzn (R) = Sn (R).
(8) R satisfies (Sn ), and Syzn (R) is closed under extensions.
Then the implications (1) ⇔ (2) ⇔ (3) ⇔ (4) ⇔ (5) ⇔ (6) ⇒ (7) ⇒ (8) hold. If R islocal, all the eight conditions are equivalent.
Proof. It is obvious that the implications (3) ⇐ (4) ⇒ (7) hold. The implication (2) ⇒ (4)and the equivalence (1) ⇔ (6) are nothing but Theorem 1.2 and Theorem 1.3, respectively.
Note by Proposition 2.2.4 that each of the conditions (3) and (7) implies that the Rmodule R belongs to Sn (R), i.e., R satisﬁes (Sn ). Thus the implication (7) ⇒ (8) follows
by Proposition 2.2.5(b). The combination of Proposition 2.2.5(a) with [1, Theorem (2.17)]
shows the inclusions
TFn (R) ⊆ Syzn (R) ⊆ Sn (R)if R satisﬁes (Sn ), which yields the implication (3) ⇒ (4). Proposition 2.2.5(b) also gives(4) ⇒ (5). When R is local, the equivalences (2) ⇔ (7) ⇔ (8) follow from [4, Theorem
B]. Consequently, it suﬃces to prove that the implications (1) ⇒ (2) and (5) ⇒ (1) hold.
e n−1 ). Then R satisﬁes (Gn−1 ) since for each prime ideal P ofAssume that R satisﬁes (GR it holds that ht P = dim RP ≥ depth RP . Fix a prime ideal p of R. If Rp has depth atleast n, then we have depth Rp ≥ n ≥ min{n, ht p}. Suppose that Rp has depth at mostn − 1. Then Rp is Gorenstein by assumption, and in particular it is Cohen–Macaulay.
Hence we getdepth Rp = dim Rp = ht p ≥ min{n, ht p}.

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Therefore the inequality depth Rp ≥ min{n, ht p} holds for all prime ideals p of R, that is,
R satisﬁes (Sn ). The implication (1) ⇒ (2) now follows.
Let us assume that the equality TFn (R) = Syzn (R) holds and that Syzn (R) is closed
under extensions. Let p be a prime ideal p of R such that Rp has depth at most n − 1.
Using the equality TFn (R) = Syzn (R) and [1, Proposition (2.26) and Corollary (4.18)],
we obtain the inequality
gradeR ExtiR (R/p, R) ≥ i − 1for all 1 ≤ i ≤ n. From the assumption that Syzn (R) is closed under extensions and [4,Theorem A], it follows that the R-module ExtnR (R/p, R) has grade at least n. Hence
ExtjR (ExtnR (R/p, R), R) = 0
for all integers j < n. Localization at p gives rise toExtjRp (ExtnRp (κ(p), Rp ), Rp ) = 0,
where κ(p) denotes the residue ﬁeld of the local ring Rp . If ExtnRp (κ(p), Rp ) does notvanish, then it contains κ(p) as a direct summand, which implies ExtjRp (κ(p), Rp ) = 0 for
all j < n. This means that Rp has depth at least n, which contradicts our choice of p.
Thus, we must have ExtnRp (κ(p), Rp ) = 0. Applying [3, Theorem 1.1], we observe that thelocal ring Rp is Gorenstein. Now the implication (5) ⇒ (1) follows, and the proof of thetheorem is completed.
■Acknowledgments. The authors are grateful to Tokuji Araya and Kei-ichiro Iima for theirvarious comments. They also thank the referee very much for his/her useful suggestions.
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Mathematical Society, Providence, R.I., 1969.
[2] E. G. Evans; P. Griffith, Syzygies, London Mathematical Society Lecture Note Series, 106,Cambridge University Press, Cambridge, 1985.
[3] R. Fossum; H.-B. Foxby; P. Griffith; I. Reiten, Minimal injective resolutions with applications
to dualizing modules and Gorenstein modules, Inst. Hautes Études Sci. Publ. Math. No. 45 (1975),
193–215.
[4] S. Goto; R. Takahashi, Extension closedness of syzygies and local Gorensteinness of commutative
rings, Algebr. Represent. Theory (to appear), arXiv:1502.06653.
[5] G. Leuschke; R. Wiegand, Ascent of ﬁnite Cohen-Macaulay type, J. Algebra 228 (2000), no. 2,674–681.
Graduate School of Mathematics, Nagoya University, Furocho, Chikusaku, Nagoya,Aichi 464-8602, Japan
E-mail address: m14037f@math.nagoya-u.ac.jp
Graduate School of Mathematics, Nagoya University, Furocho, Chikusaku, Nagoya,
Aichi 464-8602, Japan
E-mail address: takahashi@math.nagoya-u.ac.jp
URL: http://www.math.nagoya-u.ac.jp/~takahashi/
Graduate School of Mathematics, Nagoya University, Furocho, Chikusaku, Nagoya,
Aichi 464-8602, Japan
E-mail address: m15034w@math.nagoya-u.ac.jp