## 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.

## Linear Algebra – Real Orthogonal and Sign

I came across the following conjecture when I read a recent article in the Monthly, an Orthogonal Matrix of Order $$n neq 0 pmod 4$$ has a non-negative (up to a scalar) raw vector.
The dimensions should be straight $$2$$ and $$3$$ and
much impossible for the rest.
Create a relationship within a specific result.

## 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.