Koszul algebras and Gröbner bases of quadrics
Aldo Conca
我们目前的结果，旧的和新的，关于Koszul和G-二次性质的代数与点，曲线，三次和空间的二次低余维。
We present results, both old and new, concerning Koszul and G-quadratic properties of algebras associated with points, curves, cubics and spaces of quadrics of low codimension.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
Let $I$ be an ideal generated by quadrics in a standard graded polynomial ring $S$ over a field. A question of Avramov, Conca, and Iyengar asks whether the Betti numbers of $R = S/I$ over $S$ can be bounded above by binomial coefficients on the minimal number of generators of $I$ if $R$ is Koszul. This question has been answered affirmatively for Koszul algebras defined by three quadrics and Koszul almost complete intersections with any number of generators. We give a strong affirmative answer to the above question in the case of four quadrics by completely determining the Betti tables of heig…
Avramov, Conca, and Iyengar ask whether $β_i^S(R) \leq \binom{g}{i}$ for all $i$ when $R=S/I$ is a Koszul algebra minimally defined by $g$ quadrics. In recent work, we give an affirmative answer to this question when $g \leq 4$ by completely classifying the possible Betti tables of Koszul algebras defined by height-two ideals of four quadrics. Continuing this work, the current paper proves a structure theorem for Koszul algebras defined by four quadrics. We show that all these Koszul algebras are LG-quadratic, proving that an example of Conca of a Koszul algebra that is not LG-quadratic is min…
The Koszul property for spaces of quadrics of codimension three
本文证明了：若$\mathbb{k}$是特征不等于2的代数闭域，则几乎所有使得$\dim_{\mathbb{k}}R_2 = 3$的二次标准分次$\mathbb{k}$ -代数$R$都是Koszul.更精确地说，直到分次$\mathbb{k}$ -代数同态和平凡纤维扩张，我们发现只有两个（或三个，当$\mathbb{k}$的特征为3时）这类代数是非Koszul的.此外，我们还证明了存在非平凡的二次标准分次$\mathbb{k}$ -代数与$\dim_{\mathbb{k}}R_1 = 4$，$\dim_{\mathbb{k}}R_2 = 3$是Koszul，但不承认Gröbner基础的二次，即使在坐标的变化，从而解决了负面的一个问题，问孔卡。
The absolutely Koszul and Backelin-Roos properties for spaces of quadrics of small codimension
设$\kk$是域，$R$是具有$\dim_{\kk}R_2\le 3$的标准分次二次$\kk$ -代数，$\ov\kk$表示$\kk$的代数闭包.我们构造了一个分次满Golod同态$φ\colon P\to R\otimes_{\kk}\ov{\kk}$，使得$P$是余维的完全交，至多$3$ .进一步证明了$R$是绝对Koszul的（即每个生成的$R$ -模都有有限线性亏损）当且仅当$R$是Koszul的当且仅当$R$不是标准分次$\kk$ -代数的平凡纤维扩张，其Hilbert级数为$(1+2t-2t^3)(1-t)^{-1}$ .特别是，我们恢复早期的结果Koszul属性的Backelin，孔卡和D 'Alberti。
Koszul almost complete intersections
设$R = S/I$是标准分次多项式环$S$与由二次曲面生成的理想$I$的商。如果$R$是Koszul，Avramov，Conca和Iyengar的问题是询问在$S$上的$R$的贝蒂数是否可以由$I$的生成元的最小数目上的二项式系数上界。受前人关于三个二次曲面定义的Koszul代数的结果的启发，我们给出了Koszul几乎完全交的结构的完整分类，并在此过程中对所有这类环的上述问题给出了肯定的回答.
Quadratic numerical semigroups and the Koszul property
设$H$是一个数值半群.当伴随分次环$\operatorname{gr}_\mathfrak{m} K[H]$由二次型定义时，给出了重数$e(H)$的有效界.本文利用粘集对Koszul完全交半群进行了分类。此外，对于文献中考虑的几类数值半群（算术，复合，特殊几乎完全交，$3$ -半群，对称或伪对称$4$ -半群），我们分类那些是Koszul。
Let $R$ be a standard graded Gorenstein algebra over a field presented by quadrics. Conca, Rossi, and Valla have shown that such a ring is Koszul if $\mathrm{reg}\, R \leq 2$ or if $\mathrm{reg}\, = 3$ and $c= \mathrm{codim}\, R \leq 4$, and they ask whether this is true for $\mathrm{reg}\, R = 3$ in general. We determine sufficient conditions on a non-Koszul quadratic Cohen-Macaulay ring $R$ that guarantee the Nagata idealization $\tilde{R} = R \ltimes ω_R(-a-1)$ is a non-Koszul quadratic Gorenstein ring. We use this to negatively answer the question of Conca-Rossi-Valla, constructing non-Kos…
Homological properties of rings defined by $n+1$ general quadrics in $n$ variables
研究由n+1个一般二次曲面定义的环R和其链接Gorenstein环A的同调性质，证明其具有扩展Koszul环的同调性质，并给出Poincaré级数的具体公式。
On strongly Koszul algebras and tidy Gröbner bases
本文研究强Koszul代数的性质，证明了二次Gröbner基的整洁条件是强Koszul性的充分条件，并探讨了其在代数运算中的行为。
Given a monomial $m$ in a polynomial ring and a subset $L$ of the variables of the polynomial ring, the principal $L$-Borel ideal generated by $m$ is the ideal generated by all monomials which can be obtained from $m$ by successively replacing variables of $m$ by those which are in $L$ and have smaller index. Given a collection $\mathcal{I}=\{I_1,\ldots,I_r\}$ where $I_i$ is $L_i$-Borel for $i=1,\ldots,r$ (where the subsets $L_1,\ldots,L_r$ may be different for each ideal), we prove in essence that if the bipartite incidence graph among the subsets $L_1,\ldots,L_r$ is chordal bipartite, then t…
We prove that the multi-Rees algebra $\mathcal{R}(I_1 \oplus \cdots \oplus I_r)$ of a collection of strongly stable ideals $I_1, \ldots, I_r$ is of fiber type. In particular, we provide a Gröbner basis for its defining ideal as a union of a Gröbner basis for its special fiber and binomial syzygies. We also study the Koszulness of $\mathcal{R}(I_1 \oplus \cdots \oplus I_r)$ based on parameters associated to the collection. Furthermore, we establish a quadratic Gröbner basis of the defining ideal of $\mathcal{R}(I_1 \oplus I_2)$ where each of the strongly stable ideals has two quadric Borel gene…
Let $A$ be a noetherian Koszul Artin-Schelter regular algebra, and let $f\in A_2$ be a central regular element of $A$. The quotient algebra $A/(f)$ is usually called a (noncommutative) quadric hypersurface. In this paper, we use the Clifford deformation to study the quadric hypersurfaces obtained from the tensor products. We introduce a notion of simple graded isolated singularity and proved that, if $B/(g)$ is a simple graded isolated singularity of 0-type, then there is an equivalence of triangulated categories $\underline{\text{mcm}}\,A/(f)\cong\underline{\text{mcm}}\,(A\otimes B)/(f+g)$ of…
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。