论文标题
非交通性线性系统和非交通性椭圆曲线
Noncommutative linear systems and noncommutative elliptic curves
论文作者
论文摘要
在本文中，我们介绍了线性系统概念的非共同类似物，我们称之为螺旋$ \下划线{\ MathCal {\ Mathcal {l}}}：=（\ Mathcal {l} _ {i} _ {i}）_ {i \ in \ Mathb {z}} $ $ \ mathbb {z} $ - 索引代数$ a $。我们表明，在自然假设下，螺旋从$ {\ sf proj} \ operatoRatorName {end}（\ suespline {\ mathcal {l}}）$诱导非交换空间的形态。我们在椭圆曲线上构建了矢量束的螺旋螺旋的示例，该曲线概括了由Bondal-Polishchuk构建的线束的椭圆螺旋，其中$ a $是$ b：= \ operatatorname {end} {end}（\ undessline {\ nunderline {\ naselline {\ nasionline {\ mathcal {l}}}）的二次部分。在这种情况下，我们将$ b $确定为3级常规元素的Koszul代数$ a $的商，并表明$ {\ sf proj} b $是polishchuk的意义上的非合并椭圆曲线。一个人将其解释为将非交通椭圆形曲线嵌入到某些非共同投影平面中的立方除数，从而概括了Artin-Tate-Van den Bergh的一些众所周知的结果。
In this paper we introduce a noncommutative analogue of the notion of linear system, which we call a helix $\underline{\mathcal{L}} := (\mathcal{L}_{i})_{i \in \mathbb{Z}}$ in an abelian category ${\sf C}$ over a quadratic $\mathbb{Z}$-indexed algebra $A$. We show that, under natural hypotheses, a helix induces a morphism of noncommutative spaces from ${\sf Proj }\operatorname{End}(\underline{\mathcal{L}})$ to ${\sf Proj }A$. We construct examples of helices of vector bundles on elliptic curves generalizing the elliptic helices of line bundles constructed by Bondal-Polishchuk, where $A$ is the quadratic part of $B:= \operatorname{End}(\underline{\mathcal{L}})$. In this case, we identify $B$ as the quotient of the Koszul algebra $A$ by a normal family of regular elements of degree 3, and show that ${\sf Proj }B$ is a noncommutative elliptic curve in the sense of Polishchuk. One interprets this as embedding the noncommutative elliptic curve as a cubic divisor in some noncommutative projective plane, hence generalizing some well-known results of Artin-Tate-Van den Bergh.
