Reference requirement: The dimension of the Zariski-Tangent space is limited to finitely generated algebra

can anyone suggest a published reference for the following fact

For a given finitely generated algebra over an algebraically closed field, the dimension of the Zariski tangent space is bounded by maximum ideals from above.

? I can not find it in the beloved Stacks project.

ct.category theory – Definition of $ E_n $ modules for an $ E_n $ -algebra

The category $ Mod ^ {E_n} _A ( mathcal {C}) $ from $ E_n $Modules for one $ E_n $algebra in a symmetrical monoidal $ infty $-Category $ mathcal {C} $ is defined in Lurie's Higher Algebra as a special case of a more general definition (just replace the small one) $ n $-cubes operad with your favorite $ infty $-operad). However, the definition is pretty ambiguous and requires a lot of terminology that I'm not familiar with. I was wondering if there is a more intuitive way to define / think this category. For example, what is the best way to think about it? $ E_n $Modules for one $ E_n $Algebra in the category of spaces?

There is also a proof in Lurie that $ Mod ^ {E_1} _A ( mathcal {C}) $ for a $ E_1 $-Algebra $ A $ corresponds to the category of bimodules for $ A $ in the $ mathcal {C} $But again, the proof is pretty long and difficult. Is there an intuitive way to think about this result? I would also be happy if there was a short proof $ A $ an ordinary associative algebra.

linear algebra – The rank of a special matrix

Suppose that $ P $ is a polynomial of degree $ d: = deg P $ over a field $ mathbb F $ the zero characteristic, which divides completely into pairwise different linear factors, and $ B, C subset mathbb F $ are sets of size $ n: = | B | = | C | $; the area of ​​interest is $ n sim d ^ {2/3} $, Consider the matrix
$$ M: = begin {pmatrix}
P ^ {(d)} (c) , b ^ d \
P ^ {(d-1)} (c) , b ^ {d-1} , \
vdots \
P (c)
end {pmatrix} _ {b in B, c in C} $$

(This is indeed only one of the $ | B || C | $ Columns has the entire matrix size $ (d + 1) times n ^ 2 $).

  1. Is it true that usuallythis matrix has at least one rank $ d + 2-n $? Are there reasonably adequate conditions for $ mathrm {rk} , M ge d + 2-n $ hold?

  2. At least you can (under reasonable assumptions) exclude the scenario in which each of the last $ n $ Lines is a linear combination of the first $ d + 1-n $ Ranks?

Note that every line $ P ^ {(s)} (c) , b ^ s $ can be made proportional to every other line $ P ^ {(t)} (c) , b ^ t $ With $ t le d-n $: that happens when $ B $ is contained in a level set of the mapping $ x mapsto x ^ {t-s} $, and $ C $ is contained in the null location of the polynomial $ P ^ {(t)} – ​​KP ^ {(s)} $ with a corresponding $ K in mathbb F $,

abstract algebra – characteristic direct product of the rings

Leave the rings, $ R_i $ and its direct product, $ R = Pi_1 ^ {n} R_i = R_1 times R_2 times … times R_n $

say that $ Char (R_i) = m_i $

(1) $ Char (R) $ = $ lcm (m_1, m_2, … m_n) $

If all rings, $ R_i $ Commutative, the statement (1) is certainly true.

But what if there are some rings that are not commutative?

Is statement (1) true?

Many Thanks.

(IEE is still up $ (1) $ True, that respect for the $ R_i $ is a commutative or not?)

linear algebra – rank mod $ p $ of a non-singular matrix with given determinant

To let $ A $ be a non-singular $ n $-by-$ n $ Matrix with integer entries. Accept that $ p ^ r nmid det (A) $, Does that follow? $ A $ has a $ (n-r + 1) $-by-$ (n-r + 1) $ Minor, which is not singular modulo $ p $?

If the answer is no, what if we have some additional conditions – say, $ p $ big compared to $ n $and or $ A $ Restricted entries?

linear algebra – fastest algorithm for multiplying two matrices (not necessarily square matrices)

I looked at the road algorithm, but the online resources just show that it works for square matrices (with dimensions) $ 2 ^ n { times} 2 ^ n $ from where $ n $ is a natural number)? But what if there are two non-square matrices with different dimension lengths (ie $ A { times} B $ Matrix of a $ B { times} C $ Matrix). What is the fastest algorithm then?

Linear Algebra – Does an antisymmetric matrix of high rank have a minor with disjoint rows and columns and high rank?

This is a generalized version of Does a non-singular matrix have a big minor with disjoint lines and columns and full rank?

To let $ A $ Bean $ n $-by-$ n $ Antisymmetric rank matrix $ r geq epsilon n $, Is there a minor of $ A $ with disjunctive row and column indices $ I, J subset {1,2, dotsc, n } $ and rank $ k geq lfloor r / 1000 rfloor $?

(Some arguments concerning a Pfaffian may work, but it does not seem obvious to me.)

Abstract algebra – Normal subgroup and inclusion of the ideal

I am a student who has just started abstract algebra.

It's a simple question for you.

To let $ X_1 le Y $ and $ X_2 $ is a (normal or ideal) of $ Y $

Loud $ 3rd $ Isomorphism Theorem, We can easily conclude that $ X_1 cap X_2 $ is an ideal of $ X_1 $,

But the question is

$ (1) $ does $ X_1 cap X_2 $ is an (ideal or normal subgroup) of $ X_2 $?

$ (2) $ does $ X_1 cap X_2 $ is an (ideal or normal subgroup) of $ Y $?

Whenever you try to prove (1) and (2), they look like true.

But I have no confidence that both are true.

Any help would be appreciated.

Many Thanks.

oa.operator algebra – track extension on the von Neumann subalgebra

No, if $ Gamma $ is a group without torsion and with the property Infinite conjugacy class then the algebra $ L Gamma $ is a typical factor $ II $ and only has a normal traffic condition.

Now let it go $ x $ in the $ Gamma $ different from the identity. The von Neumann algebra $ S $ generated by $ x $ is isomorphic too $ L ^ infty (S ^ 1) $, the algebra of measurable functions on the circle. It has many normal traffic conditions.