All Questions
Tagged with ac.commutative-algebra ag.algebraic-geometry
2,098 questions
1
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82
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Relative 1 form of Frobenius morphism
Let $X$ be a connected smooth algebraic variety over a field $k$ of characteristic $p > 0$. We can consider the Frobenius morphism associated to $X \mapsto Spec(k)$. I'd like to show $\Omega^{1}_{X/...
- 73
1
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0
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125
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Confusion regarding change of variable and irreducibility
Let $\mathbb{K}$ be an algebraically closed field of characteristics zero. Let $X$ be an irreducible affine variety, with a rational action of a linearly reductive algebraic group $G$. Also, assume ...
- 839
6
votes
1
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1k
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Discovery of Hilbert polynomial
Presumably it was Hilbert who discovered Hilbert polynomials - where did they first appear?
The basic theorem is that for a finitely generated graded module $M = \bigoplus_k M_k$ over the ring of ...
- 5,339
1
vote
1
answer
96
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On "minimal presentation" of local rings essentially of finite type over a field
Let $k$ be a field of characteristic $0$. Let $(R,\mathfrak m)$ be a local ring essentially of finite type over $k$ (https://stacks.math.columbia.edu/tag/07DR). Then, $R$ is the homomorphic image of ...
- 361
3
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2
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395
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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 ...
- 839
10
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1
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599
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Isbell Duality and Dualizing Scheme Objects
I'm not sure if this question is too elementary for MO; nevertheless, I have seen many helpful discussions surrounding this topic here. I'm interested in studying the adjunction $\operatorname{Spec}\...
- 223
2
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0
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112
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Understanding normalization algorithms
Let $R$ be a commutative and reduced ring, finitely presented over $\mathbb Z$. Let $\overline R$ be the integral closure of $R$ in its total ring of fractions. In https://arxiv.org/abs/alg-geom/...
- 211
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169
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The dimension of the representation ring
Let $G$ be a compact Lie group. I am trying to characterize the algebraic properties of the representation ring $R(G)$ of $G$. In the case of the $n$-torus, the representation ring $R(T)$ is ...
10
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1
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851
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Is it a valuation ring?
It is known that a one-dimensional Noetherian local ring is a discrete valuation ring if it is integrally closed.
Then, even if it is not Noetherian, would a one-dimensional local ring become a ...
- 328
4
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1
answer
280
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Existence of module with periodic resolution
Let $R$ be a Gorenstein local ring. Does there always exist a module $M$ having eventually periodic minimal free resolution?
Any reference is also appreciated.
- 81
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1
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333
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Alterations and smooth complete intersections
Let $k$ be an algebraically closed field, and $X$ a projective variety over $k$. Let $i : X\subset \mathbf{P}^d_k$ be a closed immersion into a projective space of high enough dimension.
Is there a ...
user501361
2
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1
answer
191
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Cohen-Macaulay fiber products
Let $R$ be a regular local ring, $X$ and $Y$ smooth $R$-schemes, $T\to Y$ a regular closed immersion over $R$ with $T$ smooth over $R$, and $f: X\to Y$ an $R$-morphism.
Is the fiber product scheme $...
user501361
2
votes
1
answer
250
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Images of smooth schemes under lci morphisms
Let $S$ be a Noetherian scheme, $f : X\to S$ a smooth quasi-projective morphism, $g : X\to Y$ a morphism of finite type, and $h : Y\to S$ a smooth projective morphism with $h\circ g =f$.
Can we say ...
user505967
3
votes
1
answer
227
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Van der Waerden's classical paper on Hilbert polynomials and Bezout's theorem
Is it possible to obtain an electronic copy of Van der Waerden's "On Hilbert series of composition of ideals and generalisation of the Theorem of Bezout", Proceedings of the Koninklijke ...
- 5,339
5
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0
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181
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The precise relationship between (moduli space of) finite-dimensional commutative local $\kappa$-algebras and number theory?
In [BP08] Poonen constructs and studies $\mathfrak{B}_n$, the moduli space of all based $n$-dimensional commutative associative unital $\kappa$-algebras, where $\kappa$ is an algebraically closed ...
- 7,127
2
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1
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326
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Krull dimension of the smooth locus
Let $R$ be a normal complete local domain of dimension $n \geq 4$. Does there exist a prime ideal $\mathfrak{p}$ of height $\dim(R) - 1$ such that $R_{\mathfrak{p}}$ is a regular local ring? In ...
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1
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329
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Finite subschemes of projective bundles
$\DeclareMathOperator\Spec{Spec}\DeclareMathOperator\Proj{\mathbf{Proj}}$Let $S$ be a Noetherian scheme and $X$ finite $S$-scheme. The finite morphism $X \to S$ is projective in the sense of the ...
user505967
2
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0
answers
188
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Does going down property imply a corresponding map is open without "finiteness"?
Does the following proposition hold?
Proposition
Let f:A$\rightarrow$B be a ring homomorphism
If f has going down property then the corresponding map
$f^*$:Spec B$\rightarrow$Spec A is open map.
I ...
- 328
2
votes
1
answer
340
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flatness and exact sequences
Let $R$ be a commutative ring (with unit). Then if
$$0\longrightarrow M'\longrightarrow M\longrightarrow M''\longrightarrow 0$$
is an exact sequence of $R$-modules, with $M''$ $R$-flat, $M$ is flat if ...
- 739
2
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2
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369
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Can a non-zero non-prime ideal become prime in a smaller ring?
All rings are assumed commutative and unital.
Some context (feel free to skip right to the questions below). I am trying to understand to what extent the property "being prime" for an ideal $...
- 7,127
3
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1
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530
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Is the spectrum of this ring Noetherian?
Let $R = \mathbb{Z}/2\mathbb{Z}[X,X^{1/2},X^{1/4},\cdots,X^{1/{2^n}},\cdots]$
Is $\operatorname{Spec}R$ a Noetherian topological space?
Here is what I know.
$R$ is integral over $\mathbb{Z}/2\mathbb{...
- 328
4
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1
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287
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The $1$-dimensional Jacobian Conjecture over $\mathbb{Z}$-torsion free rings
I am looking for further proofs, preferably in the literature, of the following result:
Proposition: Let $R$ be a unital commutative $\mathbb{Z}$-torsion free ring. If $f(x) \in R[x]$ with $f'(x) \in ...
- 7,127
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0
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215
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On linear schemes
Let $X$ be a smooth projective curve and $T$ is any smooth variety. Let $E$ be a family of vector bundles over $X\times T$ which is flat over $T$. Then there exists a scheme $Y$ over $T$ such that for ...
- 494
4
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1
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327
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Example of a certain type of Cohen-Macaulay ring
Let $k$ be a perfect field. I am looking for a $k$-algebra $R$ with the following properties.
$R$ is of finite type over $k$ and is a domain;
for all ${\mathfrak p}\in{\rm Spec}(R)$, the local ring $...
- 3,076
2
votes
1
answer
302
views
Normal forms of ADE singularities
Given a surface $X:f(x,y,z)=0\subset \mathbb{A}^{3}_{\mathbb{C}}$ with only ADE singularities, how does one determine the correct singularity type of $X$ by computing the normal forms?
Does a similar ...
- 481
2
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1
answer
290
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Flat scheme-theoretic closure
Suppose $R$ is a discrete valuation ring with fraction field $K$. Let $X\subset \mathbf{P}^n_{C_K}$ be a closed subscheme, flat over $C_K$, a smooth projective curve over $K$.
Let $C_R$ be a flat ...
user501361
0
votes
0
answers
91
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Comparison of depth of two monomial ideals
Let $I$ and $J$ be two monomial ideals of $R=K[x_1,\dotsc,x_n]$ where $K$ is a field.
Could we say $\mid\operatorname{depth} (R/I)- \operatorname{depth}(R/(I:J))\mid\leq 1$, where $(I:J)$ is colon ...
- 113
3
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375
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On the analogy between $p$-derivations and derivations
$\DeclareMathOperator\Spec{Spec}$Let $p$ be a prime number, and $A$ a commutative ring. Recall that a $p$-derivation on $A$, or a $\delta$-ring structure on $A$ is a set map $\delta : A \to A$ such ...
- 63.9k
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1
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147
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Are maps into a smooth curve equivalent to relative effective Cartier divisors?
Let $X$ be a smooth curve and $S$ an affine scheme, both over an algebraically closed field $k$.
Given a morphism $f : S \to X$, is it true that the graph morphism $\Gamma_f : S \to X \times S$ is a ...
- 426
2
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2
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261
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Examples of stretched artinian local ring
In Sally's Paper stretched artinian local ring is defined as :
Let $(R, \mathfrak{m})$ be an Artin local ring of length $\lambda.$ If $\nu$ is the embedding dimension of $R$, that is, $\nu$ is the ...
- 81
7
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2
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784
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Is there a Hopf algebra-style description of chain complexes?
An old chestnut is that filtered objects are the same as sheaves over $\mathbb A^1 / \mathbb G_m$.
Question: Is there a similar description of chain complexes?
More precisely, if $\mathcal C$ is a ...
- 63.9k
2
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0
answers
104
views
Kouchnirenko's theorem for non-generic polynomials
In Polyèdres de Newton et nombres de Milnor (Theorem 1.18), Kouchnirenko proved that given Laurent polynomials $f_1, \dotsc, f_k$ in $k$ variables, the number of isolated solutions is less than or ...
- 315
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116
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Intersection numbers of moduli spaces and noncrossing partitions
The coefficients of the monomials $u_1^{e_1}u_2^{e_2} \ldots u_n^{e_n}$ of the partition polynomials (ParPs) $[M=M1]$ on pg. 831 of The Handbook of Mathematical Functions by Abramowitz and Stegun are ...
- 10.5k
2
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0
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195
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Interpretation of completed tensor product of algebras over lower base
Let $\mathbb F$ be a finite field of order $q = p^n$. It is known that
$$\mathbb F[[x_1]] \mathbin{\widehat{\otimes}_{\mathbb F}} \mathbb F[[x_2]] = \mathbb F[[x_1, x_2]].$$ Geometrically, this is the ...
- 383
5
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1
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479
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Alternative description of strict henselization
Let $R$ be a local ring with field of fractions $K$, maximal ideal $\mathfrak{m}$ and residue field $\kappa = R/\mathfrak{m}$. Let $R^\mathrm{sh}$ be a strict henselization of $R$, and let $L$ be the ...
- 1,329
4
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1
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295
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Finite type injective ring map between domains preserves the open point $(0)$
I am looking for a proof of the following statement without using full power of Chevalley's theorem on constructible sets. We say a domain $A$ is $0$-open if $\{(0)\}$ is open in $\operatorname{Spec}(...
- 143
4
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1
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181
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Effective bound on "Jacobian rank" for (regular) planar algebraic curves
Let an irreducible, square-free complex polynomial $f\in \mathbb C[x,y]$ be given. It is well known that the curve $\mathcal C:=\{f=0\}$ is nonsingular if and only if $\mathbb C[x,y]=<f,\partial_xf,...
- 5,428
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1
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165
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Sufficient conditions to guarantee finite intersection points in Bezout's Theorem
Bezout's Theorem concludes that if $f_1,f_2,\cdots,f_n\in k[x_1,x_2,\cdots,x_n]$ have finite intersection points, then they have at most $d_1d_2\cdots d_n$ intersection points, where $d_i$ is the ...
- 25
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2
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117
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When do there exist $ n $ commuting, derivations which locally generate $ T_{A/k} $ as an $ A $-module?
Let $ \operatorname{Spec}(A) $ be a non-singular, $ n $-dimensional, affine variety over a field $ k $ of arbitrary characteristic. For the definition of "variety" I use Hartshorne's ...
- 912
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0
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61
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A sequence of polynomials that the variety defined by every $n$ of them is small
Let $\mathbb{C}[x_1,x_2,\cdots,x_n]_{= d}$ denote the set of polynomials in $\mathbb{C}[x_1,x_2,\cdots,x_n]$ of total degree $d$.
Is there exists a sequence of polynomials $f_1,f_2,f_3,\cdots$ in $\...
- 25
1
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0
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154
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Software for computing invariant rings
I have an linearly reductive algebraic group $G$ acting regularly on an affine variety $X$(over an algebraically closed field of characteristic 0). I want to compute the invariant ring $\mathbb{K}[X]^{...
- 839
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119
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Germs of holomorphic functions and invariant functions
Consider a complex vector space $V \cong {\mathbb C}^n$. Consider the ring of germs of holomorphic functions ${\mathcal O}_0 (V)$ at $0\in V$. We know that this ring is Noetherian.
Now consider a ...
- 803
5
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2
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555
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Another characterization of tensor products of modules
It is known that the tensor product is characterized by its universality in the category of $A$-modules.
Does the following proposition hold?
Proposition There exists only one operation $\otimes$ ...
- 328
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0
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226
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Geometric interpretation of normalization inside a finite extension of function field
$\DeclareMathOperator\Spec{Spec}$Suppose $X = \Spec A$ is a smooth affine variety over $\mathbb C$ and suppose $L/K$ is a finite extension of its function field. Let $Y = \Spec B$, where $B$ is the ...
- 1,761
3
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0
answers
249
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Grothendieck schemes and the Sheffer differential op calculus (Rota, Roman, et al. finite operator calculus)
In "Left differential operators on non-commutative algebras" on p. 4, Michiel Hazewinkel displays "precisely the right definition of differential operator" as
$$D\; X^n = F(\tfrac{...
- 10.5k
3
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0
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264
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The points of $\operatorname{Spa}\mathbb{Z}_p$
$\DeclareMathOperator\Spa{Spa}$What are the points of $\Spa\mathbb{Z}_p$? I read in Scholze-Weinstein that this adic spectrum consists of 2 points, a special point, which corresponds to the pullback ...
- 2,907
5
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0
answers
160
views
Cohen-Macaulayness of rings of polynomials vanishing at points
Let $V$ be a finite dimensional vector space, let $L_1$, $L_2$, ..., $L_r$ be subspaces and let $w_1$, $w_2$, ..., $w_r$ be positive rational numbers. Define a graded ring $R$ where $R_d$ is those ...
- 156k
3
votes
0
answers
280
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The closed unit adic disk
I am reading the Scholze-Weinstein Berkeley lecture notes on "Perfectoid Spaces", and in particular I am stuck trying to understand the closed adic unit disk, which is the second example of ...
- 2,907
2
votes
0
answers
264
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What is the residue field of the integer ring of $\mathbb{C}_p$?
Fix a prime $p$. Let $\mathbb{C}_p$ be the completion of the algebraic closure of $\mathbb{Q}_p$, and $\mathcal{O}_{\mathbb{C}_p}$ the integer ring of $\mathbb{C}_p$. I know $\mathcal{O}_{\mathbb{C}_p}...
- 519
0
votes
0
answers
53
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When a given set of primes of height 1 is a set associated primes of an element
Let $R$ be a Noetherian local ring of dimension $\geq 3$ and $\{p_1,\ldots , p_n\}$ be a collection of prime ideals of height $1$. Does there exist an element $x\in R$ such that $Ass(R/xR)=\{p_1,\...
- 1,713