# Tagged Questions

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

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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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, ...
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### 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 ...
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### 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). ...
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### 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 ...
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### 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 ...
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### 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 ...
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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 ...
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### 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} ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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?
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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. ...
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### 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
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### 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 ...
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### 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 $M\oplus N$ ...
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### 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 ...