2
votes
1answer
58 views

Does the category PCM (partial commutative monoids) have a closed symmetric monoidal product?

A partial commutative monoid (PCM) is, roughly speaking, a set with a partially defined binary operation that is as associative as it can be (given that not all products are defined) and commutative. ...
2
votes
0answers
107 views

Localization and Direct limit [migrated]

Let $ A $ be a ring. Let $ I $ be a preordered set, filtering. Let $ \Sigma $ a multiplicative subset of $ A $. Suppose for any given $ i \in I $ a multiplicative subset $ S_i $ of $ A $ contained ...
0
votes
0answers
138 views

A question about the unbounded derived category of the polynomial ring in infinitely many variables

In this moment I am trying to understand the derived category of the polynomial ring in infinitely many variables over a field $k$, $R=k[x_{1},x_{2},\dots]$ and I am wonder if it is true that ...
5
votes
1answer
168 views

formally smooth functor

Let $p$ be a prime number, $\mathcal{O}$ the integers of a finite extension of $\mathbb{Q}_p$ with residue field $k$. Let $\mathcal{C}$ be the category of complete, local, noetherian ...
12
votes
3answers
426 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 ...
23
votes
5answers
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
5answers
731 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
3answers
401 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
2answers
739 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
1answer
428 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
0answers
175 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
1answer
172 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
2answers
484 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
0answers
423 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
2answers
565 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
5answers
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
1answer
409 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
3answers
568 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
1answer
406 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 ...
10
votes
2answers
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
2answers
284 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
1answer
266 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
0answers
575 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
1answer
287 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
0answers
610 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
1answer
365 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
2answers
493 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.
33
votes
1answer
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
0answers
636 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
1answer
365 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
1answer
350 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
1answer
420 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
2answers
536 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
0answers
700 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
2answers
251 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
0answers
610 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
2answers
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
1answer
618 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
6answers
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
3answers
975 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
2answers
403 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
2answers
1k 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 ...