Author:
(1) Tejas Rao. Author: Author: (1) Tejas Rao. Table of Links 1 N-Burch Ideals 1 N-Burch Ideals 2 Burch Duality and Burch Closure 2 Burch Duality and Burch Closure 3 Proof of Generalized Burch Index Theorem 3 Proof of Generalized Burch Index Theorem 4 Untwisting 4 Untwisting 5 Future Work and References 5 Future Work and References Consider a minimal free resolution of a module M over a local Noetherian ring R. Over such rings, resolutions are often infinite, for example by the The Auslander-Buchsbaum formula when depth(R) = 0 [1]. The question of periodicity in infinite resolutions is the subject of intensive research for example in the works of Eisenbud, Peeva, and Gasharov, and the central survey of Avramov [6, 5, 7]. We will consider iterated Burch Indices 1 N-Burch Ideals In this section we will flesh out some of the details of the introduction, and prove some initial results. Throughout this paper, we let (S, n, k) be a regular local ring, and for an ideal I ⊂ S write R = S/I as a local ring (R, m, k). The main thrust of the paper of Eisenbud and Dao is to consider ideals of the form BI(I) = In : (I : n) called the Burch Ideal [3]. Eisenbud and Dao restrict to the case where depth(S/I) = 0 and I 6= 0 so that Burch Ideal This allows us to define the Burch index as We begin weakening the restriction on the original Burch Ideal definitioin. However, we still restrict to the case where I 6= 0 for non-triviality, and remark that periodicity of 1×1 minors is well understood in the regular local ring case. We initially care about cases where depth(S/I) = 0. The reason is twofold. First, from the Auslander-Buchsbaum formula, if the projective dimension of M is finite, then pd(M) + dim R = depth(M) Thus if the projectve dimension of M is finite, M is free. Second, this condition, along with I 6= 0, ensures that (I : n) is a proper ideal of R, as it is well known a local Noetherian ring R is depth 0 iff xm = 0 for some nonzero x ∈ R. In particular this allows us to form the bounds However, we will also consider positive depth rings R in this paper, in which case BI(I) = R since (I : n) = I. When definitions and theorems differ for positive depth rings, we will make a disclaimer. Let I, N ⊂ S be ideals. We introduce ote that unlike in the normal Burch ideal case, BIN (I) is not necessarily contained in N, even in the case of depth 0. This is because (I : N) ⊃ (I : J) for all J ⊃ N, and this containment need not be strict. Thus let J ′ = ∪J for all J with (I : N) = (I : J). We have that Remark 1. If Burch(I) 6= 0, then depth(R) = 0. Remark 1. 2 Burch Duality and Burch Closure 3 Proof of Generalized Burch Index Theorem We are now equipped to prove the result on generalized Burch Index. First we need a helper lemma. This allows us to prove the following major lemma. After reading through the lemma, consider again this remark: 4 Untwisting In other words, under certain conditions, we can check whether the module M has N-periodicity by considering column-wise Burch periodicity of the ideals Q, which have unmixed columns as the minimal generators of Q. Proof of Lemma 5. We follow a similar proof as Lemma 4. We first show Proof of Lemma 5. 5 Future Work References [1] Maurice Auslander and David A Buchsbaum. “Homological dimension in local rings”. In: Transactions of the American Mathematical Society 85.2 (1957), pp. 390–405. [2] Michael K Brown, Hailong Dao, and Prashanth Sridhar. “Periodicity of ideals of minors in free resolutions”. In: arXiv preprint arXiv:2306.00903 (2023). [3] Hailong Dao and David Eisenbud. “Linearity of free resolutions of monomial ideals”. In: Research in the Mathematical Sciences 9.2 (2022), p. 35. [4] Hailong Dao, Toshinori Kobayashi, and Ryo Takahashi. “Burch ideals and Burch rings”. In: Algebra & Number Theory 14.8 (2020), pp. 2121–2150. [5] David Eisenbud. “Homological algebra on a complete intersection, with an application to group representations”. In: Transactions of the American Mathematical Society 260.1 (1980), pp. 35–64. [6] Juan Elias et al. Six lectures on commutative algebra. Vol. 166. Springer Science & Business Media, 1998. [7] Vesselin N Gasharov and Irena V Peeva. “Boundedness versus periodicity over commutative local rings”. In: Transactions of the American Mathematical Society 320.2 (1990), pp. 569–580. [8] Daniel R. Grayson and Michael E. Stillman. Macaulay2, a software system for research in algebraic geometry. Available at http://www2.macaulay2.com. This paper is available on arxiv under CC by 4.0 Deed (Attribution 4.0 International) license. This paper is available on arxiv under CC by 4.0 Deed (Attribution 4.0 International) license. available on arxiv available on arxiv

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On the Structure of Burch Ideals and Their Duals in Local Rings

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