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Brauer groups – Azumaya algebra and separable central algebra on rings

leave $ R $ a commutative ring

Auslander defines a central separabl $ R $-algebra $ A $ as faithful $ R $-module with $ Z (A) = R $ and such that $ A $ is projective $ A otimes_R A ^ { mathrm {op}} $-module

Following Milne; an algebra of Azumaya $ A $ over a local ring $ R $ it's an algebra (with $ Z (A) = R $) which is a free finite range module on $ R $ such that the natural map
$ A otimes_R A ^ { mathrm {op}} to mathrm {End} _R (A) $ is an isomorphism

How to prove (as Auslender noted) that a central separable algebra on a local ring is just an Azumaya algebra.

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design patterns: pointers versus maintaining indexes of objects stored in a central (associative) matrix?

Until recently, I thought it was better to refer to objects by pointers or references than to keep the objects in a kind of central or authorized matrix or dictionary and only keep indexes or keys of the members of that matrix.

However, recently I started reading something opposite. I began to read that it was preferable to avoid pointers / references and follow the last approach because (a) it allows painless serialization / deserialization of data; (b) it is the only reasonable approach if the functional paradigm is followed (and FP is praised today); (c) others?

Is what I wrote so far correct?

My problem with keeping only object identifiers instead of references to objects is that, at least for me, this leads to illegible or messy code.

Let me show you an example. This is a code I wrote for the turn-based game that I am trying to do as a hobby:

for (let i = 0; i < monsAndMoves.mons(monname).moves.length; i++) {
  for (let j = 0; j < monsAndMoves.moves(monsAndMoves.mons(monname).moves(i)).prerequisites.length; j++) {
    g.setEdge(monsAndMoves.moves(monsAndMoves.mons(monname).moves(i)).prerequisites(j), monsAndMoves.mons(monname).moves(i), {

What happened here?

  • monsAndMoves - an object that stores information about species of monsters and movements that monsters can perform.
  • monsAndMoves.moves - A matrix that stores information about movements. (It must be a dictionary whose keys must be movement names to maintain consistency with the following)
  • monsAndMoves.mons - a dictionary that maps names of monster species to obtain information about these species.
  • monname - should be called currentMon.speciesName - Identification of the current monster species.
  • prerequisites - movement indices that a monster must know to learn the given movement

The previous code cannot be read.

We can make it readable at the cost of saturating it:

const currentSpecies = monsAndMoves.mons(currentMon)
const moves = currentSpecies.moves
for (const moveName in moves) {
  const move = moves(moveName)
  for (const prereqName in move.prerequisites) {
    g.setEdge(moveName, prereqName, {
      /* ... */

The problem is that, given this approach, we will have to repeat things like const currentSpecies = monsAndMoves.mons(currentMon) at the beginning of almost all functions. And if we stick to the general advice that the functions should be as short as possible ("do one thing and one thing only") Then we will have many functions to saturate in this way. This seems annoying and very repetitive.

And this is what this code would look like if the references were kept instead of the identifiers:

for (const move of currentMon.species.moves) {
  for (const prereq of move.prerequisites) {
    g.setEdge(prereq.name, move.name, {
        /* ... */

It seems readable and without repetition.

Therefore, let me ask you:

  • Is it correct that, for the reasons I described above, avoiding references and embracing object identifiers is considered the preferred approach?
  • If so, is there any way to get the benefits of keeping identifiers against references and at the same time avoiding the pain of this approach, that is, writing messy or illegible code?

central processing unit: base clock frequency of server CPUs lower than desktop CPUs, how is this possible?

When comparing the base clock frequency of the CPUs of a given microarchitecture, there are generally two correlations:

  1. the base clock frequency is inversely proportional to the number of cores
  2. the base clock speed of a server CPU is slower than the base clock speed of a desktop CPU with the same number of cores

The first correlation can be explained by the challenges of TDP: adding more cores to a chip means having to transport more heat from basically the same area, therefore, a reduced base clock speed helps reduce the amount of heat produced.

But how do you explain the second correlation?

An example: AMD Zen 2 microarchitecture

The base clock speed of the 16-core Epyc CPU is 2.8 or 3 GHz, while the base clock speed of the 16-core Ryzen CPU is 3.5 GHz.

Is it because the core of a server CPU generally contains more transistors (since it contains more / larger functional units / caches) and therefore occupies a larger chip area? IOW, does the Epyc exchange the base clock speed with more transistors while remaining in a given TDP envelope?

The central search administration service is not activated

When I navigate to the sharepoint search administrative center I see Error

enter the description of the image here

sharepoint online: show the central sites. Do not show

Battery exchange network

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[Politics] Open question: Why do Mexicans and Central Americans receive such bad treatment in the United States? .