# Tagged Questions

**11**

votes

**3**answers

408 views

### Varieties where every algebra is free

I'd like to know more about varieties (in the sense of universal algebra) where every algebra is free. Another way to state the condition is that the comparison functor from the Kleisli category to ...

**21**

votes

**5**answers

1k views

### (Short) Exact sequences with no commutative diagram between them

This question was asked by a student (in a slightly different form), and I was unable to answer it properly. I think it's quite interesting.
The problem is to produce an example of the following ...

**13**

votes

**5**answers

689 views

### is the category of coherent sheaves some kind of abelian envelope of the category of vector bundles?

This might be obvious to experts, but I'm not sure where to look for the answer. On a reasonably nice, at least noetherian, scheme (or variety, algebraic space, stack), can the category of coherent ...

**5**

votes

**3**answers

390 views

### Exponentials in the opposite category of finite separable algebras

Let $K$ be a field and $G=Gal(K_s/K)$ is its absolute Galois group. Then, by Galois theory, the category of finite separable algebras over $K$ (denoted by $Sep(K)$) and the category of finite ...

**10**

votes

**2**answers

719 views

### Fields aren't group objects in Ab, so what are they?

This might be a vague question, but I am troubled by the fact that fields do not admit a nifty categorical definition. An obvious attempt such a definition would be to say that fields are commutative ...

**4**

votes

**1**answer

426 views

### Is the category of quasi-coherent $\mathcal{O}_X$-algebras cocomplete?

Let $X$ be a scheme. Is the category of quasi-coherent (commutative) $\mathcal{O}_X$-algebras cocomplete?
Remark.
The same question was asked in MSE last year. Since nobody has answered it, I post ...

**10**

votes

**0**answers

174 views

### Smallest class of rings closed under familiar operations

Suppose I start out with the ring $\mathbb{Z}$, and call $\mathcal{C}$ the smallest collection of (commutative, unital) rings closed under the following list of operations (which I am aware has some ...

**0**

votes

**1**answer

161 views

### Poset axioms of Boolean algebra [closed]

I found in Awodey's "Theory Category", second edition p34, a set of poset axioms to define Boolean algebras, whereas I don't see how they can be sufficient.
Here are the axioms:
A Boolean algebra ...

**3**

votes

**2**answers

449 views

### Injective Modules over Group Rings

Given the group ring $\mathbb{Z}[G]$ of a finite group $G$ over $\mathbb{Z}$, is there a way to generalize the notion of the "frobenius algebra" in some cases? One can show that every group ring ...

**8**

votes

**0**answers

422 views

### Conceptual proofs for the computation of the structure sheaf

The following lemma in commutative algebra is important for the foundations of algebraic geometry:
If $A$ is a commutative ring, $U \subseteq A$ is a finite subset generating the unit ideal, then ...

**15**

votes

**2**answers

543 views

### Is a retract of a free object free?

I wonder whether this is true in the categories of groups, monoids, commutative algebras, associative algebras, Lie algebras?

**7**

votes

**5**answers

1k views

### My first question - on Affine Schemes in Algebraic Geometry

If R is a commutative ring (with unit) then we have an affine scheme Spec(R) which is an object of the category of ringed topological spaces. Is there any way of characterising this object relative to ...

**3**

votes

**1**answer

381 views

### Direct image sheaf and tensor product (is the projection formula an isomorphism?)

Assume we have two "nice" schemes $X$ and $Y$ over $k=\mathbb{C}$, a finite flat map $f:X\rightarrow Y$ and a k-algbera $A$. Then we get an induced finite flat map $f_A:X\times_k A \rightarrow ...

**8**

votes

**3**answers

562 views

### Is the restriction map an epimorphism of commutative rings?

Let $i : U \to X$ be a quasicompact open immersion of schemes. I would like to know whether the canonical morphism $i_* \mathcal{O}_U \otimes_{\mathcal{O}_X} i_* \mathcal{O}_U \to i_* \mathcal{O}_U$ ...

**4**

votes

**1**answer

387 views

### permutation of projective limits with inductive limits

Hi everybody,
I have a lack of references concerning projective limits and injective limits. Up to my faults in Bourbaki there are only proj and inj limits indexed by a partially ordered set (not a ...

**9**

votes

**2**answers

1k views

### Elements in a localization - category theoretic approach

This question is about the elements in a localization $S^{-1} A$ of a commutative ring $A$. Is it possible to derive $\frac{a}{1} = 0 \in S^{-1} A \Rightarrow \exists s \in S : sa = 0$ only using the ...

**1**

vote

**2**answers

282 views

### Example of a commutative algebra object in a braded monoidal category C

Hi,
I am looking for an example of a commutative algebra object in a braided monoidal category C which it can also be turned into a commutative Frobenius algebra. If you have any examples could you ...

**4**

votes

**1**answer

262 views

### Filtrant (not necessarily totally ordered) projective system commuting with direct sums

Hello,
Let $R$ be a commutative (not necessarily Noetherian) ring.
Let $I$ be a small filtrant (not necessarily totally ordered) category.
Let $(M_i)_{i\in I}$ be a projective system of $R$-modules ...

**25**

votes

**0**answers

558 views

### Enriched Categories: Ideals/Submodules and algebraic geometry

While working through Atiyah/MacDonald for my final exams I realized the following:
The category(poset) of ideals $I(A)$ of a commutative ring A is a closed symmetric monoidal category if endowed ...

**3**

votes

**1**answer

286 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 ...

**2**

votes

**0**answers

577 views

### Grothendieck spectral sequence [duplicate]

Possible Duplicate:
Composing left and right derived functors
Hi,
probably this question is obvious. I apologize for this.
Given functors $F$ and $G$ left exact, with as good properties as ...

**7**

votes

**1**answer

359 views

### Spectrum of an algebra object and Reconstruction of Schemes

In "Au-dessous de $\text{Spec}(\mathbb{Z})$", Toen and Vaquié define schemes relative to a complete, cocomplete symmetric monoidal category $C$ using a functorial approach.
In the introduction the ...

**7**

votes

**2**answers

492 views

### Simple object in derived category or stable model category?

Exist any common definition of simple objects in derived categories, or even better, in stable model categories?
I was only able to find definition for abelian categories.
Thanks.

**32**

votes

**1**answer

1k views

### Categorical definition of the ideal product within the category of rings

This is an extension of this question. Let $I,J$ be ideals of a ring $R$; every ring is commutative and unital here. Is it possible to define $R \to R/(I*J)$ out of $R \to R/I$ and $R \to R/J$ in ...

**4**

votes

**0**answers

625 views

### $Ext$ functor, filtered complexes and spectral sequences

Let $\mathcal{A}$ an abelian category. Take $M$ an object of $\mathcal{A}$, and $K_*$ a bounded complex in $\mathcal{A}$ equipped with a bounded increasing filtration $F$. By using homological and ...

**3**

votes

**1**answer

359 views

### Center of the category of $R$-algebras

Let $R$ be a ring (assumed to be commutative and with unit). What is the center of the category of $R$-algebras, i.e. $\text{Z}(R\text{-Alg})$? This is a commutative monoid. See this recent question ...

**4**

votes

**1**answer

347 views

### “extend a functor”

Hi,
I have probably a basic question. I have a functor $F: Sch \rightarrow Set$, an algebraic stack $M$ with a "universal family" $G\rightarrow M$ and a representability property like this: for every ...

**7**

votes

**1**answer

416 views

### Why is the symmetric monoidal structure on invertible modules strict?

Let $N$ be an object in a symmetric monoidal category. Then the braid map $N\otimes N\to N\otimes N$ is almost never the identity, and this is the obstruction to making a symmetric monoidal category ...

**5**

votes

**2**answers

534 views

### Is the category of affine schemes (over a fixed field) Cartesian closed?

This is probably a trivial question, but I don't see the answer, and I haven't found it on Wikipedia, nLab, nor MathOverflow.
Let $\text{ComAlg}$ denote the category whose objects are commutative ...

**2**

votes

**0**answers

690 views

### Pushouts of noetherian rings

Does the category of noetherian rings has pushouts?
Background: If $X/S$ is an abelian scheme, then the relative Picard functor $Pic_{X/S}$, is only defined on the category of locally noetherian ...

**0**

votes

**2**answers

249 views

### Can all induced maps be described categorically.?. (or at least as generally as possible)

Hi: I am new here. I went over the fAQ's, still, sorry if I break protocol.
I am pretty confused about induced maps in different areas of algebraic
topology; I do know how these induced maps are ...

**9**

votes

**0**answers

603 views

### monomorphisms and epimorphisms of local rings

I want to understand the structure of monomorphisms/epimorphisms in the category of local rings (with local homomorphisms), or dually in the category of local schemes. Let $LR$ denote this category.
...

**26**

votes

**2**answers

1k views

### How can I define the product of two ideals categorically?

Given a commutative ring $R$, there is a category whose objects are epimorphisms surjective ring homomorphisms $R \to S$ and whose morphisms are commutative triangles making two such epimorphisms ...

**9**

votes

**1**answer

609 views

### Is ΩΣ in {simplicial commutative monoids} group completion?

Let C be the model category of simplicial commutative monoids (with underlying weak equivalences and fibrations), or equivalently the (∞,1)-category PΣ(Top), where T is the Lawvere theory ...

**21**

votes

**6**answers

4k views

### Algebraic stacks from scratch [closed]

I have a pretty good understanding of stacks, sheaves, descent, Grothendieck topologies, and I have a decent understanding of commutative algebra (I know enough about smooth, unramified, étale, and ...

**0**

votes

**3**answers

973 views

### equality of elements in localization via universal property [unsolved!]

I've been studying universal objects of universal algebra in a quite general setting and try to exhibit the structure of their elements just using the universal property. a very nice example for this ...

**8**

votes

**2**answers

398 views

### Which commutative rigs arise from a distributive category?

A rig is an algebraic object with multiplication and addition, such that multiplication distributes over addition and addition is commutative. However, instead of requiring that the set forms an ...

**8**

votes

**2**answers

980 views

### Exactness of filtered colimits

Are filtered colimits exact in all abelian categories?
In Set, filtered colimits commute with finite limits. The proof carries over to categories sufficiently like Set (i.e. where you can chase ...