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I'm reading the book "Rigid Analytic Geometry and its Applications" by Fresnel-van der Put, and I'm confused by their example 4.8.5. In the first two parts of the example, they define the analytic reduction corresponding to two pure affinoid coverings of $\mathbb{P}_k^{1,an}$ by explicitly writing down the rings involved. This I am happy with, but they then give a "basefree" version of this which confuses me. I'll try to give more detail.

Let $K$ be a complete nonarchimedean field with ring of integers $\mathcal O_K$, residue field $k$, let $V$ be a two dimensional vector space $V$ over $K$. We pick some lattice $M\subset V$, this is a free $\mathcal O_K$-module of rank 2. Tensoring with $K$ we get back to $V$, tensoring with $k$ we get to a two dimensional vector space over k. The claim seems to be that this allows us to define a reduction map from $Proj(V)\rightarrow Proj(k\otimes M)$. My question is how this map is defined.

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  • $\begingroup$ At the level of points: any point of $Proj(V)$ is given by a vector $v$ non zero in $V$ modulo homothety. It has a representative in $M$ whose image in $k\otimes M$ is non-zero. This imatge, modulo homothety in $k$, is the image of the reduction map. $\endgroup$
    – Xarles
    Commented Jun 21, 2017 at 15:47
  • $\begingroup$ Thank you! It seemed like it should be something simple, but I just wasn't seeing it! $\endgroup$
    – pw1
    Commented Jun 21, 2017 at 16:03

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