Resolution and Tor Algebra Structures of Grade 3 Ideals Defining Compressed Rings.

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Resolution and Tor Algebra Structures of Grade 3 Ideals Defining Compressed Rings.

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Resolution and Tor Algebra Structures of Grade 3 Ideals Defining Compressed Rings.

14 Apr 2020-arXiv: Commutative Algebra-

TL;DR: In this article, it was shown that all ideals of the above form are resolved by an iterated trimming complex, which is a new conjecture of Avramov not already constructed by Christensen, Veliche, and Weyman.

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Abstract: Let $R=k[x,y,z]$ be a standard graded $3$-variable polynomial ring, where $k$ denotes any field. We study grade $3$ homogeneous ideals $I \subseteq R$ defining compressed rings with socle $k(-s)^{\ell} \oplus k(-2s+1)$, where $s \geq3$ and $\ell \geq 1$ are integers. The case for $\ell =1$ was studied in a previous paper by the author; a generically minimal resolution was constructed for all such ideals. More recently, this resolution is generalized in the guise of (iterated) trimming complexes. In this paper, we show that all ideals of the above form are resolved by an iterated trimming complex. Moreover, we apply this machinery to construct ideals $I$ such that $R/I$ is a ring of Tor algebra class $G (r)$ for some fixed $r \geq2$, and $R/I$ may be chosen to have arbitrarily large type. In particular, this provides a new class of counterexamples to a conjecture of Avramov not already constructed by Christensen, Veliche, and Weyman.

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DG Structure on Length 3 Trimming Complexes and Applications to Tor Algebras

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Keller VandeBogert, Keller VandeBogert¹•Institutions (1)

University of Notre Dame¹

24 Nov 2020-arXiv: Commutative Algebra

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.

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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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3 citations

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Minimal Free Resolutions of Certain Equigenerated Monomial Ideals

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Keller VandeBogert

05 Jul 2020-arXiv: Commutative Algebra

TL;DR: In this paper, the authors study certain classes of equigenerated monomial ideals with the property that the complementary ideal has no linear relations on the generators and use iterated trimming complexes to deduce Betti numbers for such ideals.

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Abstract: Let $R = k[x_1, \dotsc , x_n]$ denote the standard graded polynomial ring over a field $k$. We study certain classes of equigenerated monomial ideals with the property that the so-called complementary ideal has no linear relations on the generators. We then use iterated trimming complexes to deduce Betti numbers for such ideals. Furthermore, using a result on splitting mapping cones by Miller and Rahmati, we construct the minimal free resolutions for all ideals under consideration explicitly and conclude with questions about extra structure on these complexes.

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3 citations

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Linear Strands and Minimal Free Resolutions of Certain Equigenerated Monomial Ideals

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Keller VandeBogert

05 Jul 2020

TL;DR: In this paper, the author deduced Betti tables for equigenerated monomial ideals and used these Betti numbers to construct an explicit linear strand, and, in the case where the ideals under consideration have linear resolution, explicit minimal free resolutions.

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Abstract: Let $R = k[x_1, \dotsc , x_n]$ denote the standard graded polynomial ring over a field $k$. We study certain classes of equigenerated monomial ideals and use a previous construction of the author to deduce Betti tables for such ideals. Using these Betti numbers, we are then able to construct an explicit linear strand, and, in the case where the ideals under consideration have linear resolution, explicit minimal free resolutions.

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2 citations

Cites methods from "Resolution and Tor Algebra Structur..."

...These complexes have previously been used to resolve homogeneous grade 3 ideals I ⊂ k[x, y, z] defining compressed rings with socle k(−s)⊕k(−2s+1) (l > 1) in [15], in which case these complexes are generically minimal....
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References

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Journal Article•DOI•

Algebra Structures for Finite Free Resolutions, and Some Structure Theorems for Ideals of Codimension 3

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David A. Buchsbaum, David Eisenbud

01 Jun 1977-American Journal of Mathematics

671 citations

"Resolution and Tor Algebra Structur..." refers background in this paper

...A result of Buchsbaum and Eisenbud (see [3]) established that any quotient R/I of R with projective dimension 3 admits the structure of an associative commutative differental graded (DG) algebra....
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Journal Article•DOI•

Poincaré series of modules over local rings of small embedding codepth or small linking number

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Luchezar L. Avramov¹, Andrew R. Kustin², Matthew J. Miller²•Institutions (2)

Sofia University¹, University of South Carolina²

01 Oct 1988-Journal of Algebra

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 ).

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116 citations

"Resolution and Tor Algebra Structur..." refers methods in this paper

...Later, a complete classification of the multiplicative structure of the Tor algebra Tor• (R/I, k) for such quotients was established by Weyman in [8] and Avramov, Kustin, and Miller in [2]....
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Journal Article•DOI•

On the structure of free resolutions of length 3

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Jerzy Weyman¹, Jerzy Weyman²•Institutions (2)

Northeastern University¹, Polish Academy of Sciences²

01 Oct 1989-Journal of Algebra

44 citations

"Resolution and Tor Algebra Structur..." refers methods in this paper

...Later, a complete classification of the multiplicative structure of the Tor algebra Tor• (R/I, k) for such quotients was established by Weyman in [8] and Avramov, Kustin, and Miller in [2]....
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Journal Article•DOI•

A cohomological study of local rings of embedding codepth 3

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Luchezar L. Avramov¹•Institutions (1)

University of Nebraska–Lincoln¹

01 Nov 2012-Journal of Pure and Applied Algebra

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.

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33 citations

Journal Article•DOI•

Trimming a Gorenstein ideal

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Lars Winther Christensen, Oana Veliche, Jerzy Weyman

01 Aug 2019-Journal of Commutative Algebra

TL;DR: In this paper, the authors show how to trim a Gorenstein ideal in a regular local ring of dimension 3 to obtain an ideal that defines a quotient ring that is close to the ideal in the sense that its Koszul homology algebra is a Poincare duality algebra.

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Abstract: Let $Q$ be a regular local ring of dimension $3$. We show how to trim a Gorenstein ideal in $Q$ to obtain an ideal that defines a quotient ring that is close to Gorenstein in the sense that its Koszul homology algebra is a Poincare duality algebra $P$ padded with a nonzero graded vector space on which $P_{\ge 1}$ acts trivially. We explicitly construct an infinite family of such rings.

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13 citations

"Resolution and Tor Algebra Structur..." refers background or methods in this paper

...This process is used by Christensen, Veliche, and Weyman (see [4]) in the case that R/I is a Gorenstein ring to produce ideals defining rings with certain Tor algebra classification, negatively answering a question of Avramov in [1]....
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...In particular, these ideals provide a class of counterexamples to the conjecture of Avramov that is not already contained in [4]....
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...In particular, this provides a new class of counterexamples to a conjecture of Avramov not already constructed by Christensen, Veliche, and Weyman in [4]....
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