## Extensions of rings

Let $$K$$ be a not necessarily unital, commutative ring. Let $$C$$ be a unital commutative ring.

An extension is the data of a unital commutative ring $$R$$, a homomorphism $$\phi:K\rightarrow R$$, a unital homomorphism $$\psi:R\rightarrow C$$ such that $$\mathrm{Im}\:\phi=\mathrm{Ker}\:\psi$$. Two extensions are said to be isomorphic if there is an isomorphism $$R_1\approx R_2$$ making the rhomb-like diagram commute.

Is there a standard name for the algebraic gadget parametrizing the isomorphism classes of extensions? Is there a textbook or an article where I can read about it?

## Families over Artin Rings and Deformations

Let us work with a class of schemes over an algebraically closed field $$k$$ such that any two schemes in this class are isomorphic.

An example of such a class would be genus zero nonsingular curves over $$k$$. Let $$X$$ act as a representative of this class.

Is the set of isomorphism classes of such families over an Artin ring $$A$$ equivalent to the set of isomorphism classes deformations of $$X$$ over $$A$$?

The only problem I am encountering with verifying such a statement is we allow families $$\mathcal{X}$$ over $$B$$ to have varying fibers while a deformation fixes the fiber.

** Please assume any nice property you may wish. I did not want to specify the class to be nonsingular genus zero curves because I do not want $$H^1(X, \mathcal{T}X) =(0)$$., i.e I am interested in scheme which have nontrivial deformations.**

## Conjectured count of monogenic rings of fixed rank

By a monogenic ring of rank $$n$$ we mean a ring $$R = \mathbb{Z}[\theta]$$, where $$\theta$$ is an algebraic integer of degree $$n$$ over $$\mathbb{Q}$$. Put $$f_\theta(x)$$ for the minimal polynomial of $$\theta$$; it is necessarily a monic polynomial with integer coefficients.

By the discriminant of $$R$$ we mean the discriminant of $$\Delta(f_\theta)$$ of $$f_\theta$$. Counting monogenic rings of fixed rank ordered by discriminant is a difficult problem for all $$n \geq 3$$; for example, the $$n = 3$$ case is roughly equivalent to the problem of counting elliptic curves $$E/\mathbb{Q}$$ of bounded discriminant, a notorious open problem.

Is there a conjectured asymptotic formula for the number $$M_n(X)$$ of monogenic rings of rank $$n$$ ordered by discriminant?

## What’s the logic behind the the organization of Hamburg’s bus transport into “rings”?

I’m staying in Hamburg for a while for business reasons and I’ve been told to get an HVV card for rings A and B as that would cover most of the major destinations I might need to travel to. Now I take it that the bus numbers (e.g., “take the number 2 from Altona”) signifies the route (and hence the stops) the bus will cover but what do the rings signify? Is it a collection of bus numbers? A set of routes? A particular loop in the bus network? Something else entirely? I can’t find enough info on the HVV website to clarify what a ring might signify.

I’m asking because there are some pubs in Hamburg I’d like to visit and in planning this trip, it’d be great to know if they are reachable from rings A and B too.

## Properties of rings of global functions of open subschemes

It is known (although maybe not so well) that there are nice algebraic varieties whose ring of global functions is not finitely generated over the ground field. One can find examples on the web, but they are somewhat scarce. In thinking about this question, one can try to describe the ring of functions as an algebra over the function ring of another variety. Eventually this lead me to the following question:

Let $$S$$ be an affine noetherian scheme and let $$U\subset S$$ be an open subscheme. When is the map of function rings $$f:\mathcal{O}(S)\to\mathcal{O}(U)$$ finitely generated? flat? étale?

1) example: the map $$f$$ is étale if $$S$$ is integral, S2 and every divisor is locally set-theoretically principal. Indeed, in this case by the S2 condition only the codimension 1 part of $$S\setminus U$$ contributes to the function ring, and then locally $$U$$ is a principal open. This includes locally factorial schemes, reduced curves, and the affine quadric cone $$xy=z^2$$.
2) I’m interested also in reducible and/or nonreduced examples. I’m interested in all three properties (f.g., flat, étale). All kinds of examples and counterexamples are very welcome.
3) The similar MO question MO329902 did not receive answers…

## Surjectivity of pushforward on Chow rings for stacks

Let $$f:X\rightarrow Y$$ be a proper morphism of smooth Deligne-Mumford stacks of finite type over $$\mathbb{C}$$ that is birational, but not flat. The coarse spaces of $$X$$ and $$Y$$ are both not smooth. Is proper pushforward $$f_*:A^*(X)_{\mathbb{Q}}\rightarrow A^*(Y)_{\mathbb{Q}}$$ is surjective?

The way I want to prove a statement like this is using a projection formula argument on the coarse spaces, but I don’t have a reasonable way to define pullback in the absence of flatness and the fact that the coarse spaces aren’t smooth. Another idea is to take a cycle on $$Y$$ and try to move it out of the exceptional locus, take its preimage, and then push that forward, but I don’t know if there is a moving lemma on stacks. I can’t apply it on the coarse space because the coarse space of $$Y$$ is not regular. Can I remedy one of these arguments by working on the level of the stack instead of the coarse space?

## On Hopfian modules over commutative rings

Let $$R$$ be a commutative Noetherian ring with unity. Let us call an $$R$$-module Hopfian if every surjective endomorphism $$M \to M$$ is injective.

1) If $$M_1$$ and $$M_2$$ are Hopfian modules, is $$M_1 \oplus M_2$$ Hopfian ?

2) If we have an exact sequence $$0\to M_1 \to M\to M_2\to 0$$ with $$M$$ Hopfian, then are $$M_1$$ and $$M_2$$ Hopfian ?

If these are not true in general, are there any additional conditions on $$R$$ that would make it true ?

## Structure of Complete Local Rings

Let $$X$$ be a proper $$n$$-dimensional $$k$$-scheme and $$x \in X$$ a closed point. Consider the stalk $$\mathcal{O}_{X,x}$$. We consider now it’s completion $$O_{X,x}^{\wedge}$$ wrt it’s maximal ideal $$m_x$$.

My QUESTIONS concerns the analysis of the structure of $$\mathcal{O}_{X,x}$$:

When $$O_{X,x}^{\wedge}$$ has the shape $$k[[x_1,x_2,…, x_m]]/I$$?

When – more specifically – it has the shape $$k[[x_1,x_2,…, x_{n+1}]]/(f)$$ for $$f \in k[[x_1,x_2,…, x_{n+1}]]$$ non zero divisor?

Are there any reconmendable reference with thread the structure of such local complete rings rigoriously?

My CONSIDERATIONS /starting point:

The main tool to treat this problem is of course the Cohen strucure theorem. It provides especially that $$O_{X,x}^{\wedge} \cong \Lambda [[x_1, \ldots , x_ n]]/I$$ where $$\Lambda$$ is the “mysterious” Cohen ring.

Which criterions are neccessary & sufficient to settle that here $$\Lambda=k$$ (references?)?

Another point is under which conditions we obtain a more “concretely” form $$O_{X,x}^{\wedge}= k[[x_1,x_2,…, x_{n+1}]]/(f)$$

?

This question arises from following former thread of mine: Intuition behind RDP (Rational Double Points)

## Codes over rings

A code of length n over a ring R is a submodule of R^n. The dual of a code is with respect of the standard inner product on R^n. Is the dual of a free code over a finite ring R always free as an R-module? For which classes or rings R is this true?

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