Questions tagged [ac.commutative-algebra]
Commutative rings, modules, ideals, homological algebra, computational aspects, invariant theory, connections to algebraic geometry and combinatorics.
5,335
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Tensor product of field extensions
Let $K$ be a field of characteristic 0 and $L$ a finite extension of $K$. Denote by $m$ the natural multiplication map from $L \otimes_K L$ to $L$. Denote by $I$ the kernel of the morphism $m$. Is $I$ ...
- 1,949
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2
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536
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How a "sequentially Cohen–Macaulay" simplicial complex relates to "Cohen–Macaulay" simplicial complex?
Let $\Delta$ be a simplicial complex on $[n]$ of dimension $d − 1.$ Let $0\le i\le d-1.$ One defines the pure i_th skeleton of $Δ$ to be the pure
subcomplex $\Delta(i)$ of $\Delta$ whose facets are ...
- 1,355
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2
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474
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Application of sheaves theory in ring theory
Is there any text that gives some applications of sheaves theory in commutative ring theory? In the other word, is any results in commutative ring theory that be verified by sheaves method?
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Counterexample to Openness of Flat Locus
Let $A$ be a commutative Noetherian ring and $B$ a finitely generated $A$-algebra. Then the set $$U\colon=\{P\in\operatorname{Spec}B\mid B_P\ \mathrm{is\ flat\ over}\ A\}$$ is open in $\operatorname{...
- 3,101
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Can a zerodivisor reduce both the depth and the dimension?
In this question $R$ is a commutative noetherian local ring with unity.
One can construct examples of rings $R$ and zerodivisors $z$ such that $\dim R/(z)=\dim R-1$, e.g., $S\colon=k[a,b,c],\ \...
- 3,101
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votes
2
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515
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System of generators of a homogenous ideal
Let $I$ be a homogenous ideal in the ring $k[x_{1},\dots,x_{n}]$. My question is:
If $\lbrace f_{1},\dots,f_{r}\rbrace$ is a minimal system of generators of $I$, then are the integers $r$ and $\...
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477
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Noetherian descent extension for a given ring
For a homomorphism of rings $R \to S$, the following are equivalent:
a) $(-) \otimes_R S : \mathrm{Mod}(R) \to \mathrm{Mod}(S)$ reflects isomorphisms
b) $R \to S$ satisfies effective descent with ...
- 61.5k
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Does a regular pair of elements in a noetherian domain remain regular if their order is switched?
Recall that in a commutative ring $A$ an ordered pair of elements (a,b) is said to form a regular sequence if the ideal $\langle a,b\rangle $ is strictly included in $A$ ,if $a$ is not a zero-divisor ...
- 46.8k
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Normality and rational singularities via Hilbert series
Let $A$ be a finitely generated ${\mathbb Z}_{\geq 0}$-graded algebra over a field without zero divisors;
assume that all graded components are finite-dimensional and that $Spec(A)$ is smooth
outside ...
- 6,957
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How exotic can DVRs be in the ring of rational functions over a local field?
Suppose that $R$ is a complete DVR with field of fractions $K$, uniformiser $\pi$ and residue field $k$.
Let $B$ be a subring of the ring $K(t)$ of rational functions over $K$. Moreover assume that $...
- 3,475
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267
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When is a blow-up a non-trivial product?
Suppose $X$ is an algebraic variety and let $Z \subset X$ be a subvariety. Are there some useful criteria under which the blow-up $Bl_Z X$ becomes a nontrivial product $V \times W$ of the algebraic ...
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2
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363
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How to compute the ring of invariants of SO_3(k) acting on a polynomial ring
Let $k$ be a field and let $A$ be the polynomial ring over $k$ in $3n$ variables: $A = k[X_{ij} \vert i=1,2,3 \quad j=1,2,\cdots,n]$.
${\rm SO}_3(k)$ acts on $A$ in the following way: Given $g \in {\...
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Finitely-generated algebra over Z
Let A be an artin ring which is also a finitely generated algebra over Z.
Show that $|A|<\infty$.
If A would have been a field then I know how to prove it. I know that A is a product of local ...
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Non-existence ofintegral basis of integral closure in a finite extension of Frac(A), A Dedekind.
Let $A$ be a Dedekind domain, $K:=\text{Frac}(A)$ and $L/K$ finite so that the integral closure $B$ of $A$ in $L$ is Dedekind. If $A$ is a PID, for example, then there exists an integral basis : $B$ ...
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Subalgebras of quadratic algebras that are not quadratic
Suppose $A=k\oplus A_1 \oplus A_2\oplus \cdots$ is a quadratic algebra over a field $k$. Let $B$ be the subalgebra generated by a subspace $V\subseteq A_1$. What are the examples of such subalgebras $...
3
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4
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$R$ is a UFD iff $R_{\frak{m}}$ is a UFD?
Let $R$ be a graded ring such that $R_0$ is a field and let $\frak{m}$ be the maximal ideal generated by all the elements of positive degree. Then, is it true that $R$ is a UFD iff $R_{\frak{m}}$ is a ...
- 767
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482
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Quotients of Gorenstein rings
Let $R$ be a reduced Noetherian ring. Assume $R$ is quasi-excellent and Cohen-Macaulay.
Is $R$ the quotient of a Gorenstein ring?
If the answer is yes, then $R$ has a dualizing complex. The question ...
user322153
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answer
183
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Solutions to nonhomogeneous quadratic equation mod $N$
Is there any way to find non-trivial solutions to the equation $x^2 + y^2 - x \equiv 0 \mod{N}$? There are clearly several trivial solutions, for example $(x, y) = (0, 0), (1, 0), (2^{-1}, 2^{-1}), (2^...
- 1,693
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Local ring of infinite dimension
Short version: Let $R$ be a commutative ring such that all chains of primes of $R$ with the same extremities have the same finite cardinality. Is $R$ locally finite-dimensional?
Longer version: Let $R$...
- 6,670
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When every ideal containing $J(R)$ is an intersection of maximal ideals
Let $R$ be a commutative ring with $1$ such that every ideal containing $J(R)$, the intersection of all maximal ideals, is an intersection of maximal ideals. Is there any characterization for such a ...
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When is an almost geometric quotient flat?
All varieties here are over $\Bbb C$. Let $G$ be a reductive algebraic group acting algebraically on affine $n$-space $\Bbb A^n$. Let $R$ be the coordinate ring of $\Bbb A^n$. Assume that the natural ...
- 3,031
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Irreducibility of family of polynomials
Consider the following family of polynomials over $\mathbb{Q}$:
$$f_n = x^n - x^{n-1} - \dots - 1$$
Notice that these polynomials satisfy the recurrence
$$ f_{n+1} = x f_n - 1 $$
I would like to ...
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1
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441
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Milnor patching for general modules
The Milnor patching theorem for projective modules is the following statement. Given a pullback diagram of rings
$$
\begin{array}{}
R & \xrightarrow{f_2} & R_2 \\
\downarrow{f_1} & &...
- 197
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1
answer
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Completion of a finite field extension is also finite?
Let $(L,w)/(K,v)$ be a finite extension of valuation fields, and let $L_w$, $K_v$ be the respective completions of $(L,w)$, $(K,v)$. Is the field extension $L_w/K_v$ finite?
For nonarchimedean ...
- 5,129
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822
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A strengthened version of Noether's normalisation lemma?
Noether's normalisation lemma says that if $R$ is an integral domain, finitely generated over a field $k$, with transcendence degree $n$ over $k$, then there exist elements $x_{1}, x_{2}, \ldots x_{n} ...
- 2,005
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Can non-isomorphic field extensions be isomorphic fields?
This is related to my earlier question on isomorphism of general quotients of $\:F\hspace{.02 in}[x]\:$.
Let $F$ be a field, let $p$ and $q$ be (non-zero) monic irreducible polynomials, let $I$ and $...
user5810
3
votes
3
answers
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Dimension of a ring after localization
Let $R$ be a Noetherian domain of dimension $\ge 1$. Let $\mathfrak{p}_i$, $i = 1, 2, ...$ be prime ideals of height one. Let $T = R[[X]]$ with $X$ is a indeterminate. For each $i \ge 1$ we set $\...
- 2,773
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471
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Does $\mathbb P^1 \times \mathbb P^1$ admit an Ulrich bundle?
In an answer to a MathOverflow question on the following link
Vector bundles on $\mathbb{P}^1\times\mathbb{P}^1$, it is mentioned that $\mathbb P^1 \times \mathbb P^1$ has an Ulrich sheaf. However, ...
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501
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Generic methods to check irreducibility of polynomials in $K[[X,Y]]$
I usually find it difficult to check irreducibility of polynomials in $K[[X,Y]]$ ($K$ algebraically closed). Does anyone know about generic methods that can be used ? And especially of ones that can ...
- 1,136
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2
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431
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Cohen-Macaulay and descent
Let $X$ be a Cohen-Macaulay scheme and $f:X\rightarrow Y$ a morphism. Under which "non trivial conditions" on $f$ can we conclude that $Y$ is also Cohen-Macaulay? By "trivial conditions" I mean smooth+...
- 31
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Term for an "almost regular" sequence
Let $R$ be a ring (commutative, with unit), $M$ an $R$-module, and $x_1, \dotsc, x_n \in R$. Consider the following two conditions:
For all $i$, the homomorphism $$\frac{M}{(x_1, \dotsc, x_{i-1})M}...
- 7,218
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2
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763
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Rees algebra for non-radical ideals
Today in my introductory algebraic geometry class we defined the so-called Rees algebra associated with an ideal $I$ of a ring $R$ (with strong conditions on $R$, if you like: I don't mind restricting ...
- 3,525
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1
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171
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If $B \subset C \subset B_g$, is $\mathrm{Spec} C \to \mathrm{Spec} B$ necessarily an open immersion?
Let $B \subset C$ be Noetherian integral domains, and $g \in B$. Thus, $\mathrm{Spec} B \to \mathrm{Spec} B_g$ is an open immersion.
If furthermore $C \subset B_g$, does it follow that $\mathrm{Spec}...
- 7,218
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votes
1
answer
130
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A question about minimal primes in the integral group ring of a finite abelian group
Let $G$ be a finite abelian group of order $n$ and let $R=\mathbb{Z}G$ denote the integral group ring of $G$. Let $R'$ denote the localization of $R$ with respect to the multiplicatively closed set ...
- 31
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A question about freeness of a certain class of abelian groups
Lets call an abelian group $G$, to be semi-free (or SF) if every nonzero subgroup of $G$ is isomorphic to $\mathbb{Z}\times H$ for some abelian group $H$.
Is every semi-free group, a free group? If ...
- 4,454
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377
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Cohen-Macaulay Representations
I came across the book "Cohen-Macaulay Representations" by Graham J. Leuschke and Roger Wiegand, and now I'm wondering if this is an active area of research.
If yes, then
what are some of ...
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264
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Is this functor $\mathcal{F}: \text{Sch}/\mathbb{Q}\longrightarrow \text{Sets}$ a sheaf?
Consider the functor $\mathcal{F}: \text{Sch}/\mathbb{Q}\longrightarrow \text{Sets}$, defined by sending a scheme $X$ with coordinate ring $\mathcal{O}(X)$ to the set of orbits $B(\mathcal{O}(X))\...
- 1,641
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377
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What is the name for algebras generated by elements, all of whose cubes vanish?
Given a ring $R$ with identity $1$, we can define the exterior algebra of order $k$ over $R$ to be the algebra over $R$, generated by elements $x_1, \dots, x_k$ satisfying $x_i^2 = 0$ for each index $...
- 455
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What is a PBW algebra? (I.e., an algebra generalising properties of $U(\frak{g})$)
I am reading a paper where they refer to a certain algebra as a PBW algebra. What does this mean exactly? I would infer from the $U(\frak{g})$ setting that this means the existence of an ordered ...
3
votes
1
answer
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Characterized maximal ideal [closed]
$\DeclareMathOperator\Alg{Alg}$Let $A$ be a commutative associative algebra with $1$ over $\mathbb{C}$. We define $\Alg(A,\mathbb{C}) $ to be the set of $\mathbb{C}$-algebra maps from $A$ to $\mathbb{...
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K-projectivity for rings of finite homological dimension
Let $R$ be a Noetherian commutative ring. A complex of $R$-modules $P^{\bullet}$ is K-projective if for any acyclic complex $A^{\bullet}$, the complex of abelian groups $ Hom(P^{\bullet}, A^{\bullet})$...
- 449
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votes
1
answer
184
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Non-abelian isomorphic absolute Galois groups of fields of different characteristic
Let $K$ be a field of positive characteristic and $L$ be a field of characteristic zero.
Assume the absolute Galois groups of $K$ and $L$ are non-abelian and isomorphic as profinite groups.
Can $L$ ...
- 55
3
votes
2
answers
330
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Isomorphism between finite algebras over ${\Bbb Z}_p$
Let $\pi \colon R \twoheadrightarrow {\Bbb T}$ be a surjective ring homomorphism between finite algebras over ${\Bbb Z}_p$. Further, we suppose the following three conditions$\colon$
$R$ is a ...
- 563
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1
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190
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Finitely generated sheaf of algebras over geometric points
I would like to ask if the following is true or not: Let $S$ a scheme and $X$ a $S$-scheme which is proper and flat. Let $\mathcal{F}$ a sheaf of $\mathcal{O}_{X}$-algebras over $X$. Let's suppose ...
- 439
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votes
1
answer
146
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A field having higher Witt vectors as completions
Given a finite field $F$ there is a unique (up to isomorphism) absolutely unramified complete DVR $W(F)$ of mixed characteristic that has $F$ as its residue field.
Fix a positive integer $n$. Does ...
- 31
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votes
1
answer
406
views
Are quotient varieties local complete intersections?
Let $G$ be a reductive group acting on the smooth affine variety $X$ such that the stabilizers are finite. Is it true that the quotient $X/G$ is a local complete intersection (LCI)? In particular, is ...
- 2,777
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1
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571
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question about infinite global dimension
I ask a question about $\prod k$ in Mathematics about several days.https://math.stackexchange.com/q/2766054/453628.
And I have the following question:
1.What is the global dimension about $\prod k$?
...
- 496
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1
answer
780
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Glue morphism of schemes defined over irreducible components
Let $k$ be an algebraically closed field and $X, Y$ be quasi-projective $k$-scheme.
Let $X_1, X_2$ be two irreducible components of $X$ and $f_i:X_i \to Y$ be morphims such that $f_1|_{X_1 \cap X_2}=...
- 1,949
3
votes
1
answer
382
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Stone topological Boolean algebras
I am looking for an initial reference for a theorem which is known, namely:
Theorem: A Boolean algebra $A$ admits a Stone space topology (i.e. is the underlying algebra of a Stone topological ...
- 774
3
votes
1
answer
763
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Reduced scheme structure on locally complete intersection
This question concerns reduced scheme structure on locally complete intersection, and I guess the answer is related to the number of generators of a radical ideal.
I am confused about the following ...
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