关于偏序集上模的微积分基础
Foundations of Differential Calculus for modules over posets
AI总结：
本文研究了偏序集上模的微积分基础，定义了梯度、散度等概念，并给出了梯度为零的充分必要条件及应用。
摘要：
设$k$为一个域，$C$为一个小范畴。$C$的$k$-线性表示，或$kC$-模，是从$C$到$k$上有限维向量空间范畴的函子。当$C$比线性序更一般时，其表示类型通常为无限，大多数情况下为野类型。因此，理解此类表示的不可分解因子变得困难。本文提出新的想法，旨在使研究模块局部化。受图上离散微积分工作的启发，我们设定了研究$kC$-模的微积分类型分析的基础，某些限制下。作为起点，对于$kC$-模$M$，我们定义其梯度$\nabla[M]$为同构类$kC$-模的Grothendieck群中的虚拟模。推动与普通微积分和图上离散微积分的类比，我们通过适当的左和右Kan扩展以及模块上的两种双线性配对，定义了左和右散度，并研究其性质，特别是梯度与左、右散度之间的对偶关系。左、右散度在有利情况下易于计算。设定场景后，我们特别研究$C$为有限偏序集的情况。主要结果是，在某些偏序集假设下，模块$M$的梯度为零的充要条件。我们进一步研究了梯度相等的两个模块的含义。最后，我们考虑由此得到的左、右拉普拉斯算子，即散度与梯度的复合，并研究拉普拉斯算子消失与梯度之间的关系。
英文摘要：
Let $k$ be a field and let $C$ be a small category. A $k$-linear representation of $C$, or a $kC$-module, is a functor from $C$ to the category of finite dimensional vector spaces over $k$. When the category $C$ is more general than a linear order, then its representation type is generally infinite and in most cases wild. Hence the task of understanding such representations in terms of their indecomposable factors becomes difficult at best, and impossible in general. This paper offers a new set of ideas designed to enable studying modules locally. Specifically, inspired by work in discrete calculus on graphs, we set the foundations for a calculus type analysis of $kC$-modules, under some restrictions on the category $C$. As a starting point, for a $kC$-module $M$ we define its gradient \emph{gradient} \(\nabla[M]\) as a virtual module in the Grothendieck group of isomorphism classes of $kC$-modules. Pushing the analogy with ordinary differential calculus and discrete calculus on graphs, we define left and right divergence via the appropriate left and right Kan extensions and two bilinear pairings on modules and study their properties, specifically with respect to adjointness relations between the gradient and the left and right divergence. The left and right divergence are shown to be rather easily computable in favourable cases. Having set the scene, we concentrate specifically on the case where the category $C$ is a finite poset. Our main result is a necessary and sufficient condition for the gradient of a module $M$ to vanish under certain hypotheses on the poset. We next investigate implications for two modules whose gradients are equal. Finally we consider the resulting left and right Laplacians, namely the compositions of the divergence with the gradient, and study an example of the relationship between the vanishing of the Laplacians and the gradient.
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