论文标题
通过$ a _ {\ infty} $ - fortriangulated DG类别之间的福利 - 穆凯（Fourier-Mukai）的建设性方法用于投影空间
A constructive approach to Fourier-Mukai transforms for projective spaces via $A_{\infty}$-functors between pretriangulated dg categories
论文作者
论文摘要
我们讨论以下问题：任意傅立叶 - mukai如何转换$ ϕ：\ mathrm {d}^{\ mathrm {b}}}}（\ Mathbb {p}^a）\ rightArrow \ rightArrow \ rightArrow \ rightArrow \ rightArlm {d}^{\ mathrm {\ mathrm {\ mathrm {b} ndrive timer尺寸$ a $和$ b $的空间以明确的术语表示为同型类别之间的确切函数$ \ mathrm {k}^{\ Mathrm {b}}（\ \ Mathbb {b}^a）由该行的完整强例序列生成$ \ MATHBB {B}^a = \ {\ MATHCAL {o {O}（ - a），\ dots，\ Mathcal {o} \} \} $ and $ \ \ \ \ \ Mathbb {b}^b} \ Mathcal {O} \} $？我们表明，此问题可以简化为以下任务，该任务与任何规定的傅里叶 - 木核内核：找到$ a _ {\ infty} $ - 函数$ p $明确术语中的函数在同型上的诱导函数会产生$ \ \ \ {\ nathcal {\ boxtimes \ boxtimes \ beximes \ beximes \ beximes \ beximes \ beximes \ beximes \ beximes \ beximes { -2a，\ dots，0，~~ j = -2b，\ dots 0 \} $ in $ \ mathrm {d}^{\ mathrm {b}}}（\ Mathbb {p}^{p}^{a} \ times times \ times \ times \ mathbb {p}^p}^{b}^{b}）$。 作为我们的主要技术工具，我们提供了一个明确的公式，用于提升$ a _ {\ infty} $ - 函数$ f：\ mathbf {a} \ rightarrow \ Mathbf {b} $之间的dg类别$ \ \ \ \ m mathbf {a} $和praytriangualited的dg catem $ \ mathbf $ \ mathbf { $ \ mathbf {a} $由前赫尔的通用属性给出。作为该工具的进一步应用，我们提供了一个简单的示例，讲述了三角形类别之间的两个非晶格精确函子，这些函数重合由完整强的特殊序列生成的完整子类别。
We discuss the following problem: how can an arbitrary Fourier-Mukai transform $ϕ: \mathrm{D}^{\mathrm{b}}( \mathbb{P}^a ) \rightarrow \mathrm{D}^{\mathrm{b}}( \mathbb{P}^b )$ between the bounded derived categories of two projective spaces of dimensions $a$ and $b$ be expressed in explicit terms as an exact functor between the homotopy categories $\mathrm{K}^{\mathrm{b}}( \mathbb{B}^a ) \rightarrow \mathrm{K}^{\mathrm{b}}( \mathbb{B}^b )$ generated by the full strong exceptional sequences of the line bundles $\mathbb{B}^a = \{\mathcal{O}(-a), \dots, \mathcal{O}\}$ and $\mathbb{B}^b = \{\mathcal{O}(-b), \dots, \mathcal{O}\}$? We show that this problem can be reduced to the following task which is independent of any prescribed Fourier-Mukai kernel: finding an $A_{\infty}$-functor $P$ in explicit terms whose induced functor on homotopy categories yields the embedding of $\{ \mathcal{O}(i) \boxtimes \mathcal{O}(j) \mid i = -2a, \dots, 0, ~~j = -2b, \dots 0 \}$ into $\mathrm{D}^{\mathrm{b}}( \mathbb{P}^{a} \times \mathbb{P}^{b})$. As our main technical tool we provide an explicit formula for the lift of an $A_{\infty}$-functor $F: \mathbf{A} \rightarrow \mathbf{B}$ between a dg category $\mathbf{A}$ and a pretriangulated dg category $\mathbf{B}$ to the pretriangulated hull of $\mathbf{A}$ given by the universal property of pretriangulated hulls. As a further application of this tool, we provide a simple example of two non-isomorphic exact functors between triangulated categories that coincide on the full subcategory generated by a full strong exceptional sequence.
