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DG Structure on Length 3 Trimming Complexes and Applications to Tor Algebras
TL;DR: In this article, the iterated trimming complex associated to data yielding a complex of length $3 was considered, and an explicit algebra structure in this complex was computed in terms of the algebra structures of the associated input data.
Abstract: In this paper, we consider the iterated trimming complex associated to data yielding a complex of length $3$. We compute an explicit algebra structure in this complex in terms of the algebra structures of the associated input data. Moreover, it is shown that many of these products become trivial after descending to homology. We apply these results to the problem of realizability for Tor-algebras of grade $3$ perfect ideals, and show that under mild hypotheses, the process of "trimming" an ideal preserves Tor-algebra class. In particular, we construct new classes of ideals in arbitrary regular local rings defining rings realizing Tor-algebra classes $G(r)$ and $H(p,q)$ for a prescribed set of homological data.
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TL;DR: In this article, it was shown that a product of ideals defines a Golod ring in a regular local ring and a proper ideal with grade 4 is not Golod, but is a complete intersection.
Abstract: In this paper, we study conditions guaranteeing that a product of ideals defines a Golod ring. We show that for a $3$-dimensional regular local ring (or $3$-variable polynomial ring) $(R , \m)$, the ideal $I \m$ always defines a Golod ring for any proper ideal $I \subset R$. We also show that non-Golod products of ideals are ubiquitous; more precisely, we prove that for any proper ideal with grade $\geq 4$, there exists an ideal $J \subseteq I$ such that $IJ$ is not Golod. We conclude by showing that if $I$ is any proper ideal in a $3$-dimensional regular local ring and $\mfa \subseteq I$ a complete intersection, then $\mfa I$ is Golod.
TL;DR: In this article , an explicit free resolution of R/J with a DG algebra structure is presented. But the DG algebra resolution does not address the realizability question for ideals of class G .
Abstract: . Let ( R, m , k ) be a regular local ring of dimension 3. Let I be a Gorenstein ideal of R of grade 3. Buchsbaum and Eisenbud proved that there is a skew-symmetric matrix of odd size such that I is generated by the sub-maximal pfaffians of this matrix. Let J be the ideal obtained by multiplying some of the pfaffian generators of I by m ; we say that J is a trimming of I . In this paper we construct an explicit free resolution of R/J with a DG algebra structure. Our work builds upon a recent paper of Vandebogert. We use our DG algebra resolution to prove that recent conjectures of Christensen, Veliche and Weyman on ideals of class G hold true in our context and to address the realizability question for ideals of class G .
11 Apr 2022
TL;DR: In this article , the authors construct an explicit free resolution of R/J and compute a partial DG algebra structure on this resolution, which they use to study the Tor algebra of such trimmed ideals and use the information obtained to prove the realizability question for ideals of class G .
Abstract: . Let ( R, m , k ) be a regular local ring of dimension 3. Let I be a Gorenstein ideal of R of grade 3. Buchsbaum and Eisenbud proved that there is a skew-symmetric matrix of odd size such that I is generated by the sub-maximal pfaffians of this matrix. Let J be the ideal obtained by multiplying some of the pfaffian generators of I by m ; we say that J is a trimming of I . Building on a recent paper of Vandebogert, we construct an explicit free resolution of R/J and compute a partial DG algebra structure on this resolution. We provide the full DG algebra structure in the appendix. We use the products on this resolution to study the Tor algebra of such trimmed ideals and we use the information obtained to prove that recent conjectures of Christensen, Veliche and Weyman on ideals of class G hold true in our context. Furthermore, we address the realizability question for ideals of class G .
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671 citations
01 Jan 1998
TL;DR: The notes for a series of five lectures to the Barcelona Summer School in Commutative Algebra at the Centre de Recerca Matematica, Institut d'Estudis Catalans, July 15-26, 1996.
Abstract: This text is based on the notes for a series of five lectures to the Barcelona Summer School in Commutative Algebra at the Centre de Recerca Matematica, Institut d’Estudis Catalans, July 15–26, 1996
378 citations
TL;DR: In this paper, a factorization of the canonical map R → S as a composition of a complete intersection R → C with a Golod map C → S is presented, which is accomplished by invoking a theorem of Avramov and Backelin once one has existence of a DGΓ-algebra structure on a minimal R -free resolution of S together with detailed knowledge of the structure of the induced homology algebra Tor R (S, k ) = H ( K S ).
116 citations
"DG Structure on Length 3 Trimming C..." refers methods in this paper
...Later, a complete classification of the multiplicative structure of the Tor algebra Tor• (R/I, k) for quotient rings of projective dimension 3 was established by Weyman in [14] and Avramov, Kustin, and Miller in [3]....
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...We begin with the Tor-algebra classification provided by Avramov, Kustin, and Miller in [3]....
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44 citations
"DG Structure on Length 3 Trimming C..." refers methods in this paper
...Later, a complete classification of the multiplicative structure of the Tor algebra Tor• (R/I, k) for quotient rings of projective dimension 3 was established by Weyman in [14] and Avramov, Kustin, and Miller in [3]....
[...]
TL;DR: In this paper, it was shown that there is a real number γ > 1, such that μ R d + i ≥ γ μ Rd + i − 1 holds for all i ≥ 0, except for i = 2 in two explicitly described cases.
33 citations