The Auslander-Reiten conjecture
for non-Gorenstein Cohen-Macaulay rings
Conference on Commutative Algebra and its Interaction with Algebraic Geometry In Honor ofBernd Ulrich
Shinya Kumashiro (Chiba University, Japan)

1. Introduction

3. Sketch of proofs

Auslander-Reiten conjecture (AR conjecture)
Let R be an Artin algebra and M a f.g. R-module. If>0
ExtR (M, M ⊕ R) = 0,then M is a projective R-module.

3-1. Proof of Theorem A

Fact (Araya, Huneke-Leuschke)
The AR conjecture holds for Gorenstein rings which arecomplete intersections in codimension one.
⇒ How about the AR conjecture for non-Gorenstein rings?

2. Main Results
Key Theorem
Suppose that R is a Gorenstein local ring. Let
Q = (x1, x2, . . . , xn) be an ideal of R generated by a regularsequence on R. Then TFAE.
1. The AR conjecture holds for R.
ℓ
2. ∃ℓ > 0 s.t. the AR conjecture holds for R/Q .
ℓ
3. 1 ≤ ∀ℓ ≤ n, the AR conjecture holds for R/Q .

Proposition
Let 2s ≤ t + 1 be positive integers. Suppose that▶ R is a Gorenstein local ring and▶ {xij }1≤i≤s,1≤j≤t forms a regular sequence on R.
Let I denote an ideal of R generated by maximal
minors of (xij ). Thenthe AR conjecture holds for R ⇔ it holds for R/I .

3-2. Proof of Theorem BProposition
With the assumption of Theorem B, we have thefollowing.
▶ d + r ≤ v.
▶ ∃S: a c.i. local ring of dimension r and2∼
∃Q: a parameter ideal of S s.t. R/q = S/Q .
∗ d : dimension, r : CM type, v :embedding dimension.

3-3. Proof of Key Theorem

This theorem provides two applications, i.e., Theorem A and Theorem B.

We focus on a proof of 1. ⇒ 3. for the case where2-1. First application (determinantal rings)ℓ = 2 and n = dim R.

Let s ≤ t be positive integers. Set▶ A[X] = A[Xij ]1≤i≤s,1≤j≤t : poly. ring over a commutative ring A▶ Is (X) : ideal of A[X] generated by the maximal minors of (Xij ).
With these notations, we have the following.

Theorem A (determinantal rings)
Suppose that▶ 2s ≤ t + 1 and▶ A is a Gorenstein ring which is a c.i. in codimension one.
Then the AR conjecture holds for the determinantal ringA[X]/Is (X).

2-2. Second application (Ulrich ideals)
Theorem B (Ulrich ideals)
Let (R, m) be a CM local ring. Suppose that
∃I : Ulrich ideal s.t. R/I is a c.i.
Then the AR conjecture holds for R.
Here is the deﬁnition of Ulrich ideals.
Let (R, m) be a CM local ring and I an m-primary ideal.
Then I is an Ulrich ideal if▶ I 2 = qI for some parameter ideal q of R and2
▶ I /I is a free R/I -module.

(1. ⇒ 3.) Set R = R/Q 2. Let M be a f.g. R-module s.t.
>0ExtR (M, M ⊕ R) = 0. Consider the exact sequence
0 → Q/Q → R → R/Q → 0i+1i⊕n∼of R-modules. Then Ext (M, R/Q) = Ext (M, R/Q) for2RR≫0all i > 0. Hence ExtR (M, R/Q) = 0, whence>0
ExtR (M, R/Q) = 0. Thus
RTor>0(M, R/Q) = 0

since R/Q is an Artinian Gorenstein ring. Hence we have theexact sequence⊕n0 → (M/QM)→ M → M/QM → 0of R-modules. Then, similarly to the above,>0RExtR (M, M/QM) = 0. Thus, since Tor>0(M, R/Q) = 0, wehave>0
ExtR/Q (M/QM, M/QM ⊕ R/Q) = 0.
Hence M/QM is a free R/Q-module since the AR conjectureholds for R/Q. This concludes that M is a free R-module since
R
Tor1 (M, R/Q) = 0.
□

4. References▶ [M. Auslander, I. Reiten], On a generalized version of theNakayama conjecture, Proceedings of the American MathematicalSociety, 52 (1975), 69–74.
▶ [S. Kumashiro], Auslander-Reiten conjecture for non-Gorenstein
Cohen-Macaulay rings, arXiv:1906.02669.
This work was supported by JSPS KAKENHI Grant Number JP19J10579.