The $\gamma$-linear projected barcode was recently introduced as an alternative to the well-known fibered barcode for multiparameter persistence, in which restrictions of the modules to lines are replaced by pushforwards of the modules along linear forms in the polar of some fixed cone $\gamma$. So far, the computation of the $\gamma$-linear projected barcode has only been studied in the functional setting, in which persistence modules come from the persistent cohomology of $\mathbb{R}^n$-valued functions. Here we develop a method that works in the algebraic setting directly, for any multiparameter persistence module over $\mathbb{R}^n$ that is given via a finite free resolution. Our approach is similar to that of RIVET: first, it pre-processes the resolution to build an arrangement in the dual of $\mathbb{R}^n$ and a barcode template in each face of the arrangement; second, given any query linear form $u$ in the polar of $\gamma$, it locates $u$ within the arrangement to produce the corresponding barcode efficiently. While our theoretical complexity bounds are similar to the ones of RIVET, our arrangement turns out to be simpler thanks to the linear structure of the space of linear forms. Our theoretical analysis combines sheaf-theoretic and module-theoretic techniques, showing that multiparameter persistence modules can be converted into a special type of complexes of sheaves on vector spaces called conic-complexes, whose derived pushforwards by linear forms have predictable barcodes.

翻译：γ-线性投影条形码是作为多参数持久性中著名的纤维化条形码的替代方案被提出的，其核心思想是将模在线上的限制替换为沿固定锥γ的极锥中线性形式对模的前推。迄今为止，γ-线性投影条形码的计算仅在函数设定下得到研究，即持久性模来源于ℝⁿ值函数的持久上同调。本文提出了一种直接在代数设定中适用于任意通过有限自由分解给出的ℝⁿ上多参数持久性模的计算方法。我们的方法类似于RIVET：首先对分解进行预处理，在ℝⁿ的对偶空间中构建一个排列结构，并在该排列的每个面中建立条形码模板；其次，对于γ的极锥中任意查询线性形式u，通过将u定位到排列结构中，高效生成对应的条形码。尽管理论复杂度边界与RIVET相似，但由于线性形式空间的线性结构，我们构建的排列更为简洁。理论分析结合了层论与模论技术，证明多参数持久性模可转化为向量空间上一类特殊的层复形——锥复形，其沿线性形式的导出前推具有可预测的条形码结构。