0
votes
1answer
57 views

Question regarding Vakil's algebraic geometry notes

Exercise 1.3 D of Vakil's lecture notes on algebraic geometry asks: "Verify that $A \to S^{−1}A$ satisfies the following universal property: $S^{−1}A$ is initial among $A$-algebras $B$ where every ...
4
votes
0answers
77 views

Functorial approach to Ideals and Quotients, Multiplicative Sets and Localizations

I have been playing with substructures of commutative rings today and noticed that there is a strong analogy between the formation of quotients and kernels with the formation of localizations with ...
3
votes
1answer
84 views

Universal property of polynomial ring in $\mathbf{CRING}$

I know that the polynomial ring $A[x]$ is the free $A$-algebra on $\{x\}$; this is its universal property in the category of $A$-algebras. Is there also a universal property for $A[x]$ considered as a ...
3
votes
0answers
29 views

Localization and Direct limit [duplicate]

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 ...
1
vote
1answer
134 views

Localization and direct limit

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 ...
1
vote
1answer
41 views

Set maps given by a polynomial & Yoneda Lemma

This Exercise 4.1. from the book Algebraic Geometry I, by Gortz. Problem Let $R$ be a ring, and for every $R$-algebra $A$ let $\alpha_A:A\rightarrow A$ be a map of sets such that for every ...
3
votes
0answers
56 views

Topological characterisations of freeness and separability?

According to wikipedia, a spectral space is defined to be homeomorphic to the spectrum of some commutative ring. They form a category $Spec$ where we take morphisms to be those whose preimage of open ...
2
votes
1answer
71 views

Exercise from Rotman: formal power series ring as inverse limit

Let $A$ be a commutative ring with unit, $J = (x)$ an ideal of $A[x]$. Thus we can consider the inverse system defined as $$\psi_{n,m}: A[x]/J^m \to A[x]/J^n$$ $$g(x) + J^m \to g(x) + J^n$$ $$\forall ...
0
votes
0answers
41 views

Alternative proof for localization isomorphism

Let $f$ be an $A$-module morphism and $\operatorname{res}_{A_m}^A$ be the restriction of scalars functor from $A_m$-mod to $A$-mod. I'm curious if you have proven that for every maximal ideal ...
1
vote
1answer
88 views

Does localization functor have both sides adjoint functors?

Let $A$ be commutative ring, and $S$ a multiplicative set. The localization $S^{-1}$: $A$-module $\rightarrow$ $S^{-1}A$-module. Functor $F$: $S^{-1}A$-module $\rightarrow$ $A$-module, regard $A$ as ...
3
votes
1answer
34 views

Is the Derivation Algebra functorial

Suppose $A$ is a commutative, associative $k$-algebra with unit and $Der(A)\subset End_k(A,A)$ is the algebra of derivations on $A$, that is the subalgebra of endomorphisms, such that ...
8
votes
2answers
324 views

A proof using Yoneda lemma

Martin Brandenburg pointed out elsewhere in the comments that he could give a one line proof, using the Yoneda lemma, of $$\frac{\mathbf{C}[x_1,\ldots,x_{n+m}]}{I(X)^e+I(Y)^e} \cong ...
1
vote
1answer
45 views

Map induced by localization on categories

I have been doing some reading in Hartshorne's Algebraic Geometry on derived functors and subsequent results in cohomology. Given $A$ an abelian category of groups, I have seen that the map ...
3
votes
0answers
54 views

Regular monomorphisms of commutative rings

What are the regular monomorphisms of $\mathsf{CRing}$? Is there a purely algebraic characterization? Since regular monomorphisms coincide here with effective monomorphisms (see Prop. 1. here), the ...
11
votes
1answer
137 views

$B\otimes_A A[x]=B[x]$

Let $A\rightarrow B$ be a homomorphism of commutative rings. Then $B\otimes_A A[x]\cong B[x]$ as $B$-algebras. How can one demonstrate this nicely, i.e. using universal properties alone and the Yoneda ...
1
vote
1answer
47 views

Moving tensor products inside homs

Suppose that $(\mathcal C, \otimes, I)$ is a closed symmetric monoidal category with $\hom(A,B)$ the hom-sets and $[A,B]$ the internal hom (where $[A,-]$ is right adjoint to $-\otimes A$). Is there ...
1
vote
2answers
72 views

Are coproduct exact functors?

Are coproducts left exact or right exact functors in general? Let k be a commutative ring (unital assosiative). Specifically in the category of k-algebras is the tensor exact. (This is not the case ...
7
votes
3answers
196 views

Does free functor preserve monomorphism?

The free functor is left adjoint to the forgetful functor so it preserves epimorphism. In the category of modules and algebras, it also preserves monomorphisms (the free functors being free modules ...
2
votes
2answers
101 views

Is $R$ initial in the category of $R$-algebras?

Let $R$ be an arbitrary unital associative ring. In the category of $R$-algebras $\mathfrak{Alg_R}$, if we consider $R$ as an $R$-algebra over itself (trivially), what type of object is it then in ...
2
votes
0answers
54 views

Stable fiber products of commutative rings

Let $R_1 \to T$ and $R_2 \to T$ be homomorphisms of commutative rings. Consider the fiber product $R=R_1 \times_T R_2$. Let $R \to R'$ be a homomorphism of commutative rings, and define $R'_i$ to be ...
5
votes
0answers
103 views

When is $K_0(i)$ an injection?

Suppose that $\mathcal A$ and $\mathcal B$ are two abelian categories such that $\mathcal A$ is a full subcategory of $\mathcal B$. If $i: \mathcal A\rightarrow\mathcal B$ is the inclusion functor, ...
7
votes
1answer
161 views

Understanding the right-exactness of the tensor product using *only* its universal property and the Yoneda lemma

I would like to get an intuition for why $(-)\otimes N$ is right-exact using its universal property involving bilinear maps, not by appealing to higher-level observations such as "left-adjoints ...
4
votes
1answer
162 views

Could I see that the tensor product is right-exact using its universal property and the Yoneda lemma?

I have been doing some review with the goal of trying to understand as much as I can via universal properties and category theory (already feeling comfortable with the mundane way of doing things). ...
8
votes
2answers
158 views

What about a module of rank $\frac{1}{2}$?

Let $R$ be a commutative ring. The possible ranks of free $R$-modules are $0,1,2,\dotsc$. But what about a generalized notion of an $R$-module where ranks may be rational numbers such as ...
0
votes
1answer
150 views

Localization and initial objects

Let $A$ be a ring and let $S$ be a multiplicative subset of $A$. Why is the map from $A$ to $S^{-1}A$ initial among all $A$-algebras $B$? Why does localization not have to commute with respect to ...
7
votes
1answer
95 views

Can we really understand $R$ by studying $R$-modules? [duplicate]

According to Algebra: Chapter 0, the category $R\operatorname{-Mod}$ reveals a lot about $R$. However after completing the first eight chapters, I still found no examples where this happens. Can ...
5
votes
3answers
803 views

Right-adjoint functors are left-exact?

As a final exercise to VIII.1 in Algebra: Chapter 0, we are asked to prove If $\mathcal{F}\colon\operatorname{R-Mod}\to\operatorname{S-Mod}$ is a right-adjoint operator, then $\mathcal{F}$ is ...
2
votes
1answer
114 views

Is this square diagram cocartesian for every regular local ring?

Let $K$ be a field and $R=\{f\in K[X]\mid f(0)=f(1)\}$ the $K$-algebra obtained by pulling back $K[X]\to K\times K$, $X\mapsto (0,1)$ along the diagonal. Is the induced square \begin{eqnarray} ...
10
votes
1answer
277 views

Completion as a functor between topological rings

In the following all rings are assumed to be commutative and unitary. Preliminaries: For any topological ring $R$ we can form its completion $\widehat{R}$ by taking all Cauchy sequences modulo null ...
7
votes
1answer
146 views

Projective objects in the category of rings

What are the projective objects in the category of rings with identity ? Remarks: The only projectives I could find so far are $\{ 0\}$ and $\mathbb{Z}$. If $R$ is projective and ...
2
votes
2answers
229 views

Direct limit in category theory

$\newcommand{\al}{\alpha}$Let $(M_\alpha)_\alpha$ be a direct system of abelian groups, and $\varinjlim M_\alpha$ its direct limit. Then one can show that every element of $\varinjlim M_\alpha$ can be ...
1
vote
1answer
92 views

On colim $Hom_{A-alg}(B, C_i)$

We assume all rings considered are commutative. Let $A$ be a ring. Let $B$ be an $A$-algebra of finite presentation. Let $I$ be a small filtered category. Let $C\colon I \rightarrow$ $A$-alg be a ...
4
votes
2answers
112 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?
2
votes
1answer
335 views

Does the ring of global sections functor on the category of locally ringed spaces have an adjoint functor?

Let $Rng$ be the category of commutative rings. Let $Loc$ be the category of locally ringed spaces. Let $(X, \mathcal{O}_X)$ be an locally ringed space. Then $\Gamma(X) = \Gamma(X, \mathcal{O}_X)$ is ...
3
votes
0answers
170 views

Reflection of Exact Sequences

Consider the category of $R$-modules. I am trying to see how i can express a short exact sequence in terms of kernels and cockerels, and how this description can be used to prove that a conservative ...
5
votes
1answer
218 views

Injective Cogenerators in the Category of Modules over a Noetherian Ring

Let $R$ be a Noetherian ring and let $\mathcal{A}$ be an injective $R$-module. The injectivity of $\mathcal{A}$ is equivalent to the exactness of the functor $Hom_R(-,\mathcal{A})$, i.e. whenever we ...
4
votes
1answer
252 views

Tensor products commute with inductive limit

How to prove, that tensor products commute with direct limits, if the main ring is not the same? For every $i$ we have modules $L_i$ and $M_i$ over a ring $A_i$, and for every $i \geq j$ homomorphisms ...
3
votes
1answer
88 views

Does tensoring by a flat module preserve pullbacks of pairs of monos?

Let $k$ be a commutative ring and let $C$ be a flat module over $k$. Let $M$ be a module and let $A,B \subseteq M$ be two submodules. We get a pullback diagram: where $s, i, j, t$ are inclusions. ...
0
votes
1answer
42 views

The image of the spec functor under a restriction

What is the image of the restriction of the Spec functor (the functor from commutative rings to affine schemes) to commutative rings with the trivial monoid under multiplication? Thanks very much
4
votes
1answer
199 views

Why are these two functors isomorphic?

Let $A$ be a local noetherian ring, $M$ an $A$-module finitely generated. Let $f$ be an $A$-regular and $M$-regular element (i.e. $f$ is not a zero divisors on $A$ nor on $M$). Then inside the ...
0
votes
2answers
185 views

Coproduct of two modules

Suppose that $M$ is an $A$-module, and $N$ is a $B$-module. The coproduct of $A$ and $B$ is $A\otimes_{\mathbb{Z}}B$, and the coproduct of $M$ and $N$ is $M\oplus N$. I was wondering if $M\oplus N$ ...
7
votes
1answer
468 views

Infinite coproduct of rings

I just learned from Wikipedia that coproduct of two (commutative) rings is given by tensor product over integers, and that coproduct of a family of rings is given by a "construction analogous to the ...
2
votes
1answer
268 views

Definition of a universal example

I'm not sure how the term is being used here: Let $R$ be a commutative ring and $X_1,\ldots, X_n$ indeterminates over $R$. Set $P = R[X_1, \ldots, X_n]$. Given a ring homomorphism $\phi: R ...
1
vote
2answers
405 views

What is a lift?

What exactly is a lift? I wanted to prove that for appropriately chosen topological groups $G$ we can show that the completion of $\widehat{G}$ is isomorphic to the inverse limit ...
7
votes
0answers
144 views

Question about definition of $\mathrm{Ext}$

One can define $\mathrm{Ext}^n(M,N)$ (where $M,N$ are $R$-modules) in two ways, either by taking an injective resolution of $N$ and applying $\mathrm{Hom}(M,-)$or by taking a projective resolution and ...
3
votes
1answer
179 views

What is meant by the Grothendieck group being the “best possible” construction of an abelian group from a commutative monoid?

On the wiki page for Grothendieck group, the first sentence says the Grothendieck group is the "best possible" way to construct an abelian group from a commutative monoid. What actually does this ...
3
votes
1answer
87 views

Fields as a reflective subcategory of integral domains?

A subcategory $\mathbf A$ is reflective subcategory of $\mathbf B$ if for every $B\in\mathbf B$ there exists an $A_B\in\mathbf A$ and a $\mathbf B$-morphism $r_B \colon A \to A_B$ such that: for any ...
20
votes
2answers
603 views

What is the coproduct of fields, when it exists?

This is a slightly more advanced version of another question here. Let $\textbf{CRing}$ be the category of commutative rings with unit. Let $\textbf{Dom}$ be the category of integral domains – by ...
4
votes
1answer
251 views

Koszul Complex Homology

I'm attempting to understand Eisenbud's proof that: If $x_1,x_2,\ldots,x_i$ is an $M$-sequence, then $H^i(M\otimes K(x_1,...,x_n))=((x_1,\ldots,x_i)M:(x_1,\ldots,x_n))/(x_1,\ldots,x_i)M$. Here ...
30
votes
2answers
1k views

What are exact sequences, metaphysically speaking?

Why is it natural or useful to organize objects (of some appropriate category) into exact sequences? Exact sequences are ubiquitous - and I've encountered them enough to know that they can provide a ...