持续模块的Künneth定理连续体
A continuum of Künneth theorems for persistence modules
AI总结：
本文发展了持续模块的同调代数理论，提出新的张量积结构并证明了Künneth短正合列，为持续同调计算提供了新的工具。
摘要：
我们发展了持续模块的同调代数理论，针对单参数和多参数情况。对于偏序集P和顺序保持映射φ:P×P→P，引入了新的张量积结构⊗_φ。证明了每个⊗_φ都有右伴随Hom^φ，即依赖于φ的持续模块内部同态。证明了每个⊗_φ产生持续模块链复形的Künneth短正合列。对偶地，Hom^φ也产生同调的Künneth短正合列。作为特例，这些短正合列产生通用系数定理。我们展示了如何将这些应用于由过滤的CW复形产生的持续模块链复形。对于P=R_+的特殊情况，每个p∈(0,∞]的p-准范数产生不同的⊗_{ℓ^p_c}及其伴随Hom^{ℓ^p_c}。我们明确计算了其导出函子Tor^{ℓ^p_c}和Ext_{ℓ^p_c}。我们展示了通用系数定理可用于计算非紧空间的持续Borel-Moore同调。最后，我们展示了对于每个p∈[1,∞]，相应的Künneth短正合列可用于显著加速和近似产品度量空间(X×Y,d^p)的持续同调计算。
英文摘要：
We develop new aspects of the homological algebra theory for persistence modules, in both the one-parameter and multi-parameter settings. For a poset $P$ and an order preserving map $φ:P\times P\to P$, we introduce a novel tensor product of persistence modules indexed by $P$, $\otimes_φ$. We prove that each $\otimes_φ$ has a right adjoint, $\mathbf{Hom}^φ$, the internal hom of persistence modules that also depends on $φ$. We prove that every $\otimes_φ$ yields a Künneth short exact sequence of chain complexes of persistence modules. Dually, the $\mathbf{Hom}^φ$ also has an associated Künneth short exact sequence in cohomology. As special cases both of these short exact sequences yield Universal Coefficient Theorems. We show how to apply these to chain complexes of persistence modules arising from filtered CW complexes. For the special case of $P=\mathbb{R}_+$, the $p$-quasinorms for each $p\in (0,\infty]$ yield a distinct $\otimes_{\ell^p_c}$ and its adjoint $\mathbf{Hom}^{\ell^p_c}$. We compute their derived functors, $\mathbf{Tor}^{\ell^p_c}$ and $\mathbf{Ext}_{\ell^p_c}$ explicitly for interval modules. We show that the Universal Coefficient Theorem developed can be used to compute persistent Borel-Moore homology of a filtration of non-compact spaces. Finally, we show that for every $p\in [1,\infty]$ the associated Künneth short exact sequence can be used to significantly speed up and approximate persistent homology computations in a product metric space $(X\times Y,d^p)$ with the distance $d^p((x,y),(x',y'))=||d_X(x,x'),d_Y(y,y')||_p$.
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。
