# Questions tagged [ac.commutative-algebra]

Commutative rings, modules, ideals, homological algebra, computational aspects, invariant theory, connections to algebraic geometry and combinatorics.

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### completed tensor products and filtered limit

Let's start with two inverse systems $\{A_p\}_{p \in \mathbb{Z}}$, and $\{B_p \}_{p \in \mathbb{Z}}$ of $C$ modules. Give each $A_p$, $B_p$, and $C$ discrete topology. Consider inverse limit topology ...
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### Spectral theorem for unital $C^{*}$-algebras

Let $A$ be a unital $C^{*}$-algebra and $a \in A$ be normal, with spectrum $\sigma(a)$. Let $B = C^{*}(a)$ be the $C^{*}$-algebra generated by $1$ and $a$, which is abelian. Let $\hat{B}$ be the space ...
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### Construction of a symmetric polynomial in the roots that acts like the discriminant

The discriminant $\Delta(P)$ of a monic polynomial $P(x)=x^n + a_{n-1} x^{n-1} + \dotsb + a_0$ of degree $n$, when expanded (using elementary symmetric polynomials), is a symmetric polynomial of ...
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### Homomorphism group of the additive group of rational numbers $\mathbb{Q}$ into quasi-cyclic group $\mathbb{Z}(p^\infty)$ [duplicate]

I read somewhere that the group of homomorphisms from the additive group of rational numbers $\mathbb{Q}$ into the quasi-cyclic group $\mathbb{Z}(p^\infty)$ is isomorphic to the additive group of ...
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### Are smooth irreducible affine varieties set theoretical complete intersection?

I'm trying to prove a lemma, so I wanted to use the next claim (which I do not know if it is true or if there is a counter example): every smooth irreducible affine variety is set theoretical complete ...
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Is there an injective homomorphism of commutative rings $A \to B$ such that the induced map $\mathbb{P}^1(A) \to \mathbb{P}^1(B)$ is not injective? Here, $\mathbb{P}^1(A) = \mathrm{Hom}(\mathrm{Spec}(... • 58.2k 6 votes 0 answers 196 views ### Is there a relevant universal property for Fitting ideals? Let$S$be a scheme and$\mathscr F$a quasi-coherent sheaf on$S$, locally finitely presented. Set$S_{-1}=S$and for all$n\geqslant 0$, let$S_n$be the closed subscheme of$S$defined by the$n$-... • 1,544 3 votes 1 answer 235 views ### Regular ring is smooth when the field is perfect Take$A$a (not necessarily local) commutative algebra over a field$k$which is essentially of finite type (i.e. a localization of a finitely generated algebra). In simple words, I just want to know ... 2 votes 0 answers 70 views ### How to decompose a matrix over a ring$F[X_1,\ldots,X_k]$as a product of two matrices Let$F$be a field. Assume any reasonable conditions if needed, such as$F=\mathbb R$,$F=\mathbb C$,$F$is a finite field, or$F$has a specific characteristic, etc. Let$C$be an$n\times1$matrix ... • 121 7 votes 2 answers 653 views ### Classfication of vector bundles on projective line over a local ring Let$R$be a local ring. Let$\mathbb{P}^1_R=\rm{Proj}~R[x_0, x_1]$be the Projective line over$R$. Is there a classification of vector bundles of rank$n$on$\mathbb{P}^1_R$in terms of splitting ... • 421 6 votes 0 answers 113 views ### Rings with epimorphism from a finitely generated ring For a commutative ring with unit$R$let's say it has property$(*)$if there is an epimorphism in the category of rings${\mathbb Z}[X_1,\dots,X_n]\to R$, where the former is the polynomial ring in$...
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Let $P[X_1, X_2, \ldots, X_N]$ be a reducible polynomial in $\mathbb{Z}[X_1, X_2, \ldots, X_N]$ such that $P[X_1, X_2, \ldots, X_N] + 1$ is also reducible. What (if anything) can we say about $P$? One ...
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### completion and tensor product

Let $A$ be a commutative ring, consider the map $Spec(A[[t]])\rightarrow Spec(A)$, does it have geometrically connected fibers? If $A$ is noetherian, it is clear because one has for $k$ a residue ...
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### Example of a periodic free resolution over a hypersurface

I'm reading "HOMOLOGICAL ALGEBRA ON A COMPLETE INTERSECTION, WITH AN APPLICATION TO GROUP REPRESENTATIONS" by David Eisenbud I'm wondering what would be a nice example illustrating Theorem 6....
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### Derivation of positively graded domain

We know that if $f\in k[X_1,{...},X_n]$ is quasi-homogeneous polynomial and $R :=k[X_1,{...},X_n]/(f)$, then any minimal generating set of $\operatorname{Der}_k(R)$ contains the Euler derivation. Is ...
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### Cyclic homology can be recovered from topological cyclic homology?

Let $R$ be a commutative ring, $S$ a $R$-algebra of finite type. By an equivalence of ring spectra $$\operatorname{HH}(S/R) \simeq \operatorname{THH}(S)\otimes_{\operatorname{THH}(R)} HR,$$ ...
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### A profinite group such that any deformation has finite image

Let $p$ be a fixed prime and $n\geq 2$ a fixed integer. Suppose that $G$ is a profinite group such that any continuous homomorphism $\rho:G\to {\rm GL}_{n}(F)$ has finite image for all non-...
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### Testing whether a finite dimensional algebra is a complete intersection with QPA

Let $A$ be a local commutative finite dimensional $K$-algebra for a field $K$. We can thus assume that $A$ is given by quiver and admissible relations $KQ/I$. It is easy to check when $A$ is ...
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### Are algebras of smooth functions formally smooth?

Let $M$ be a manifold. Then is the ring of smooth functions $C^\infty(M,\mathbb{R})$ formally smooth over $\mathbb{R}$? If it helps, feel free to assume that $M$ is compact. (This is not a joke ...
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### On $\text{depth}_S\left(\dfrac {S}{JS+xS}\right)$, when $\text{depth}_R(R/J)=0$, and $R\to S$ is a certain flat map of local rings

Let $(R, \mathfrak m) \xrightarrow{\phi} (S,\mathfrak n)$ be a flat homomorphism of local rings such that $\mathfrak n=\mathfrak m S +xS$ for some $x\in \mathfrak n \setminus \mathfrak n^2$. Let $J$ ...
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