# Tagged Questions

**5**

votes

**1**answer

218 views

### A generalized K- theory via generalized idempotents

Edit After the answer by Neil Strickland, I add the word "a ring" in this new version.
In the literature, there is a concept of generalized idempotent: an n- idempotent is an element $a$ of a Banach ...

**7**

votes

**1**answer

158 views

### $K_0$ group of graph underlying an approximately finite (AF) C* algebra

Say we have an AF C* algebra $A$ described by some Bratteli diagram $E$. If $M_\infty (A)=\displaystyle{\lim_\rightarrow M_n(A)}$ and $P(A)$ are the projections in this algebra, we know that ...

**18**

votes

**5**answers

612 views

### Bass' stable range of $\mathbf Z[X]$

Let $n$ be a positive integer and $A$ be a commutative ring. The ring $A$ is said to be of Bass stable range $\mathrm{sr}(A)\leq n$ if for $a, a_1, \dots, a_n \in A$ one has the following implication:
...

**1**

vote

**0**answers

108 views

### Cotorsion theory and its relative homology

Let (F(R), Cot(R)) be a cotorsion theory, Such that F(R) is the class of flat R-modules and Cot(R) the cotorsion modules. Why this is true that, For $ N\in Cot(R) $,
$ \text{Ext}_{F(R)}^i(M, N)\cong ...

**3**

votes

**1**answer

106 views

### ring with prescribed K group

One can construct topological spaces with prescribed homotopy groups or, say, homology groups.
But is it possible to construct a ring with any given $K_0$ group? What about $K_1$ group et.c.?
I know ...

**12**

votes

**1**answer

673 views

### finite dimensional real division algebras

A celebrated theorem of Milnor and Kervaire asserts that any finite dimensional division algebra over the real numbers has dimension 1,2,4 or 8. This result is established using methods from algebraic ...

**5**

votes

**0**answers

217 views

### Roots of unity in algebraic K-theory

For any commutative ring $R$, the tensor product of (finitely generated, projective) $R$-modules equips the algebraic K-theory $K(R) = K_0(R)$ with the structure of a commutative ring with unit.
For ...

**6**

votes

**1**answer

199 views

### Significance of the vanishing of $K_{-1}(A)$

In M. Schlichting's paper, he defines the negative $K$-theory for derived categories. In this he states that for $\mathcal{A}$ an idempotent complete (see below) triangulated category, ...

**2**

votes

**2**answers

224 views

### Action of centralizer on Borel-Moore homology of Springer Fibers for Affine Hecke Algebra

In Chriss and Ginzburg's "Representation Theory and Complex Geometry", they describe a geometric construction of representations of the affine Hecke algebra, using the Borel-Moore homology of ...

**5**

votes

**3**answers

544 views

### Infinite dimensional central simple algebras

When constructing the Brauer group of a field, only the finite-dimensional central simple algebras are considered (because of Artin-Wedderburn's characterization).
But what happens to the ...

**6**

votes

**1**answer

376 views

### Projective modules over free groups

Consider the ring of Laurent polynomials $R := \mathbb{Z}[s,s^{-1}]$ with integer coefficients. Are all projective $R$-modules free? (Let's say left modules by convention.)
More generally, let $G$ ...