基于Gröbner基的自由分解
Free resolutions via Gröbner bases
摘要：
对于许多不同类别的结合代数，可以发展Gröbner基的机制。一个Gröbner基础定义关系的代数这样一个范畴提供了一个“单项式替代”这个代数。这篇文章的主要目标是演示如何使用这种机器的目的同调代数。更准确地说，我们的方法分为三个步骤。首先，我们定义了一个组合的解决方案的单项式替换的对象。其次，我们从这些决议中提取明确的代表同调类。最后，我们解释如何“变形”的差异，以处理一般情况。对于结合代数，我们恢复了Anick的一个著名结构。我们详细讨论的另一种情况是操作数，在那里我们发现了以前不知道的决议。给出了Hoffbeck的PBW准则的证明，一类来自交换代数的运算的Koszulness的证明，Batalin-Vilkovisky代数和Rota-Baxter代数的运算的同调计算.
英文摘要：
For associative algebras in many different categories, it is possible to develop the machinery of Gröbner bases. A Gröbner basis of defining relations for an algebra of such a category provides a "monomial replacement" of this algebra. The main goal of this article is to demonstrate how this machinery can be used for the purposes of homological algebra. More precisely, our approach goes in three steps. First, we define a combinatorial resolution for the monomial replacement of an object. Second, we extract from those resolutions explicit representatives for homological classes. Finally, we explain how to "deform" the differential to handle the general case. For associative algebras, we recover a well known construction due to Anick. The other case we discuss in detail is that of operads, where we discover resolutions that haven't been known previously. We present various applications, including a proofs of Hoffbeck's PBW criterion, a proof of Koszulness for a class of operads coming from commutative algebras, and a homology computation for the operads of Batalin--Vilkovisky algebras and of Rota--Baxter algebras.
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