The Stacks project

45.2 Conventions and notation

Let $F$ be a field. In this chapter we view the category of $F$-graded vector spaces as an $F$-linear symmetric monoidal category with associativity constraint as usual and with commutativity constraint involving signs. See Homology, Example 12.17.4.

Let $R$ be a ring. In this chapter a graded commutative $R$-algebra $A$ is a commutative differential graded $R$-algebra (Differential Graded Algebra, Definitions 22.3.1 and 22.3.3) whose differential is zero. Thus $A$ is an $R$-module endowed with a grading $A = \bigoplus _{n \in \mathbf{Z}} A^ n$ by $R$-submodules. The $R$-bilinear multiplication

\[ A^ n \times A^ m \longrightarrow A^{n + m},\quad \alpha \times \beta \longmapsto \alpha \cup \beta \]

will be called the cup product in this chapter. The commutativity constraint is $\alpha \cup \beta = (-1)^{nm} \beta \cup \alpha $ if $\alpha \in A^ n$ and $\beta \in A^ m$. Finally, there is a multiplicative unit $1 \in A^0$, or equivalently, there is an additive and multiplicative map $R \to A^0$ which is compatible the $R$-module structure on $A$.

Let $k$ be a field. Let $X$ be a scheme of finite type over $k$. The Chow groups $\mathop{\mathrm{CH}}\nolimits _ k(X)$ of $X$ of cycles of dimension $k$ modulo rational equivalence have been defined in Chow Homology, Definition 42.19.1. If $X$ is normal or Cohen-Macaulay, then we can also consider the Chow groups $\mathop{\mathrm{CH}}\nolimits ^ p(X)$ of cycles of codimension $p$ (Chow Homology, Section 42.42) and then $[X] \in \mathop{\mathrm{CH}}\nolimits ^0(X)$ denotes the “fundamental class” of $X$, see Chow Homology, Remark 42.42.2. If $X$ is smooth and $\alpha $ and $\beta $ are cycles on $X$, then $\alpha \cdot \beta $ denotes the intersection product of $\alpha $ and $\beta $, see Chow Homology, Section 42.62.

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