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 $\varphi:P\times P\to P$, we introduce a novel tensor product of persistence modules indexed by $P$, $\otimes_{\varphi}$. We prove that each $\otimes_{\varphi}$ has a right adjoint, $\mathbf{Hom}^{\varphi}$, the internal hom of persistence modules that also depends on $\varphi$. We prove that every $\otimes_{\varphi}$ yields a Künneth short exact sequence of chain complexes of persistence modules. Dually, the $\mathbf{Hom}^{\varphi}$ 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 $d^p((x,y),(x',y'))=||d_X(x,x'),d_Y(y,y')||_p$.

翻译：我们发展了持久模同调代数理论的新方面，涵盖单参数与多参数情形。对于偏序集$P$及其保序映射$\varphi:P\times P\to P$，我们引入一种由$P$索引的持久模新型张量积$\otimes_{\varphi}$。证明每个$\otimes_{\varphi}$均具有右伴随$\mathbf{Hom}^{\varphi}$，即同样依赖于$\varphi$的持久模内部同态。我们证明每个$\otimes_{\varphi}$都能导出持久模链复形的Künneth短正合列。对偶地，$\mathbf{Hom}^{\varphi}$在上同调中也具有关联的Künneth短正合列。作为特例，这两种短正合列均可导出万有系数定理。我们展示了如何将这些结果应用于来自滤过CW复形的持久模链复形。对于$P=\mathbb{R}_+$的特例，每个$p\in (0,\infty]$对应的$p$-拟范数产生不同的$\otimes_{\ell^p_c}$及其伴随$\mathbf{Hom}^{\ell^p_c}$。我们显式计算了区间模的导出函子$\mathbf{Tor}^{\ell^p_c}$与$\mathbf{Ext}_{\ell^p_c}$。研究表明，所建立的万有系数定理可用于计算非紧空间滤流的持久Borel-Moore同调。最后，我们证明对每个$p\in [1,\infty]$，关联的Künneth短正合列可显著加速和近似乘积度量空间$(X\times Y,d^p)$（其中距离$d^p((x,y),(x',y'))=||d_X(x,x'),d_Y(y,y')||_p$）中的持久同调计算。