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4u9ust
4u9ust
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I am trying to loosely follow Casselman's "The Bruhat-Tits Trees of SL(2)" instead using the field $F=\mathbb R_\rho$, a quotient of a subring of the hyperreals. It has a non-archimedean valuation with value group $\mathbb R$ and valuation ring $\mathcal O$. I am trying to show every bounded open subgroup of $\mathrm{GL}_2(F)$ stabilizes a lattice over $F^2$, that is a finitely generated $\mathcal O$-module which spans $F^2$ as an $F$-vector space.

To this end, I am trying to prove the following lemma: If $M$ is a lattice, $M\subseteq X\subseteq F^2$ with $X$ bounded, then the intersection of all lattices containing $X$ is finitely generated (and therefore a lattice.) Here $X$ is an $\mathcal O$-module which is not necessarily finitely generated. To this end I am trying to make a Zorn's lemma argument on the set of lattices containing $X$: If $C$ is a descending chain of lattices $$L_1\supseteq L_2 \supseteq \cdots$$ then, $\bigcap L_i$ is finitely generated, which is where I am stuck. My advisor told me to scrape the literature for lemmas about nesting intersections of lattices over Bézout domains, or more generally Prüfer domains. Can anyone recommend any sources? Thank you.

I am trying to loosely follow Casselman's "The Bruhat-Tits Trees of SL(2)" instead using the field $F=\mathbb R_\rho$, a quotient of the hyperreals. It has a non-archimedean valuation with value group $\mathbb R$ and valuation ring $\mathcal O$. I am trying to show every bounded open subgroup of $\mathrm{GL}_2(F)$ stabilizes a lattice over $F^2$, that is a finitely generated $\mathcal O$-module which spans $F^2$ as an $F$-vector space.

To this end, I am trying to prove the following lemma: If $M$ is a lattice, $M\subseteq X\subseteq F^2$ with $X$ bounded, then the intersection of all lattices containing $X$ is finitely generated (and therefore a lattice.) Here $X$ is an $\mathcal O$-module which is not necessarily finitely generated. To this end I am trying to make a Zorn's lemma argument on the set of lattices containing $X$: If $C$ is a descending chain of lattices $$L_1\supseteq L_2 \supseteq \cdots$$ then, $\bigcap L_i$ is finitely generated, which is where I am stuck. My advisor told me to scrape the literature for lemmas about nesting intersections of lattices over Bézout domains, or more generally Prüfer domains. Can anyone recommend any sources? Thank you.

I am trying to loosely follow Casselman's "The Bruhat-Tits Trees of SL(2)" instead using the field $F=\mathbb R_\rho$, a quotient of a subring of the hyperreals. It has a non-archimedean valuation with value group $\mathbb R$ and valuation ring $\mathcal O$. I am trying to show every bounded open subgroup of $\mathrm{GL}_2(F)$ stabilizes a lattice over $F^2$, that is a finitely generated $\mathcal O$-module which spans $F^2$ as an $F$-vector space.

To this end, I am trying to prove the following lemma: If $M$ is a lattice, $M\subseteq X\subseteq F^2$ with $X$ bounded, then the intersection of all lattices containing $X$ is finitely generated (and therefore a lattice.) Here $X$ is an $\mathcal O$-module which is not necessarily finitely generated. To this end I am trying to make a Zorn's lemma argument on the set of lattices containing $X$: If $C$ is a descending chain of lattices $$L_1\supseteq L_2 \supseteq \cdots$$ then, $\bigcap L_i$ is finitely generated, which is where I am stuck. My advisor told me to scrape the literature for lemmas about nesting intersections of lattices over Bézout domains, or more generally Prüfer domains. Can anyone recommend any sources? Thank you.

Fixed mistake pointed out by @LSpice and slightly strengthened assumptions.
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4u9ust
4u9ust
  • 131
  • 4

I am trying to loosely follow Casselman's "The Bruhat-Tits Trees of SL(2)" instead using the field $F=\mathbb R_\rho$, a quotient of the hyperreals. It has a non-archimedean valuation with value group $\mathbb R$ and valuation ring $\mathcal O$. I am trying to show every bounded open subgroup of $\mathrm{GL}_2(F)$ stabilizes a lattice over $F^2$, that is a finitely generated $\mathcal O$-module which spans $F^2$ as an $F$-vector space.

To this end, I am trying to prove the following lemma: If $M$ is a lattice, $M\subseteq X\subseteq \mathrm{GL}(F)$$M\subseteq X\subseteq F^2$ with $X$ bounded, then the intersection of all lattices containing $X$ is finitely generated (and therefore a lattice.) Here $X$ is an $\mathcal O$-module which is not necessarily finitely generated. To this end I am trying to make a Zorn's lemma argument on the set of lattices containing $X$: If $C$ is a descending chain of lattices $$L_1\supseteq L_2 \supseteq \cdots$$ then, $\bigcap L_i$ is finitely generated, which is where I am stuck. My advisor told me to scrape the literature for lemmas about nesting intersections of lattices over Bézout domains, or more generally Prüfer domains. Can anyone recommend any sources? Thank you.

I am trying to loosely follow Casselman's "The Bruhat-Tits Trees of SL(2)" instead using the field $F=\mathbb R_\rho$, a quotient of the hyperreals. It has a non-archimedean valuation with value group $\mathbb R$ and valuation ring $\mathcal O$. I am trying to show every bounded open subgroup of $\mathrm{GL}_2(F)$ stabilizes a lattice over $F^2$, that is a finitely generated $\mathcal O$-module which spans $F^2$ as an $F$-vector space.

To this end, I am trying to prove the following lemma: If $M$ is a lattice, $M\subseteq X\subseteq \mathrm{GL}(F)$ with $X$ bounded, then the intersection of all lattices containing $X$ is finitely generated (and therefore a lattice.) To this end I am trying to make a Zorn's lemma argument on the set of lattices containing $X$: If $C$ is a descending chain of lattices $$L_1\supseteq L_2 \supseteq \cdots$$ then, $\bigcap L_i$ is finitely generated, which is where I am stuck. My advisor told me to scrape the literature for lemmas about nesting intersections of lattices over Bézout domains, or more generally Prüfer domains. Can anyone recommend any sources? Thank you.

I am trying to loosely follow Casselman's "The Bruhat-Tits Trees of SL(2)" instead using the field $F=\mathbb R_\rho$, a quotient of the hyperreals. It has a non-archimedean valuation with value group $\mathbb R$ and valuation ring $\mathcal O$. I am trying to show every bounded open subgroup of $\mathrm{GL}_2(F)$ stabilizes a lattice over $F^2$, that is a finitely generated $\mathcal O$-module which spans $F^2$ as an $F$-vector space.

To this end, I am trying to prove the following lemma: If $M$ is a lattice, $M\subseteq X\subseteq F^2$ with $X$ bounded, then the intersection of all lattices containing $X$ is finitely generated (and therefore a lattice.) Here $X$ is an $\mathcal O$-module which is not necessarily finitely generated. To this end I am trying to make a Zorn's lemma argument on the set of lattices containing $X$: If $C$ is a descending chain of lattices $$L_1\supseteq L_2 \supseteq \cdots$$ then, $\bigcap L_i$ is finitely generated, which is where I am stuck. My advisor told me to scrape the literature for lemmas about nesting intersections of lattices over Bézout domains, or more generally Prüfer domains. Can anyone recommend any sources? Thank you.

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4u9ust
4u9ust
  • 131
  • 4

Proving the intersection of lattices is finitely generated over non-discrete valuation ring

I am trying to loosely follow Casselman's "The Bruhat-Tits Trees of SL(2)" instead using the field $F=\mathbb R_\rho$, a quotient of the hyperreals. It has a non-archimedean valuation with value group $\mathbb R$ and valuation ring $\mathcal O$. I am trying to show every bounded open subgroup of $\mathrm{GL}_2(F)$ stabilizes a lattice over $F^2$, that is a finitely generated $\mathcal O$-module which spans $F^2$ as an $F$-vector space.

To this end, I am trying to prove the following lemma: If $M$ is a lattice, $M\subseteq X\subseteq \mathrm{GL}(F)$ with $X$ bounded, then the intersection of all lattices containing $X$ is finitely generated (and therefore a lattice.) To this end I am trying to make a Zorn's lemma argument on the set of lattices containing $X$: If $C$ is a descending chain of lattices $$L_1\supseteq L_2 \supseteq \cdots$$ then, $\bigcap L_i$ is finitely generated, which is where I am stuck. My advisor told me to scrape the literature for lemmas about nesting intersections of lattices over Bézout domains, or more generally Prüfer domains. Can anyone recommend any sources? Thank you.