Questions tagged [ac.commutative-algebra]
Commutative rings, modules, ideals, homological algebra, computational aspects, invariant theory, connections to algebraic geometry and combinatorics.
5,496 questions
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Regularity of special monomial ideals
Let $R = k[x_1\ldots x_n]$, and $a,b$ are vectors with integer entries, whose all entries of $a$ are non-negative, and say sum of coordinates of $b$ is $0$. Let $I$ be a monomial ideal generated by $x^...
- 65
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1
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256
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Heisenberg-type groups over rings with involution
Hello everyone!
In a paper "Coverings of twisted Chevalley groups over commutative rings" Eiichi Abe intoroduced the following construction:
Let $R$ be a commutative ring and $x\mapsto\overline{x}$ ...
- 3,186
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0
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54
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Elimination theory for variables packaged in a matrix
I am wondering if the elimination theory in computational algebraic geometry can be more efficiently carried out if all variables lies within some given matrices.
For instance, consider the following:
...
- 367
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1
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110
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Variety of commutative semi group [closed]
V is a variety of commutative semi group satisfying the identity $x^2 = x^3$.
I need to prove that:
$|F_V(\{x_1\dots,x_n\})|$ = $3^n -1$.
Any hints on this ?
$F_V$ is V-free algebra.
- 155
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0
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211
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Relation between Castelnuovo-Mumford regularity for coherent sheaves and modules
Let $S$ be the ring $\mathbb{C}[X_0,...,X_n]$. Let $X$ be a smooth projective scheme of the form $\mathrm{Proj}(S/I_X)$ for some ideal $I_X$. Let $C$ be a scheme associated to a Cartier divisor on $X$....
- 1,593
3
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1
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201
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How to estimate a local hilbert samuel funcion
Let $X$ be a reduced hypersurface in the projective variety $\mathbb{P}^n(K)$, where $K$ is a number field. Select $\xi$ is a $F_{\mathfrak{p}}$-rational point of $X$ where $\mathfrak{p}$ is a prime ...
- 403
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166
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Can the zero-degree part of $M_f \otimes_{S_f} N_f$ be identified with $M_{(f)} \otimes_{S_{(f)}} N_{(f)}$?
The isomorphism ${(M \otimes _ {S} N)} _ {f} = M _ {f} \otimes _ {S _ {f}} N _ {f}$ is well-known. Here, $S$ is a graded ring, and $M,N$ are graded $S$ modules.
Now, let $f$ be any homogeneous ...
- 945
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315
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Definitions for Oddness
In the thread Even Xor Odd Infinities I defined odd models of Modular Arithmetic (MA) as models satisfying the axioms of MA and two first order statements. Even XOR Odd Infinities?
$O1) \forall x(x=0 ...
- 687
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666
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Structure theorem for Finitely Generated modules over PID's using localization
I have been trying to work out a proof of structure theorem for finitely generated modules over PID's using localization. This is what my plan is:
1.Prove that every finitely generated torsion free ...
- 11
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188
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fixed point scheme in caracteristic p
Let X\rightarrow A^{n} a smooth affine scheme over an affine space. Everything is defined over a field k.
Let G a finite group acting on X and suppose that his order is divisible by the caracteristic ...
- 3,472
4
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410
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Finding purely transcendental parts of field extensions
If we have a field $K$ such that $K\cong K(t)$ (i.e. it is isomorphic to the field you get if you adjoin one transcendental) then is there necessarily a subfield $L\lt K$ such that $L\ncong L(t)$ and $...
- 83
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181
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unfolding as resolution
Has anyone described 'unfolding' as used in mathematical physics (e.g. on-shell AND off-shell) as analogous to a resolution in algebra - higher derivatives are unfolded in terms of new variables?
- 3,880
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109
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Vanishing of the module of differentials of a extension of perfect fields
Let $L|F$ be a extension of perfect fields of characteristic $p$, $\phi_F:F \to F_{\phi}$, $\phi_L:L \to L_{\phi}$ the Frobenius isomorphisms ($F_{\phi}=F$ but considered as $F$-algebra via $\phi_F$). ...
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323
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Filtrant (not necessarily totally ordered) projective system commuting with direct sums
Hello,
Let $R$ be a commutative (not necessarily Noetherian) ring.
Let $I$ be a small filtrant (not necessarily totally ordered) category.
Let $(M_i)_{i\in I}$ be a projective system of $R$-modules ...
- 41
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514
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E(n) Deformations of the infinity category Qcoh(X) with it's E(n)-tensor product
Let $X$ be a smooth scheme, then an infinity enchancement of $QCoh(X)$ has an $E_\infty$ structure and in particular an $E_n$ structure for any $n$. In this paper, http://arxiv.org/abs/0805.0157 Ben-...
- 3,758
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348
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Modules with connection over $p$-adic laurent series rings
If $X$ is a smooth rigid analytic space over a $p$-adic field $K$ (of characteristic zero), then every coherent $\mathcal{O}_X$-module with integrable connection is locally free. In his paper "...
- 1,838
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181
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A question on binary polynomials
This is probably a well-known result but I was not able to find a reference on my search. My question concerns general polynomials $f(x,y) \in \mathbb{Z}[x,y]$ such that $f$ cannot be written as a ...
- 26.9k
2
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1
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Iterated Reduced Tensor Power of Graded Vector spaces
This might be inappropriate for the MO-level. If so I'll delete it...
Suppose $V$ is a $\mathbb{Z}$-graded vector space and
$\overline{T}(V):=V \oplus V\otimes V \oplus \otimes^3 V \ldots$
is the '...
- 624
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1
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356
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Constructive Bezout cofactors in the ring of algebraic integers
We see the following in an answer to This Question : (mangled by me for my purposes, all errors my fault)
Dedekind stated (in 1871) that the ring of algebraic integers is a Bezout Domain. He calls it ...
- 30.1k
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658
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Invertible elements in generalized fields
Durov's theory of generalized rings also includes generalized fields (5.7.6), which are defined as generalized rings, which are not subtrivial and whose proper strict quotients are subtrivial. For ...
- 63.1k
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354
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"un-nil-ifying" ideals via deformation
This is perhaps a naïve question; given an irreducible scheme $X$, is there a general procedure to find a flat family $Y \to T$ such that over some point $t_0$ we have $Y \times_T {t_0} \cong X$ but ...
- 467
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1
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212
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Liftability of a submodule from an associated graded module
Let $k$ be a field, $A$ a $k$-algebra (probably noncommutative), and $M$ an $A$-module that's finite-dimensional as a vector space over $k$.
Let $Gr(M;k)$ denote the set of all $k$-subspaces of $M$, ...
- 27.9k
1
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1
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515
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Cohen Macaulay, free and finitely generated module
Here is an unsolved problem for me in Kaplansky's "Commutative rings" p. 103, no. 18.
Let $R$ be a Cohen-Macaulay ring. Let $T$ be a ring containing $R$ and suppose that as an $R$-module it is free ...
- 1,661
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443
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Does the free resolution of the cokernel of a generic matrix remain exact on a Zariski open set?
"Random" modules of the same size over a polynomial ring seem to always have the same Betti table. By a "random" module I mean the cokernel of a matrix whose entries are random forms of a fixed degree....
- 538
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209
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Is Hochschild cohomology finitely generated?
I am sorry if this is a naive question.
Let $k$ be a field, and let $A$ be a finitely generated commutative $k$-algebra.
Let $M$ be a finite $A$-module.
Consider the Hochschild cohomologies of $A$ ...
- 31
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562
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Is $gcd(zx,zy)=zgcd(x,y)$ (i.e. does the left hand side of this equality 'exist' if the right hand side does).
This should be a simple question on basic definitions.
In an integral domain $R$ we will say that $gcd(x,y)$ exists and is equal to some $r\in R$ if $r$ divides $x$ and $y$, and any common divisor $...
- 16.9k
2
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463
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Characterization of prime ideals in regular local rings
Let $R$ be a regular local ring of dimension $d$ and let $x_1,x_2,...,x_d$ be a regular system of parameters. Now, for any $y\in R$, the colon ideal $(x_1,x_2,...,x_h):y$ where $h\leq d$ is a prime ...
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2
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340
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Infinite collection of elements of a number field with very similar annihilating polynomials
Hello all, let $n$ be an integer $\geq 2$ and let $\alpha$ be an algebraic number
of degree $n$. Let $R$ be the ring of algebraic integers in ${\mathbb Q}(\alpha)$, and
let $B$ be the subset of $R$ ...
- 3,595
6
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806
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Radicals of binomial ideals
Let $R=k[x_1,x_2,...,x_n]$ be the polynomial ring in $n$ indeterminates over a field $k$. An ideal (that can be) generated by monomials is called a monomial ideal. For the monomial ideal $M=(m_1,m_2,.....
- 805
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130
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Is this duality operation on simplicial complexes/Stanley-Reisner rings previously known?
Let $K$ be an abstract simplicial complex on vertices $x_1,\ldots,x_n$, then there is the familiar construction of the face ideal $I_K=\langle x_{i_1}\cdots x_{i_r} | \{x_{i_1},\ldots,x_{i_{r}}\}\not\...
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211
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Is there a prime of height $i$ in support of $H^i_I(R)$?
$I$ is an ideal of a local Noetherian ring $R$ and $i>0$ .
Clearly the height of primes in support of $H^i_I(R)$ is at least $i$
The question is if it
contains a prime of height $i$, specially ...
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272
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About free resolutions of graded commutative algebras
Hi, I'm having troubles in adapting certain algebraic constructions to graded cases.
We know that if $A$ is a commutative ring and $a_1,...,a_k$ are elements on $A$, there is a construction of the ...
3
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1
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398
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Is the first part of Eisenbud's Proposition 15.15's proof o.k?
In the chapter on Gröbner bases from Eisenbud's "Commutative Algebra" the following statement appears as Proposition 15.15 (page 344):
Let $F$ be a free $S$ module with basis and monomial order ...
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0
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197
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Existence of flat models of a smooth finite type algebra over $R((t))$
Let $k$ be a field, $R$ a $k$-algebra (of finite type if necessary),
$B$ an algebra of finite type over ring of the formal Laurent series $R((t))$, which is smooth.
Up to this generality, can one ...
- 51
2
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1
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504
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A question arising from the Krull intersection theorem.
Let R be a local ring, I an ideal, M a finitely generated module and $N=\cap _nI^nM$. Then the Krull intersection theorem states that $N=IN$. Now if R is a local ring of characteristic $p>0$, for ...
- 1,207
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1
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374
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Composition and intersection of residue fields
Let $A$ be a normal ring with quotient field $K$. Let $L/K$ be a finite separable extension.
Let $E_1/K$ and $E_2/K$ be extensions of $K$ contained in
$L$. Let $B_1$ (resp. $B_2$) be the normalization ...
- 1,673
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105
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$\Gamma_Z(\widetilde M)\cong\widetilde{ \Gamma_Z(M)}$
Let $R$ be a Noetherian ring and let $M$ is an $R$-module. Consider the associated affine scheme $(\text{Spec R},\mathcal{O}_{\text{Spec R}})$ and Suppose $Z\subset X$ is a closed subset of $\text{...
- 21
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108
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Recovering fractional ideals from ideals
Let $R$ be a Noetherian domain; let ${I_{(i)}}$ be a set of fractional ideals in $K$, the fraction field of $R$, indexed by a lattice such that $I_{(i)} I_{(j)} = I_{(i + j)}$. Let $J_{(i)} = I_{(i)} \...
- 4,991
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396
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Dynamics of polynomial roots
Are there any good tools to understand the movement of roots of polynomials in single variable with real or rational coefficients? That is say the coefficients are of the form $a_{i} + M b_{i}$ where $...
- 13.9k
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An example of a noetherian ring in which the integral closure of a finite extension of its field of fractions is not noetherian
Similar to another question I posted. Does anyone know of an example of a noetherian ring in which the integral closure of a finite extension of it's field of fractions is not noetherian.
- 113
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138
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Profinite Local Ring inside Polynomial Ring
This is a "technical" question that I came across in my research.
Let $A = \textbf{Z}_{p}[\![t_1, \cdots, t_a ]\!]<z_1, \cdots, z_b>$ be the $(p, t_1, \cdots, t_a)$-adic completion of the ...
- 61
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1
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285
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Are there ever exotic isomorphisms between quotients of F[x]?
(This is inspired by the answer to my earlier question.)
Does there exist
a field $F$ $\:$ and $\:$ two ideals $I$ and $J$ of $F[x]$ $\:$ and $\:$ a ring isomorphism $\: \phi : F[x]/I \to F[x]/J$
...
user5810
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245
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Notation Problem, Fixed Rings and Fields
I am trying to make sense of the notation and certain sets in two articles by Annick Valibouze whose results I'm using for my bachelor's thesis, I hope it's relevant enough to merit an answer.
In one ...
- 119
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538
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Functoriality of a standard integral domain construction.
The evident forgetful functor from fields to integral domains has a left adjoint, namely the construction of the quotient field for a given integral domain. Another standard construction is taking the ...
- 21
1
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1
answer
320
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Hochschild (co)homology and Kahler differentials
Suppose $A$ is an augmented commutative algebra over a field $k$. What is the relation between Hochschild homology $H_n(A,k)$ and Kahler differential $\Omega_{A/k}$? The same question is also asked ...
- 97
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0
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197
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A similarity problem over matrices over Gaussian integers
Let $R = \mathbb Z[\sqrt{-1}]$ and
$$\Omega = \{X \in GL_4(R) : X \overline X = I_4 \text{ or } -I_4 \},$$
where $\overline X$ is the complex conjugate matrix of $X$.
Two matrices $A, B \in \Omega ...
- 1,841
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1
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579
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Is there a clean definition of the residue map in Milnor K-theory?
If K is a field, v a discrete valuation, and k the residue field, there is a residue map $\partial: K^M_n(K) \to K^M_{n - 1}(k)$. All the definitions I have seen for this map involve two pages of ...
- 467
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0
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243
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I. Kaplansky, Going up in polynomial rings, unpublished manuscript, 1972
Anyone got a copy of this article?
- 1,388
3
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0
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389
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What's the relation between henselianization and completion?
It's a short question, namely
When can one think about the henselianization $A^h$ of a local ring $A$ as the "algebraic part" of its completion $\hat{A}$?
It seems to be true for $k[\vec{x}]$, ...
- 2,040
3
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0
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251
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On the associated graded ring to a section ring
Consider a nonsingular projective variety $X$ over an algebraically closed field $k$ and let $Y \subseteq X$ be a nonsingular closed subvariety. Let $\mathcal I \subseteq \mathcal O_X$ be the ideal ...
- 3,637