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We study when $R to S$ has the property that prime ideals of $R$ extend to prime ideals or the unit ideal of $S$, and the situation where this property continues to hold after adjoining the same indeterminates to both rings. We prove that if $R$ is reduced, every maximal ideal of $R$ contains only finitely many minimal primes of $R$, and prime ideals of $R[X_1,dots,X_n]$ extend to prime ideals of $S[X_1,dots,X_n]$ for all $n$, then $S$ is flat over $R$. We give a counterexample to flatness over a reduced quasilocal ring $R$ with infinitely many minimal primes by constructing a non-flat $R$-module $M$ such that $M = PM$ for every minimal prime $P$ of $R$. We study the notion of intersection flatness and use it to prove that in certain graded cases it suffices to examine just one closed fiber to prove the stable prime extension property.
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We continue investigating the interaction between flatness and $mathfrak{a}$-adic completion for infinitely generated modules over a commutative ring $A$. We introduce the concept of $mathfrak{a}$-adic flatness, which is weaker than flatness. We prove that $mathfrak{a}$-adic flatness is preserved under completion when the ideal $mathfrak{a}$ is weakly proregular. We also prove that when $A$ is noetherian, $mathfrak{a}$-adic flatness coincides with flatness (for complete modules). An example is worked out of a non-noetherian ring $A$, with a weakly proregular ideal $mathfrak{a}$, for which the completion $hat{A}$ is not flat. We also study $mathfrak{a}$-adic systems, and prove that if the ideal $mathfrak{a}$ is finitely generated, then the limit of any $mathfrak{a}$-adic system is a complete module.
This paper has two parts. In the first part we recall the important role that weak proregularity of an ideal in a commutative ring has in derived completion and in adic flatness. We also introduce the new concepts of idealistic and sequential derived completion, and prove a few results about them, including the fact that these two concepts agree iff the ideal is weakly proregular. In the second part we study the local nature of weak proregularity, and its behavior w.r.t. ring quotients. These results allow us to prove that weak proregularity occurs in the context of bounded prisms, in the sense of Bhatt and Scholze. We anticipate that the concept of weak proregularity will help simplify and improve some of the more technical aspects of the groundbreaking theory of perfectoid rings and prisms (that has transformed arithmetic geometry in recent years).
Recently, Witten has proposed a mechanism for symmetry enhancement in $SO(32)$ heterotic string theory, where the singularity obtained by shrinking an instanton to zero size is resolved by the appearance of an $Sp(1)$ gauge symmetry. In this short letter, we consider spacetime constraints from anomaly cancellation in six dimensions and D-flatness and demonstrate a subtlety which arises in the moduli space when many instantons are shrunk to zero size.
We study two-dimensional weighted ${mathcal N}=2$ supersymmetric $mathbb{CP}$ models with the goal of exploring their infrared (IR) limit. $mathbb{WCP}(N,widetilde{N})$ are simplifi
A physical nonlinear dynamical model of a laser diode is considered. We propose a feed-forward control scheme based on differential flatness for the design of input-current modulations to compensate diode distortions. The goal is to transform without distortion a radio-frequency current modulation into a light modulation leaving the laser-diode and entering an optic fiber. We prove that standard physical dynamical models based on dynamical electron and photons balance are flat systems when the current is considered as control input, the flat output being the photon number (proportional to the light power). We prove that input-current is an affine map of the flat output, its logarithm and their time-derivatives up to order two. When the flat output is an almost harmonic signal with slowly varying amplitude and phase, these derivatives admit precise analytic approximations. It is then possible to design simple analogue electronic circuits to code approximations of the nonlinear computations required by our flatness-based approach. Simulations with the parameters of a commercial diode illustrate the practical interest of this pre-compensation scheme and its robustness versus modelling and analogue implementation errors.
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