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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Calculation of Dynkin operator on free Lie rings

The Dynkin operator on associative Lie monomials in the free generators $ \{ x_1, x_2, \dotsc \}$ generating a Lie ring $ L$ is defined as $ \delta (x_{i_1} x_{i_2} \dotsc x_{i_m}) = [\dotsc [x_{i_1}, x_{i_2}], \dotsc, x_{i_m}]$ (left normed commutator). I am trying to understand a proof of the following lemma in book by E.I.Khukhro ($ p$ -automorphisms of finite $ p$ -groups) stating : $ \delta(x_{k+1}[x_1, \dotsc, x_k]) = [x_{k+1}, [x_1, \dotsc, x_k]]$ .

The step while we prove $ \delta (x_{k+1} x_k [x_1, \dotsc, x_{k-1}]) = [[x_{k+1}, x_k], [x_1, \dotsc, x_{k-1}]]$ , the argument says “we replace $ x_{k+1}x_k$ by the commutator $ [x_{k+1}, x_k]$ and then regard this commutator as a new variable, which implies this step.

The replacement seems to change the definition of Dynkin operator by changing the variables. It is unclear how to derive this step. On another thought, a direct approach seems to look like a modified version of the lemma, which require another proof.

Is there any general formula for $ \delta(x_{k+l} \dotsc x_{k+1}[x_1, \dotsc, x_k])$ with a proof via induction on $ k+l$ ?

Are the cpu protection rings meant to protect against malicious programs, or against unintentional programming mistakes?

I am trying to understand what is the purpose of the CPU protection rings (specifically user mode and kernel mode).

I don’t think that the CPU protection rings are meant to protect against malicious programs, because a malicious program can be written to run in kernel mode instead of user mode (it can be written as a device driver for example) and then it will be able to cause whatever damage it wants.

I rather think that the CPU protection rings are meant to protect against unintentional programming mistakes, for example a programmer may unintentionally write code to access the memory of another process or the memory of the kernel, but since the program will run in user mode, the program will not be able to access the memory of another process or the memory of the kernel.

Am I correct?

Power series rings and the formal generic fibre

Let $ S = K[[S_1,\ldots,S_n]]$ and consider $ d$ elements \begin{equation*} f_1,\ldots,f_d \in S[[X_1,\ldots,X_d]] \end{equation*} and the ideal $ I \colon\!= (f_1,\ldots,f_d)$ generated by them.

Suppose that $ I$ satisfies the following three conditions$ \colon$

  1. $ I \cap S = 0$ .

  2. $ {\mathrm{ht}}(I) = d$ .

  3. Neither of $ \overline{f_1},\ldots,\overline{f_d} \in K[[X_1,\ldots,X_d]]$ is zero.

(The condition 3. is equivalent to that every $ f_i$ contains at least one term $ c_{e_1,\ldots,e_d}X_1^{e_1} \ldots X_d^{e_d}$ such that $ c_{e_1,\ldots,e_d} \notin (S_1,\ldots,S_n)$ .)

Q. Is the ring $ S[[X_1,\ldots,X_d]]/I$ necessariy finite over $ S$ ?

“Category of elements” for presheaves of rings?

(We use quasi-categories throught.)

Let $ C$ be an $ \infty$ -category, a morphism of simplicial sets $ C \xrightarrow{F} Cat_\infty$ unstraightens to some arrow $ \hat F \xrightarrow{p} C$ .

  • In general $ p$ is a coCartesian fibration
  • When $ F$ factors through $ Groupoid_\infty \hookrightarrow Cat_\infty$ , the fibers are $ \infty$ groupoids.
  • When $ F$ factors through $ N(Groupoid_1) \hookrightarrow Cat_\infty$ , the fibers are 1-groupoids.
  • When $ F$ factors through $ N(Set) \hookrightarrow Cat_\infty$ , the fibers are discrete.

Between the last two cases, we have the case where $ F$ factors through the $ N(Groups)$ , that is, full subcategory of groupoids supported on objects that are contractible. In this case it looks reasonable to guess that

  • The fibers of the $ p$ should be contractible, so these are ordinary groups.

  • If $ F$ furthermore factors through abelian groups $ N(Ab)$ , the fibers are also abelian groups.

I haven’t checked these, so part of the question is whether these are true.

The question is, suppose now I’m given a functor $ $ C \xrightarrow{F} N(Ring)$ $ landing in the category of commutative rings. This takes us out of the ordinary case described above, but it’s not too far off, as we were able to describe the unstraightening a functor into abelian groups. The question is whether we know a sort of unstraightening construction for a functor like this, so that fibers of the resulting fibration are naturally commutative rings.

Remarks

  • Apologies for any mistakes, I’m a noob.
  • In the case that C is the nerve of some 1-category, I could have stated this without mentioning infinity categories. An answer in this case would make me happy as well.
  • In the other direction, I didn’t need to restrict the RHS to nerve of some 1-category, and instead considered functors into of spaces, those into contractible spaces, and asked some final question about $ E_\infty$ rings. But that sounds harder.