Guillermo Cortiñas, Susan C. Geller, Charles A. Weibel
We propose an Artinian version of Berger's Conjecture for curves, concerning the module of Kähler differentials of an algebra. Our version implies Berger's Conjecture in characteristic 0. We establish our Artinian Berger Conjecture in a number of cases, and prove that Berger's Conjecture holds for curve singularities whose conductor ideal contains the cube of a maximal ideal.
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In this short note, we confirm a conjecture of Vasconcelos which states that the Rees algebra of any Artinian almost complete intersection monomial ideal is almost Cohen-Macaulay.
This article concerns commutative algebras over a field $k$ of characteristic zero which are finite dimensional as vectorspaces, and particularly those of such algebras which are graded. Here the term graded is applied to non-negatively graded algebras $A$ with $A_0$ reduced and finite dimensional. Thus the trivial grading $A=A_0$ is only allowed if $A$ is a product of finite field extensions of $k$. It has been conjectured (G. Cortiñas, S. Geller, C. Weibel; The Artinian Berger Conjecture. Math. Zeitschrift {\bf 228} 3 (1998) 569-588) that for all finite dimensional algebras $A$ which are not…
Telescope conjecture for quiver representations over artinian rings
本文证明了在艺术环上有限有向图表示的导出范畴中的准射线猜想，并扩展到交换完美环的非noetherian背景下。
For any finitely generated module $M$ with non-zero rank over a commutative one-dimensional Noetherian local domain, we study a numerical invariant $\operatorname{h}(M)$ based on a partial trace ideal of $M$. We study its properties and explore relations between this invariant and the colength of the conductor. Finally we apply this to the universally finite module of differentials $Ω_{R/k}$, where $R$ is a complete $k$-algebra with $k$ any perfect field, to study a long-standing conjecture due to R. W. Berger.
Let $(R,\mathfrak{m},\mathbb{k})$ be an equicharacteristic one-dimensional complete local domain over an algebraically closed field $\mathbb{k}$ of characteristic 0. R. Berger conjectured that R is regular if and only if the universally finite module of differentials $Ω_R$ is a torsion-free $R$-module. We give new cases of this conjecture by extending works of Güttes (Arch Math 54:499-510, 1990) and Cortiñas et al. (Math Z 228:569-588, 1998).This is obtained by constructing a new subring $S$ of $\operatorname{Hom}_R(\mathfrak{m},\mathfrak{m})$ and constructing enough torsion in $Ω_S$, enabling…
We study the conjecture that if $I,J$ are $μ-$primary in a regular local ring $(R,μ)$ with dim$(R)=n$, then $\frac{I \cap J}{IJ} \cong \Tor_1(R/I, R/J)$ needs at least $n$ generators, and a related conjecture about the number of generators of the first Koszul homology module of an Artinian local ring $(A,m)$. In this manuscript, we focus our attention on the complete intersection defect of the Artinian ring and its quotient by the Koszul elements. We prove that the number of generators of the first Koszul homology module of $x_1,...,x_n \in m$ on an Artinian local ring $(A,m)$ is at least $n+ …
At the 2015 Workshop on Lefschetz Properties of Artinian Algebras, Junzo Watanabe conjectured that every graded Artinian complete intersection algebra with the standard grading can be embedded into another such algebra cut out by quadratic generators. We verify this conjecture in the case where the defining polynomials split into linear factors.
Tor and Ext vanishing results for commutative Artinian rings
该研究针对交换阿廷环领域的三个猜想，通过构造特定的代数与模，对Avramov等人、Huneke等人的猜想给出反例，回答了Dao的问题，推进了Tachikawa猜想的相关研究。
Let $I$ be a homogeneous Artinian ideal in a polynomial ring $R=k[x_1,...,x_n]$ over a field $k$ of characteristic 0. We study an equivalent condition for the generic initial ideal $\gin(I)$ with respect to reverse lexicographic order to be almost reverse lexicographic. As a result, we show that Moreno-Socias conjecture implies Fröberg conjecture. And for the case $\Codim I \le 3$, we show that $R/I$ has the strong Lefschetz property if and only if $\gin(I)$ is almost reverse lexicographic. Finally for a monomial complete intersection Artinian ideal $I=(x_1^{d_1},...,x_n^{d_n})$, we prove that…
We study the WLP and SLP of artinian monomial ideals in $S=\mathbb{K}[x_1,\dots ,x_n]$ via studying their minimal free resolutions. We study the Lefschetz properties of such ideals where the minimal free resolution of $S/I$ is linear for at least $n-2$ steps. We give an affirmative answer to a conjecture of Eisenbud, Huneke and Ulrich for artinian monomial ideals with almost linear resolutions.
Let $R$ be a commutative noetherian ring, $I,J$ be two ideals of $R$, $M$ be an $R$-module, and $\mathcal{S}$ be a Serre class of $R$-modules. A positive answer to the Huneke$^,$s conjecture is given for a noetherian ring $R$ and minimax $R$-module $M$ of krull dimension less than 3, with respect to $\mathcal{S}$. There are some results on cofiniteness and artinianness of local cohomology modules with respect to a pair of ideals. For a ZD-module $M$ of finite krull dimension and an integer $n\in\mathbb{N}$, if $\lc^{i}_{I,J}(M)\in\mathcal{S}$ for all $i>n$, then $\lc^{i}_{I,J}(M)/\fa^{j}\lc^{i…
For any finitely generated module $M$ with non-zero rank over a commutative one dimensional Noetherian local domain, the numerical invariant $h(M)$ was introduced and studied in the author's previous work "Partial Trace Ideals and Berger's Conjecture". We establish a bound on it which helps capture information about the torsion submodule of $M$ when $M$ has rank one and it also generalizes the discussion in the mentioned previous article. We further study bounds and properties of $h(M)$ in the case when $M$ is the canonical module $ω_R$. This in turn helps in answering a question of S. Greco a…
We study the asymptotic behavior of the Castelnuovo-Mumford regularity along chains of graded ideals in increasingly larger polynomial rings that are invariant under the action of symmetric groups. A linear upper bound for the regularity of such ideals is established. We conjecture that their regularity grows eventually precisely linearly. We establish this conjecture in several cases, most notably when the ideals are Artinian or squarefree monomial.
We establish a link between trace modules and rigidity in modules over Noetherian rings. Using the theory of trace ideals we make partial progress on a question of Dao, and on the Auslander-Reiten conjecture over Artinian Gorenstein rings.
We study the weak Lefschetz property and the Hilbert function of level Artinian monomial almost complete intersections in three variables. Several such families are shown to have the weak Lefschetz property if the characteristic of the base field is zero or greater than the maximal degree of any minimal generator of the ideal. Two of the families have an interesting relation to tilings of hexagons by lozenges. This lends further evidence to a conjecture by Migliore, Miro-Roig, and the second author. Finally, using our results about the weak Lefschetz property, we show that the Hilbert function…
The $n$-type vectors introduced by Geramita, Harima and Shin are in 1-1 correspondence with the Hilbert functions Artinian of lex ideals. Letting $\mathbb{A} =\{a_1,..., a_n\}$ define the degrees of a regular sequence, we construct ${\rm lpp}_{\le\mathbb{A}}$-vectors which are in 1-1 correspondence with the Hilbert functions of certain lex plus powers ideals (depending on $\mathbb{A}$). This construction enables us to show that the residual of a lex plus powers ideal in an appropriate regular sequence is again a lex plus powers ideal. We then use this result to show that the Eisenbud-Green-Har…
Tensor products and solutions to two homological conjectures for Ulrich modules
本文刻画 Cohen-Macaulay 局部环上有限生成模张量积成为 Ulrich 模的条件，并据此证明 Auslander-Reiten 与 Huneke-Wiegand 猜想对 Ulrich 模成立，同时给出自由性判据、完全交刻画及 Berger 猜想的新途径。
We use the theory of resolutions for a given Hilbert function to investigate the multiplicity conjectures of Huneke and Srinivasan and Herzog and Srinivasan. To prove the conjectures for all modules with a particular Hilbert function, we show that it is enough to prove the statements only for elements at the bottom of the partially ordered set of resolutions with that Hilbert function. This enables us to test the conjectured upper bound for the multiplicity efficiently with the computer algebra system Macaulay 2, and we verify the upper bound for many Artinian modules in three variables with s…
Let $(R,\mathfrak{m}_R,k)$ be a one-dimensional complete local reduced $k$-algebra over a field of characteristic zero. R. Berger conjectured that $R$ is regular if and only if the universally finite module of differentials $Ω_R$ is torsion free. When $R$ is a domain, we prove the conjecture in several cases. Our techniques are primarily reliant on making use of the valuation semi-group of $R$. First, we establish a method of verifying the conjecture by analyzing the valuation semi-group of $R$ and orders of units of the integral closure of $R$. We also prove the conjecture in the case when ce…
In this note we study trace ideals of canonical modules. Characterizations of the trace ideals in terms of annihilators of certain Ext modules are given. We apply our results to study many classes of rings close to being Gorenstein that appear in recent literature. We discuss the behavior of nearly Gorensteinness, introduced by Herzog-Hibi-Stamate, under reductions by regular sequences. A nearly Gorenstein version of the Tachikawa conjecture is posed, and an affirmative answer given for type 2 rings. We also introduce the class of weakly almost Gorenstein rings using the canonical module, and …
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