论文标题
Maclane同源性和拓扑Hochschild同源性的乘法比较
A multiplicative comparison of MacLane homology and topological Hochschild homology
论文作者
论文摘要
令$ q $表示麦克莱恩的$ q $ - 构建，$ \ otimes $表示Spectra的Smash产品。 In this paper we construct an equivalence $Q(R)\simeq \mathbb Z\otimes R$ in the category of $A_\infty$ ring spectra for any ring $R$, thus proving a conjecture made by Fiedorowicz, Schwänzl, Vogt and Waldhausen in "MacLane homology and topological Hochschild homology".更确切地说，我们的构造是$ q $（在$ \ infty $ - 分类含义上）上的一种对称单体结构，扩展了通常的单型结构，我们证明了等价$ q（ - ）\ simeq \ simeq \ simeq \ mathbb z \ otimes- $作为对称的单型函数，来自对称的单体功能，从此立即推荐。 从这个结果，我们获得了等价$ \ MATHRM {hml}（r，m）\ simeq \ simeq \ mathrm {thh}（r，m）$最初由Pirashvili和Waldaushen证明的“ Maclane同源性和Topological Hochschild同源性”（与上面的不同论文）。实际上，这种等价是对称的单体，因此它也提供了等价$ \ mathrm {hml}（r）\ simeq \ mathrm {thh}（r）$ as $ e_ \ effty $ ring spectra，当$ r r $是通勤环时。
Let $Q$ denote MacLane's $Q$-construction, and $\otimes$ denote the smash product of spectra. In this paper we construct an equivalence $Q(R)\simeq \mathbb Z\otimes R$ in the category of $A_\infty$ ring spectra for any ring $R$, thus proving a conjecture made by Fiedorowicz, Schwänzl, Vogt and Waldhausen in "MacLane homology and topological Hochschild homology". More precisely, we construct is a symmetric monoidal structure on $Q$ (in the $\infty$-categorical sense) extending the usual monoidal structure, for which we prove an equivalence $Q(-)\simeq \mathbb Z\otimes -$ as symmetric monoidal functors, from which the conjecture follows immediately. From this result, we obtain a new proof of the equivalence $\mathrm{HML}(R,M)\simeq \mathrm{THH}(R,M)$ originally proved by Pirashvili and Waldaushen in "MacLane homology and topological Hochschild homology" (a different paper from the one cited above). This equivalence is in fact made symmetric monoidal, and so it also provides a proof of the equivalence $\mathrm{HML}(R)\simeq \mathrm{THH}(R)$ as $E_\infty$ ring spectra, when $R$ is a commutative ring.
