localization – Algebras over a Dedekind domain with isomorphic generic fibers

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ra.rings and algebras – Cohn Localization

I’m working on my master’s thesis, part of which involves an exposition on Cohn Localization. (nlab discussion)

In Free Ideal Rings and Localization in General Rings, Sec 7.4, Cohn gives a construction for a ring $Sigma^{-1}R$. Given a set of matrices $Sigma$ (with a mild closure condition), this ring admits a homomorphism from $R$ which is universal with respect to the property that the image of each matrix in $Sigma$ is invertible over $Sigma^{-1}R$.

I understand the construction and it’s uses in finding conditions for embeddability of domains into skew fields and the existence of universal fields, but I would really like to have some concrete examples of what the construction actually gives.

There are a few trivial examples – if $R$ is commutative, then $Sigma^{-1}R$ is just the ring of quotients of $R$ with denominator set comprised of the determinants of matrices in $Sigma$. If $Sigma$ contains the zero matrix, the Cohn Localization is the zero ring.

But neither of these highlights what makes Cohn localization a novel idea or sheds any light on what “matrix inverting homomorphisms” look like away from the commutative case.

Cohn’s book also lacks examples. Where else can I see some concrete and informative examples of the ring $Sigma^{-1}R$?

oa.operator algebras – Lower bounds in the space of compact operators

Let $H$ be a separable Hilbert space, and $K(H)$ the corresponding space of compact operators. Consider the “unit sphere” $S:={Tin K(H)|Tgeq 0text{ and }||T||=1}$. Is it true that, given any pair of operators $T_1,T_2in S$, there exists another operator $Tin S$ such that $Tleq T_1,T_2$?.

oa.operator algebras – Need reference for ideals and representations of $C_0(X,A)$

Let $A$ be $C^{ast}$– Algebra and $X$ be a locally compact Hausdorff space and $C_{0}(X,A)$ be the set of all continuous functions from $X$ to $A$ vanishing at infinity. Define $f^{ast}(t)={f(t)}^{ast}$ (for $tin X$). It is well known that $C_0(X,A)$ is $C^{ast}-$ Algebra.

What’s known about ideals and representations of $C_0(X,A)$?

My guess is that it must be related with ideals and representations of $A$. Can someone give a reference or some ideas?

P.S: The same question was first posted on MSE but unfortunately I dint not get any answer so I am posting it here.

ct.category theory – In the category of sigma algebras, are all epimorphisms surjective?

Consider the category of abstract $sigma$-algebras ${mathcal B} = (0, 1, vee, wedge, bigvee_{n=1}^infty, bigwedge_{n=1}^infty, overline{cdot})$ (Boolean algebras in which all countable joins and meets exist), with the morphisms being the $sigma$-complete Boolean homomorphisms (homomorphisms of Boolean algebras which preserve countable joins and meets). If a morphism $phi: {mathcal A} to {mathcal B}$ between two $sigma$-algebras is surjective, then it is certainly an epimorphism: if $psi_1, psi_2: {mathcal B} to {mathcal C}$ are such that $psi_1 circ phi = psi_2 circ phi$, then $phi_1 = phi_2$. But is the converse true: is every epimorphism $phi: {mathcal A} to {mathcal B}$ surjective?

Setting ${mathcal B}_0 := phi({mathcal A})$, the question can be phrased as follows. If ${mathcal B}_0$ is a proper sub-$sigma$-algebra of ${mathcal B}$, does there exist two $sigma$-algebra homomorphisms $phi_1, phi_2: {mathcal B} to {mathcal C}$ into another $sigma$-algebra ${mathcal C}$ that agree on ${mathcal B}_0$ but are not identically equal on ${mathcal B}$?

In the case that ${mathcal B}$ is generated from ${mathcal B}_0$ and one additional element $E in {mathcal B} backslash {mathcal B}_0$, then all elements of ${mathcal B}$ are of the form $(A wedge E) vee (B wedge overline{E})$ for $A, B in {mathcal B}_0$, and I can construct such homomorphisms by hand, by setting ${mathcal C} := {mathcal B}_0/{mathcal I}$ where ${mathcal I}$ is the proper ideal
$$ {mathcal I} := { A in {mathcal B}_0: A wedge E, A wedgeoverline{E} in {mathcal B}_0 }$$
and $phi_1, phi_2: {mathcal B} to {mathcal C}$ are defined by setting
$$ phi_1( (A wedge E) vee (B wedge overline{E}) ) := (A)$$
and
$$ phi_2( (A wedge E) vee (B wedge overline{E}) ) := (B)$$
for $A,B in {mathcal B}_0$, where $(A)$ denotes the equivalence class of $A$ in ${mathcal C}$, noting that $phi_1(E) = 1 neq 0 = phi_2(E)$. However I was not able to then obtain the general case; the usual Zorn’s lemma type arguments don’t seem to be available in the $sigma$-algebra setting. I also played around with using the Loomis-Sikorski theorem but was not able to get enough control on the various null ideals to settle the question. (However, Stone duality seems to settle the corresponding question for Boolean algebras.)

rings and algebras – Existence of non-trivial transfinite divisibility in modules $ R $

Leave $ R $ be a unitary, possibly non-commutative ring and $ s in R $. For a right $ R $-module $ M $, define $ Ms = {ms mid m in M ​​} $; this is an additive subgroup of $ M $, which is a module on the centralizer of $ s $. For an ordinal $ alpha $, define $ Ms ^ alpha $ by transfinite induction: $ Ms ^ 0 = M $, $ Ms ^ { alpha + 1} = (Ms ^ alpha) s $and take intersections in the limit ordinals. Then the groups $ Ms ^ alpha $ it must eventually stabilize; say that the $ s $divisibility range of $ M $ is the minimum $ alpha $ such that $ Ms ^ alpha = Ms ^ { alpha + 1} $. Consider the following conditions:

  1. For some $ s in R $, the $ s $-divisibility range $ R in Mod_R $ is infinite (i.e. $ R $ it's not strongly $ pi $-regular).

  2. For some $ s in R $ and something $ M in Mod_R $, the $ s $-divisibility range $ M $ It's infinite.

  3. For some $ s in R $, there is a right $ R $– arbitrarily large modules $ s $-divisibility range.

Obviously, $ 3 Rightarrow 2 Leftrightarrow 1 $. I suspect 1,2,3 are equivalent, and I think I can show this when $ R $ it is commutative, through a kind of "generalized Prufer module" construction. But I'm not sure when $ R $ It is not commutative.

Question 1: Make $ 3 Rightarrow 1 $ when $ R $ is it non-commutative?

There is another condition that these seem to be related to. Leave $ m in M ​​$ and $ n in N $ be two elements of law $ R $-modules. Say it $ m in M, n in N $ are weakly equivalent if there are maps on the right $ R $-modules $ f: M a N $ and $ g: N a M $ such that $ f (m) = n $ and $ g (n) = m $. This is an equivalence relationship in elements of law $ R $-modules. Consider the condition:

  1. There is an adequate class of elements of law $ R $-modules, no two of which are weakly equivalent.

Why $ f (Ms ^ alpha) subseteq N s ^ alpha $ for each map on the right $ R $-modules $ f: M a N $we have that $ 3 Rightarrow 4 $.

Question 2: Make $ 4 Rightarrow 3 $?

The "compound" of questions 1 and 2, that is, the question of whether $ 4 Rightarrow 1 $, can be expressed in a contradictory way as: if $ R $ is strongly $ pi $-regular then it does $ R $ have only one set of weak equivalence classes of elements of $ R $-modules?

set theory – Rasiowa Sikorski's Lemma Test for Boolean Algebras

My version of the theorem is as follows: Let $ {E_n: n in mathbb {N} } $ be an accounting family of subsets of Boolean algebra, $ mathbb {A} $, having infs and sups. Show that there is an ultrafilter $ Q $ which $ E_n $-complete for everyone $ n in mathbb {N} $.

My attempt at a test:
Leave $ {E_n: n in mathbb {N} } $ be given as above. Leave $ a> 0 $ be repaired, for previous work I have to $$ d (E_n) = {a> mathbb {O}: forall b in E_n (a cdot b = mathbb {O}) } cup {a> mathbb {O}: there is b in E_n (a le b) } $$

it's dense we have that $ d (E_1) $ it is dense for what exists $ b_1 in d (E_1) $ such that $ b_1 leq a $. Then for everyone $ n> 1 $we have that $ d (E_n) $ it is dense then it exists $ b_n in d (E_n) $ such that $ b_n leq b_ {n-1} $. In this way we inductively obtain a decreasing sequence of $ b_n $s with $ b_n in d (E_n) $.

Leave $ F_ {b_n} = {x: b_n le x } $then leave $ Q = bigcup_ {n in mathbb {N}} F_ {b_n} $.

Suppose $ E_n subseteq Q $, so $ E_n subset F_ {b_m} $ for some $ m $. Therefore, for each $ e in E_n $, $ b_m leq e $, so $ b_m cdot e = b_m $ so we have that $ inf {b_m cdot e: e in E_n } = inf {b_m } = b_m $. But, in Exercise 2.6, this is $ inf {b_m cdot e: e in E_n }
= b_m cdot inf {e: e in E_n } $
so we have that $ b_m cdot inf {e: e in E_n } = b_m $, that is to say, $ b_m leq inf {e: e in E_n } $, so $ inf {e: e in E_n } in F_ {b_m} subset Q $. $ Q $ it's like this $ E_n $-to complete.

The only thing I don't know is how to demonstrate that $ Q $ it is an ultrafilter … I tried many things but I don't see it. Any suggestions please?

lie algebras – Real roots and imaginary roots in $ widetilde {E} _6 $

Dynkin's diagram for $ widetilde {E} _6 $ is
begin {align}
circ – circ – & circ – circ – circ \
& | \
& circ \
& | \
& circ
end {align}

We denote by $ alpha_1, ldots, alpha_5 $ the simple roots that correspond to the vertices on the horizontal line and are denoted by $ alpha_6, alpha_7 $ the second and third vertex on the vertical line respectively.

What are the real and imaginary roots of $ widetilde {E} _6 $ in terms of $ alpha_1, ldots, alpha_7 $? In particular, is the following a root in $ widetilde {E} _6 $ (the numbers denote the coefficients of $ alpha_i $)?

begin {align}
1 – 2 – and 3 – 2 – 1 \
& | \
& 2 \
& | \
& one
end {align}

Is it a real or imaginary root? Thank you.

oa. operational algebras – Dixmier trace, Wodzicki residuals and topological index

There are known facts about Dixmier's trail and Wodzicki's residue. Leave $ P $ be an elliptic pseudodifferential degree operator $ −n $ in a compact Riemannian collector $ (M, g) $, that its Dixmier trace is related to the Wodzicki residual by the formula:
$$
operatorname {Tr} ^ {+} P = frac {1} {n (2 pi) ^ {n}} operatorname {Res} P
$$

Also, for arbitrary smooth function $ a in C ^ { infty} (M) $ the next equality
$$
int_ {M} a (x) left | nu_ {g} right | = frac {n (2 pi) ^ {n}} { Omega_ {n}} operatorname {Tr} ^ {+} left (a Delta_ {g} ^ {- n / 2} right)
$$

sustains. Is it possible to associate the Wodzicki residuals and the operator topological index? $ a Delta_ {g} ^ {- n / 2} $? For example, can something like $ operatorname {Tr} ^ {+} left (a Delta_ {g} ^ {- n / 2} right) in H ^ * _ {b, dR} (M) $ occur? Where H ^ * b, dR (M) It can be understood as John Roe's limited cohomology classes.

Functional Analysis: Measure Algebra in Bohr Compaction vs. Two-Dimensional Algebras

The next question probably boils down to a standard Twister play abstract harmonic analysis, but I'd still appreciate some feedback on that.

Leave $ G $ be a locally compact abelian group and leave $ bG $ denotes its Bohr compaction (the dual Pontryagin of $ G $ with the discrete topology). We denote by $ mathfrak {A} $ space $ L_1 (G) ^ {**} $ furnished with any Arens product.

Is there a canonical action of $ M (bG) $ (the algebra of measure in $ bG $) in $ L_ infty (G) $ that would lead to an isometric homomorphism $ M (bG) a mathfrak {A} $?