The purpose of this article is to compare several versions of bivariant algebraic cobordism constructed previously by the author and others. In particular, we show that a simple construction based on the universal precobordism theory of Annala and Yokura agrees with the more complicated theory of bivariant derived algebraic cobordism constructed earlier by the author, and that both of these theories admit a Grothendieck transformation to operational cobordism constructed by Luis González and Karu over fields of characteristic 0. The proofs are partly based on convenient universal characterizations of several cobordism theories, which should be of independent interest. Using similar techniques, we also strengthen a result of Vezzosi on operational derived $K$-theory. In the appendix, we give a detailed construction of virtual pullbacks in algebraic bordism, filling the gaps in the construction of Lowrey and Schürg.

本文的目的是比较作者和其他人先前构建的几个版本的双变量代数协同曲线。特别是，我们证明了一个基于 Annala 和 Yokura 的普遍预协调理论的简单结构与作者之前构建的更复杂的双变量派生代数协调理论相一致，并且这两个理论都承认格洛腾迪克转换为构建的操作协调协调。由 Luis González 和 Karu 在特征 0 的场上进行。证明部分基于几个 cobordism 理论的方便的普遍表征，这应该是独立的兴趣。使用类似的技术，我们还加强了 Vezzosi 对操作派生的结果$ķ$-理论。在附录中，我们给出了代数边界论中虚拟回调的详细构造，填补了 Lowrey 和 Schürg 构造中的空白。