All questions with tag [math: algebraic-k-theory]


Characteristic polynomial of the unique automorphism of the zero module

Is there any convention which makes sense of the characteristic polynomial of the unique automorphism of the zero module? This might seem like an odd question but it matters to me. The background (for those who understand): I am studying the K-theory of the category of pairs $(P,f)$ where $P$ is some projective $R$-module and $f$ is an automorph...
2022-07-25 20:43:33

Idea behind the factorization of the matrix $\operatorname{diag}(a,a^{-1})$ in algebraic K-Theory

If $a \in S$ is some invertible component in a ring $S$, after that a calculation reveals $$\pmatrix{a & 0 \\ 0 & a^{-1}} = \pmatrix{1 & a \\ 0 & 1} \pmatrix{1 & 0 \\ -a^{-1} & 1} \pmatrix{1 & a \\ 0 & 1} \pmatrix{0 & -1 \\ 1 & 0}.$$ If $R \to S$ is a surjective ho...
2022-07-09 23:48:46

Solutions to equation in matrix form

Suppose $x=[x_1, x_2, \cdots, x_n]^t$, $b=[b_1,b_2,\cdots,b_n]^t$ with $b_i\in K$ and also $A\in M_n(K)$, where $K$ is an area. There are popular standards for the system of formulas $Ax=b$, by taking into consideration ranking of $A$ and also $[A|b]$. If we take into consideration the formula $x^tAx=\lambda$, for $\lambda \in K$, what are the ...
2022-06-09 18:32:54

How to compute the formal group law of K-Theory

Could any person aim me to a reference where the official team regulation of (topological or motivic) K - concept is calculated in as much information as feasible?
2022-06-09 16:48:07

Milnor $K_2$ of an inverse limit is inverse limit of Milnor $K_2$'s?

Let $\{A_n\}$ be an inverted system of rings and also $\hat{A}$ be the inverted restriction of this system. Allow $K_2(R)$ represent Milnor $K_2$ (I will certainly think that the instance I want the Milnor $K_2$ is isomorphic to algebraic $K_2$ of Quillen). Inquiry: Is $K_2(\hat{A}) \cong {\rm inverse~ lim}\ K_2(A_n)$? I want a response to thi...
2022-06-06 19:32:05

Basic question on $K$-theory

Let $h\colon A\to A'$ be a ring homomorphism in between $A,A'$ which are commutative rings with $1$. Allow $P,Q$ be $A$ - components. After that, exist any kind of voids in my adhering to argument? $A'\otimes_{A}(P\otimes_{A}Q)=(A'\otimes_{A}P)\otimes_{A}Q=((A'\otimes_{A}P)\otimes_{A'}A')...
2022-06-06 16:20:01

Question on K-theory

If $R$ is a ring with 1 which pleases $R^r=R^s$ for some $r\neq s$. Exist any kind of specific estimation of $K_0(R)$ for such $R$? I need to know such instances due to the fact that I assume that such $R$ might not have $\mathbb{Z}$ - summands.
2022-06-06 15:33:06

Chromatic Filtration of Burnside Ring

I simply participated in a workshop on the colorful purification of the Burnside ring. I recognized it reasonably well, yet at no factor did any person offer a specific definition of what a colorful purification in fact is, and also I have had some problem locating one. Thinking I have the essential specific history, i.e. expertise of purificati...
2019-12-02 02:48:27

What is the connection of the sequence 3, 4, 5/3, 2/3, 1 with deep topics?

Price Estimate from Don Zagier (Mathematicians: An Outer View of the Inner World): " I such as specific, hands - on solutions. To me they have an elegance of their very own. They can be deep or otherwise. As an instance, visualize you have a collection of numbers such that if you add 1 to any kind of number you will certainly get the item ...
2019-05-29 22:31:09

Topological vs. Algebraic $K$-Theory

Suppose I can compute the phenomenal cohomology inscribed in topological $K$ - teams of a topological room $X$ with CW framework. What details does this offer me concerning $C^{*}$ - algebras related to $X$? What is the algebraic analogue of topological suspension or the algebraic variation of Bott Periodicity? Larger Question : More usually,...
2019-05-23 07:18:07

How to show directly that two elements become equal in Grothendieck group?

Take into consideration commutative semigroup S and also its Grothendieck conclusion team G (S). Intend I demand specifying G (S) as free abelian team on basis $[a]$ (with $a\in S$) separated out by the relationships $[a+b]-[a]-[b]$. Just how do I show with that said definition that if the photos of $a,b\in S$ come to be equivalent in G (S), af...
2019-05-19 01:20:25