# Questions tagged [lambda-rings]

For questions about $\lambda$-rings which are commutative rings with operations which mimic the behavior of exterior powers of vector spaces.

**6**

votes

**0**answers

92 views

### Adams operation on Q-construction of fields

Let $F$ be a field that we want to compute its rational algebraic $K$-theory using the Quillen's $Q$-construction. Let $QF$ be the $Q$ construction of the category of finite dimensional vector spaces ...

**3**

votes

**0**answers

78 views

### Adams operation on the rational homology

The Adams operation acts on the algebraic $K$-theory of $R$ but the action as far as I know doesn't come from a endo-functor on the category of projective modules over $R$. For the $K_0$ there is an ...

**10**

votes

**2**answers

256 views

### Categories which are both monadic and comonadic over another category

I heard a professor say that $\lambda$-rings are both monadic and comonadic over commutative rings. Remark 2.11(a) on the nlab page says the same.
What does it mean, intuitively, that a category is ...

**11**

votes

**1**answer

457 views

### Is an ordinary scheme in Borger's Absolute Geometry the same as a “scheme over F1” with a map to Spec(Z)?

$\newcommand{\Z}{\mathbb{Z}}$
$\newcommand{\F}{\mathbb{F}_1}$
$\newcommand{\spec}{\operatorname{Spec}}$
If I understand correctly, in Borger's paper about the field with one element, the category of "...

**3**

votes

**1**answer

456 views

### Witt vectors and maps of $\lambda$-rings

Consider the ring $W(\mathbb{F}_p)$ of big Witt vectors of $\mathbb{F}_p$. This has a natural structure of a $\lambda$-ring (in the strong sense) since rings of big Witt vectors always do.
$\mathbb{Z}...

**8**

votes

**2**answers

404 views

### How do I find coefficients of a product expansion

Any power series $f(t) = 1 + t \mathbb{Z}[[t]]$ can be uniquely expanded in the following two ways:
$$1 + \sum_{i=1}^\infty f_i t^i =
\prod_{i=1}^\infty (1-t^i)^{-n_i}$$
Here, the $f_i$ and $n_i$ ...

**36**

votes

**3**answers

1k views

### Lambda-operations on stable homotopy groups of spheres

The Barratt-Quillen-Priddy theorem says in one interpretation that there is a weak equivalence of spectra $K(FinSet) \simeq \mathbb{S}^0$. In other words K-theory groups of finite sets are the stable ...

**8**

votes

**4**answers

955 views

### $\lambda$-ring structure defined for a graded ring in Fulton-Lang's book

Given a commutative ring $A$ with unity, Grothendieck used universal polynomials to define a special $\lambda$-ring structure on $\Lambda(A):=1+t\:A[[t]]$. Suppose $A$ is graded, say $A=\bigoplus_{i=o}...

**5**

votes

**1**answer

340 views

### Center of the category of special $\lambda$-rings

Recall that the center $\mathrm{Z}(C)$ of a category $C$ is the monoid of endomorphisms of $\mathrm{id}_C$. Thus $\eta \in \mathrm{Z}(C)$ is given by a familiy of endomorphisms $\eta_x : x \to x$, ...

**5**

votes

**3**answers

508 views

### An isomorphism of 2-Schur modules

This is the little brother of question 68071: elementary, simple-looking and probably much easier to answer. Of course, it is just a small part of question 68071, as anybody with $\lambda$-rings ...

**5**

votes

**1**answer

611 views

### K-Theory as a special $\lambda$-ring

I wonder if there is a nice and short proof that the $K$-theory of a topological space is a special $\lambda$-ring without invoking the splitting principle and alike. Is it possible to show directly ...

**3**

votes

**1**answer

478 views

### Left Adjoint to the Forgetful Functor on $\lambda$-rings?

The forgetful functor from the category of $\lambda$-rings to that of rings has a right adjoint in the form of the universal $\lambda$ functor $\Lambda$, which is isomorphic to the big Witt vectors ...

**11**

votes

**1**answer

1k views

### When are representation rings special lambda-rings? (variations of an old question)

Status: Questions 2 and 4 answered in the negative. Questions 1 and 3 ARE STILL UNANSWERED, despite previous claims.
On the third page of Wolfang K. Seiler's paper "lambda-rings and Adams operations ...

**6**

votes

**1**answer

743 views

### Is the Burnside ring a lambda-ring? + conjecture in Knutson p. 113

Warning: I'll be using the "pre-$\lambda$-ring" and "$\lambda$-ring" nomenclature, as opposed to the "$\lambda$-ring" and "special $\lambda$-ring" one (although I just used the latter a few days ago ...

**7**

votes

**2**answers

974 views

### Is every Adams ring morphism a lambda-ring morphism?

A lambda-ring $R$ is called "special" if it satisfies the $\lambda^i\left(xy\right)=...$ and $\lambda^i\left(\lambda^j\left(x\right)\right)=...$ relations, or, equivalently, if the map $\lambda_T:R\to\...