论文标题
分类非交通变形
Categorifying non-commutative deformations
论文作者
论文摘要
我们定义函数$ \ textrm {ncdef} _ {（z_1，\ ldots，z_n）} $的$ n $ tuple $ n $ tuple的$ k $ k $ k $ - $ k $ lineArian Abelian abelian类别$ \ Mathcal \ nathcal {z} $的非交换变形的$。在我们的分类方法中，我们将无限平面变形的基本空间视为deligne有限类别，即，有限长度的Abelian类别承认投影生成器，并具有$ n $ simplemorphist类别的简单对象类别。 更笼统地，我们定义了精确函数$ \ textrm {ncdef}_ζ$的非交易函数$ $ qualotor $ζ\ colon \ colon \ Mathcal {a} \ to \ Mathcal {Z} $ to \ Mathcal {Z} $。在这里，$ \ Mathcal {a} $的无限非交易性增厚的作用是由Abelian类别$ \ Mathcal {B} $包含$ \ MATHCAL {A} $的$ \ MATHCAL {B} $扮演的角色，并且$ \ Mathcal {a} $ fertens $ \ nathcal $ \ nathcal {a} $函数$ \ textrm {ncdef}_ζ$分配给此类$ \ MATHCAL {B} $ exact Exact Fightors $ \ MATHCAL {B} \ to \ Mathcal {Z} $的equivalence类的集合。我们证明，仅当它完全忠于第一个无穷小社区时，当它完全忠实时，无限扩展的确切函数才是完全忠实的。 我们表明，如果$ζ$完全忠实，则函数$ \ textrm {ncdef}_ζ$是由$ζ$的基本图像的扩展封闭来表示的。 我们证明，对于在封闭点$ c = \ bigCup_ {i = 1}^n c_i $上的纤维上的f \ f \ colon x \至y $，其中$ c_i $'s是不可修复的曲线，$ \ \ \ \ \ \ {\ nathcal {\ natercal {o} _ {c_i} _ {c_i} $} $ f $。我们得出的结论是，null类别ind-represents functor $ \ textrm {ncdef} _ {（\ Mathcal {o} _ {c_1}（ - 1），\ ldots，\ Mathcal {o}
We define the functor $\textrm{ncDef}_{(Z_1,\ldots,Z_n)}$ of non-commutative deformations of an $n$-tuple of objects in an arbitrary $k$-linear abelian category $\mathcal{Z}$. In our categorified approach, we view the underlying spaces of infinitesimal flat deformations as Deligne finite categories, i.e. finite length abelian categories admitting projective generators, with $n$ isomorphism classes of simple objects. More generally, we define the functor $\textrm{ncDef}_ζ$ of non-commutative deformations of an exact functor $ζ\colon \mathcal{A} \to \mathcal{Z}$ of abelian categories. Here the role of an infinitesimal non-commutative thickening of $\mathcal{A}$ is played by an abelian category $\mathcal{B}$ containing $\mathcal{A}$ and such that $\mathcal{A}$ generates $\mathcal{B}$ by extensions. The functor $\textrm{ncDef}_ζ$ assigns to such $\mathcal{B}$ the set of equivalence classes of exact functors $\mathcal{B} \to \mathcal{Z}$ which extend $ζ$. We prove that an exact functor on an infinitesimal extension is fully faithful if and only if it is fully faithful on the first infinitesimal neighbourhood. We show that if $ζ$ is fully faithful, then the functor $\textrm{ncDef}_ζ$ is ind-represented by the extension closure of the essential image of $ζ$. We prove that for a flopping contraction $f\colon X\to Y$ with the fiber over a closed point $C = \bigcup_{i=1}^n C_i$, where $C_i$'s are irreducible curves, $\{\mathcal{O}_{C_i}(-1)\}$ is the set of simple objects in the null-category for $f$. We conclude that the null-category ind-represents the functor $\textrm{ncDef}_{(\mathcal{O}_{C_1}(-1),\ldots,\mathcal{O}_{C_n}(-1))}$.
