Usbrth

Just kidding, it's not Noetherian. And "Emphemeralization" implies it is a physical construct, or that if it is, due to knowledge heretofore unbeknownst but recently gained by visualization of black holes via really, really big polaroid pictures taken atop Mauna Kea, that one would not accept its intrinsic life span just as it is.

I changed the title of this page to attract attention - please let me know if this is a faux pas. And please share any insider knowledge on boundaries for attempted humor that is not not PC.

I will proceed to answer the questions below, at least to my satisfaction, and hopefully to the benefit of those who care about such things. As always, any critiques, advice, or recommendations are welcome, as I live in a vacuum in the middle of the Pacific Ocean.

1. Is there a (necessarily locally Noetherian) formulation of the Noether Normalization Lemma for (generally non-Noetherian) topologically finitely generated commutative algebras over the ring of adeles (those with a continuous scalar multiplication compatible with a commutative/continuous ring multiplication with 1 for a finitely generated module over the ring of adeles, where only closed ideals are considered so that quotients are Hausdorff)?


2. Please share any references with which you are familiar that deal explicitly with $mathbbA$-schemes and/or topologically finitely generated commutative $mathbbA$-algebras.

Additional Context for Finitely Generated Commutative Topological $mathbbA$-Algebras:

Example. Let $S,colon!=fracmathbbA[x]langle x-arangle$ where $a=prodlimits_ple inftyp^r_pa_pinwidehatmathbbZtimesmathbbR$, $prodlimits_p<inftyp^r_pinmathbbS$ (supernatural numbers), $a_p$ is a unit for $pleinfty$, $p^inftymathbbZ_p,colon!=0$ and $infty^inftymathbbR,colon!=0$. Let $cong_rm t$ denote topological isomorphism (open bijective morphism of topological groups). We have $fracmathbbZ_pp^r_pmathbbZ_pcong_rm twidehatmathbbZ(p^r_p)$ where $widehatmathbbZ(p^r_p),colon!=fracmathbbZp^r_pmathbbZ$ if $r_p<infty$ and $widehatmathbbZ(p^infty),colon!=mathbbZ_p$. Also, $fracmathbbRinfty^r_inftymathbbRcong_rm tmathbbR(infty^r_infty)$ where $mathbbR(infty^r_infty),colon!=0$ if $r_infty<infty$ and $mathbbR(infty^infty),colon!=mathbbR$.

Case $r_infty=infty$ : $Scong_rm tprodlimits_p<inftywidehatmathbbZ(p^r_p)$, a procyclic algebra (the $mathbbR$ "cancels").

Case $r_infty<infty$ : $S$ is a solenoid; that is, $Scong_rm tfracprodlimits_p<inftywidehatmathbbZ(p^r_p)timesmathbbRmathbbZ(boldsymbol1,1)$ is a $1$-dimensional compact connected abelian group.

For 2 or more indeterminates, finitely generated commutative topological $mathbbA$-algebras are products of finitely generated profinite algebras and finite-dimensional compact connected abelian groups (protori). By using some tricks, one finds that any real torus, any complex torus, any elliptic curve, and any abelian variety can be represented as a protorus, whence as a finitely generated commutative topological $mathbbA$-algebra (by way of a category equivalence between finite-dimensional protori and finitely generated commutative topological $mathbbA$-algebras $mathbbA[x_1,dots,x_n]/langle f rangle$ where $langle frangle$ is free as an $mathbbA$-module).

So Questions 1 and 2 above are asking whether Noether normalization and nullstellensatz can be formulated in this setting of topological algebras. Among other things, the motivation is to introduce geometric insight into the study of protori and their duals, torsion-free abelian groups.

A couple days of introspection revealed that the objects I was attempting to describe in the original post are topologically isomorphic to $fracmathbbA_mathbbZ^mbigopluslimits_i=1^mwidehatmathbbZboldsymbolx_i+bigopluslimits_i=1^mmathbbZboldsymboly_i$ for some $boldsymbolx_i,boldsymboly_iinmathbbA_mathbbZ^m$ satisfying $overlinebigopluslimits_i=1^mmathbbA_mathbbZboldsymbolx_i+bigopluslimits_i=1^mmathbbZboldsymboly_i=mathbbA_mathbbZ^m$. (Randomly selecting free $mathbbA_mathbbZ$-modules and free $mathbbZ$-modules to topologically generate $mathbbA_mathbbZ^m$ would have a success rate comparable to obtaining an abelian variety by randomly selecting a lattice in $mathbbC^m$ to form a complex torus...hmm...I wonder if a form of Riemann bilinear relations might be derived for this setting...).

Because the realization determined above for the $mathbbA$-objects in question articulates a bijective correspondence with $m$-dimensional compact connected abelian groups (protori), an application of the Structure Theorem for Protori allows us to conclude that an $m$-dimensional protorus $G$ has a free resolution $mathbbQDelta + Xhookrightarrow mathbbA^m twoheadrightarrow G$, where $DeltasubsetmathbbA^m$ is a torsion-free profinite abelian group with non-Archimedean dimension $m$ and $XsubsetmathbbA^m$ is algebraically isomorphic to the Pontryagin dual of $G$. The $boldsymbolx_i$ above determine the isogeny class of $Delta$ and the $boldsymboly_i$ above prescribe an adjunction between the non-Archimedean and Archimedean "halves" of the topological group $G$.

When I determine how to explicitly and intuitively describe a geometric formulation of these objects, I will append it here.
It would be very helpful for a practitioner of arithmetic algebraic geometry to write out in words how to describe these objects in modern mathematical parlance.