"Cycles Premiers" In EGA

Question

I am reading through EGA IV, 3.1, but I can't figure out what "cycles premiers" translates to. I don't know any french, really. I think it means "prime cycles", but an English search for that doesn't give many results, and I am left uncertain. Can someone give a self-contained definition here?

Answer 1

I don't know if this helps, but let me translate the definition 3.1.1 given in EGA IV (which is self-contained).

Definition $3.1.1$: Let $X$ be a pre-scheme and $\mathcal{F}$ a quasi-coherent $\mathcal{O}_X$-module. An element $x\in X$ is said to be associated to $\mathcal{F}$ if the maximal ideal $\mathfrak{m}_x$ of $\mathcal{O}_{X,x}$ is associated to the $\mathcal{O}_{X,x}$-module $\mathcal{F}_x$ (id est $\mathfrak{m}_x$ is the annihilator of some element of $\mathcal{F}_x$). The set of $x\in X$ associated to $\mathcal{F}$ is denoted by $\mathrm{Ass}(\mathcal{F})$. An irreducible closed subset $Z$ of $X$ is called a cycle premier associated to $\mathcal{F}$ if its generic point is associated to $\mathcal{F}$. When $\mathcal{F}=\mathcal{O}_X$, we shall say that the points (respectively the cycles premiers) associated to $\mathcal{F}$ are associated to the pre-scheme $X$.

The translation of cycle premier is indeed prime cycle (and the one of cycles premiers is prime cycles) but this does not seem to be standard.

Answer 2

The complete expression in French translates as prime cycle associated to a quasi-coherent $\mathcal O_X$-Module $\mathcal F$, where $X$ is a scheme.

It denotes a closed irreducible subset of $X$, hence with a generic point $x$, under the condition that this generic point is associated to $\mathcal F$, i.e. that the maximal ideal $\mathfrak m_x$ of the local ring $\mathcal O_x$ is associated to the $\mathcal O_x$-module $\mathcal F_x$.