All Questions
Tagged with reference-request ac.commutative-algebra
402 questions
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265
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Algebraic closure of field of fractions of multivariate polynomial ring over $\mathbb{R}$
I am searching for good references on the topic of the behaviour of the elements in the algebraic closed field $(\mathbb{R}[x_{1},\dots,x_{n}])^{\operatorname{alg}}.$ I imagine that, when we try to ...
- 573
5
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0
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132
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Asymptotics of Hilbert series for locally finite free graded-commutative algebras?
Let $A^\bullet$ be an $\mathbb N$-graded algebra over a field $k$, and let $d_A(n) = \dim A^n$ be the dimension of the $n$-th graded piece, so that $P^A(t) = \sum_n d_A(n) t^n$ is the Hilbert-Poincare ...
- 63.9k
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1
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383
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Euler characteristic and rational Poincaré series
$\DeclareMathOperator\len{len}\DeclareMathOperator\Tor{Tor}$Let $(A,\mathfrak{m})$ be a regular local ring, and $x \in \mathfrak{m}^2$ be a non-zero prime element. So $R:=A/(x)$ is a non-regular Cohen-...
- 125
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324
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Earliest reference for infinitesimal neighborhoods of the diagonal
Where was $I_x/I_x^2$ first introduced? (DG or AG) asks about the algebraic cotangent space. The paper First neighborhood of the diagonal and geometric distributions by Kock claims Grothendieck ...
- 10.5k
2
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111
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If $\mathfrak{p}\subset R$ is a minimal prime divisor of $\mathrm{Ann}_R(M)$, then $\mathrm{Ann}_R(M/\mathfrak{p}M)=\mathfrak{p}$
$\DeclareMathOperator\Ann{Ann}$Let $R=\mathbb{C}[x_1,\dots,x_n]$. I am looking for a reference of the following statement.
$(*)$ Let $M$ be an $R$-module, and let $\mathfrak{p}$ is a minimal prime ...
- 2,788
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2
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256
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Reference for integral extensions of $\mathbb{Z}/p^k\mathbb{Z}$
I was looking for a reference which discusses the structure of finite integral extensions of $\mathbb{Z}/p^k\mathbb{Z}$. In particular, I am interested in understanding what the abelian group of its ...
- 145
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126
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Identity relating iterated determinant line bundles
Suppose that $R$ is a (commutative, unital) ring and that $A$ is a (commutative, unital) $R$-algebra that is projective of constant rank $n$ as an $R$-module. Then $A$ has a "determinant line ...
- 2,356
8
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1
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855
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What is the motivation for excellent rings?
First of all I am not formally educated in mathematics so pardon my ignorance if this is obvious and I am skipping something vital, but I am interested nonetheless in what the original motivation and ...
- 81
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430
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Is the Pierce spectrum useful elsewhere in Mathematics?
In Borceaux and Janelidze's Galois Theories, a construction of the Pierce spectrum is given. It is the poset of ideals in a Boolean ring. It's construction is reminiscent of the Zariski spectrum in ...
- 2,369
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491
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Examples of almost Dedekind domains that are not Dedekind
All I know about almost Dedekind domains (which I have come to learn about only recently) is that they are integral domains whose localization at every prime is a discrete valuation ring. In other ...
- 739
3
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1
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606
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Prime ideals and localizations of the ring $\mathbb Z[\{\sqrt p: p \text{ prime}\}]$
I have been trying to study the prime ideals of the ring $R:=\mathbb Z [\{ \sqrt{p_n}\}_{n=1}^\infty]$, where $p_n$ denotes the $n$-th prime. This is how far I got: I could conclude, by means of the ...
- 739
3
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0
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243
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Quick proof of the first part of Kaplansky's Theorem on characterization of Noetherian domains with all maximal ideals principal
I have been reading section 12 of this paper "Elementary Divisors and Modules" by I. Kaplansky (https://www.ams.org/journals/tran/1949-066-02/S0002-9947-1949-0031470-3/S0002-9947-1949-...
- 739
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2
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650
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Definition of subcoalgebra over a commutative ring
Let $k$ be commutative ring and $(C, \Delta)$ be a coalgebra over $k$. Let $D$ be a $k$-submodule of $C$.
Notes I'm reading give the following definition:
$D$ is called subcoalgebra of $C$ if the ...
user167952
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236
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Functions on Stone spaces as "enveloping algebra" of Boolean algebra
I'm looking for references for the following closely related facts:
Given a Boolean algebra $B$, I denote by $\mathbb{Z}[B]$ the free ring generated by symbols $e_b$ such that $e_b e_{b'} = e_{b \cap ...
- 42.4k
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130
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Sources for describing the characteristic polynomial of a nonintegral hyperplane arrangement in terms of point counting?
I have a family of hyperplane arrangements, and I'd like to describe their characteristic polynomials. When the hyperplanes are defined over the integers, the easiest way for me to do this is to use ...
- 453
3
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69
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On Ext-duals of injective modules for commutative rings
Let $R$ be a commutative noetherian ring and $I=E(R/p)$ the injective hull of the module $R/p$ for a prime ideal $p$.
Question: Is there a (more) explicit description of the $R$-modules $Ext_R^i(I,R)$...
- 26.5k
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2
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Original proof of Hilbert's syzygy theorem
Does anyone know an English reference for the original proof of Hilbert's syzygy theorem? The three proofs that I know use either:
the theory of projective dimension and change of rings (plus a step ...
- 14.7k
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132
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On the dual version of an isomorphism of spectral sequence term (from Cartan and Eilenberg)
I'm trying to take spectral sequences as a black box for application in commutative algebra and I admit that I haven't really gone through (or understand) all the proofs of all the isomorphisms ...
- 642
4
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1
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367
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Condition such that the fibres of a polynomial map $p :\mathbb{C}^n\rightarrow \mathbb{C}^n$ are finite
I was told that if $A$ is the subring of $\mathbb{C}[x_1,\ldots, x_n]$ generated by the polynomials $p_1(x_1,\ldots, x_n),\ldots, p_1(x_1,\ldots, x_n)$, then the preimage $p^{-1}(c)$ via the map $p = (...
- 173
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On a reference for computing global spectrum of $A_n$-curve singularities, by H.Dao and E.Faber
This question is about chasing down a reference in a paper relating to non-commutative crepant resolutions and Cohen-Macaulay representation theory.
Allow me to first give a minor introduction.
Let $(...
- 51
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1k
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Summing infinitely many infinitesimally small variables makes sense in algebra
There is an identity $e^x=\lim_{n\to \infty} (1+x/n)^n$, and I always thought it is a purely analytic statement. But then I discovered its curious interpretation in pure algebra:
Consider the ring of ...
- 3,772
6
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1
answer
275
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Algebraic geometry additionally equipped with field automorphism operation
I am looking for some facts on theory, which is essentially algebraic geometry but with field automorphisms added as 'basic' operations. (Precisely, I mean universal algebraic geometry for (universal) ...
- 63
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223
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Intrinsic characterisation of a class of rings
This may be well known, but I was unable to find an answer browsing literature. Let us temporarily call a commutative (unital) ring $R$ an O-ring if there exists an integer $n \ge 1$, a local field of ...
- 6,214
3
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98
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Hales' generalization of the stacked bases theorem (seeking a proof)
In his paper Analogues of the stacked bases theorem, published in the proceedings of a 1976 conference, A.W. Hales claimed some interesting generalizations of the stacked bases theorem for abelian ...
- 2,992
3
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1
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332
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Algebraic vector bundles on the punctured spectrum: an exact reference for a result
Let $(R, \mathfrak m)$ be a Noetherian local ring of depth at least $2$. Let $X=Spec(R)$ denote the affine- scheme with structure sheaf $\mathcal O_X$ and $U=Spec(R)\setminus \{\mathfrak m\}$ be the ...
- 642
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763
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Further developments of Cartier–Gabriel–Kostant–Milnor–Moore Structure Theorem for cocommutative Hopf algebras
A very well-known theorem in Hopf algebra theory (see, for example, Lorenz - A tour of representation theory or the EGNO book (Etingof, Gelaki, Nikshych, and Ostrik - Tensor categories)) states that ...
- 718
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1
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836
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Solving multilinear equations
Let $N=\{1,2,\ldots,n\}$. Suppose we are given $n$ equations, with each equation taking the form $\sum_{A\subseteq N}\left(c_A \prod_{i\in A}x_i \right) = 0$, where each $c_A$ is a real number ...
- 239
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2
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543
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Rings $R$ such that every [regular] square matrix with entries in $R$ is equivalent to an upper triangular matrix
Let $\text{M}_n(R)$ be the ring of $n$-by-$n$ matrices with entries in a commutative unital ring $R$. Theorem III in
C.R. Yohe, Triangular and Diagonal Forms for Matrices over Commutative ...
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1
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379
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Splitting of short exact sequence in the category of finitely generated modules over a commutative Noetherian ring
In the category of finitely generated modules over a commutative Noetherian ring, the splitting of a short exact sequence can be checked locally at the maximal ideals of the ring. One reference for ...
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123
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Dimension of the socle of the first local cohomology module
Let $M$ be a graded $\mathbb{C}[z_0,\dots,z_n]$-module. Using local duality one can show that
$$
\dim_\mathbb{C} (\text{soc} H_\mathfrak{m}^1(M))_k = \beta_{n,k+n+1}(M).
$$
Here $H_\mathfrak{m}^1(M)$ ...
- 294
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1
answer
229
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Generic Galois alteration of an arithmetic model with semistable special fiber
Let $R$ be a DVR of char $0$ and $S=Spec(R)$. Let $X\longrightarrow S$ be a proper flat morphism. Assume $X$ is integral. De Jong's Theorem 8.2 in his paper Smoothness, semi-stability and alterations ...
- 43
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1
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100
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category of non-welldefined linear maps
I was wondering whether the following category already has been used somewhere and whether it already has been named.
Let us fix a field $k$ (or more generally a ring). An object is just a $k$-vector ...
- 11.1k
3
votes
0
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186
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Is cup product of cycle classes on Noetherian regular excellent scheme compatible with intersection
Let $\mathcal{X}$ be a Noetherian regular integral excellent scheme. Let $Y$ and $Z$ be algebraic cycles of codimension $c$ and $d$ on $\mathcal{X}$.
Let $n$ be a positive integer invertible on $\...
- 43
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3
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714
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Cohomology of elementary abelian $p$-groups, i.e. $H(G,{\mathbb F}_p)$ with $G\cong{\mathbb F}_p^r$
I have two questions.
$\bf 1.$ First, a reference request. Let $G\cong{\mathbb F}_p^r$ for some integer $r\geq 0$ and let $V=G^*={\rm Hom}(G,{\mathbb F}_p)$. Then $(H(G,{\mathbb F}_p),+,\cup )$ is a ...
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1
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656
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Closed points of a closed subscheme of $\mathbb{P}^n$ over the residue field and the fraction field of a valuation ring $R$
Let $(R, M)$ be a valuation ring with algebraically closed fraction field $k$. Let $L = R/M$ be the residue field of $R$; it follows that $L$ is algebraically closed. I would like to understand the ...
- 3,625
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A question about the Buchsbaum-Eisenbud-Horrocks Conjecture
It's known that Mark E. Walker proved the "weaker" version of Buchsbaum-Eisenbud-Horrocks' Conjecture (BEH). Although the claim was stated to hold in arbitrary field $k$, Walker's proof does not seem ...
- 43.2k
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225
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Tachikawa conjecture for finite dimensional commutative monomial algebras
Let $A=K[x_1,...,x_n]/I$ be a finite dimensional local algebra with a monomial ideal $I$.
The Tachikawa conjectures are conjectures for finite dimensional algebras. Im interested in them here for such ...
- 26.5k
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126
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Algebraic structures on graphs
There are many algebraic structures linked to graphs.
For example one can find zero divisor graphs $[1]$, $[2]$ and many other graphs.
Does there exist any survey paper which characterizes all the ...
- 444
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1
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691
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Picard group of hypersurfaces in $\mathbb{P}^r\times\mathbb{P}^s$
Let $k$ be an algebraically closed field, say $k=\mathbb{C}$. Let $r,s$ be sufficiently large integers.
Is it true that, for any irreducible hypersurface $X$ of bi-degree $(d,1)$ in $\mathbb{P}^r\...
user39380
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0
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122
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Are affinoid algebras over nontrivially valued fields Jacobson?
It is well-known that for any field $k$ with valuation the Tate algebra $k\{T_1,\dots,T_n\}$ is Jacobson (see Bosch-Güntzer-Remmert for nontrivial valuations; for trivial valuations those are just ...
- 28.2k
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166
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When do the spectra of overrings glue to a proper morphism?
This question is motivated by the construction of blowups.
Let $A \subset K$ be a commutative domain and its fraction field, and let $\{A_i\}$ be some finite collection of overrings in between.
Let $X ...
- 1,338
3
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1
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157
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Projecting onto the span of a generic Veronese variety
Let $\sigma_d:\mathbb{P}^2\to\mathbb{P}^n$ be the d-th Veronese map and let $X=\sigma_d(\mathbb{P}^2)$. Let $W\subset\mathbb{P}^n$ be a 2-plane such that $W\cap X=\emptyset$. For a line $L\subset \...
- 41
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Rings of $S$-integers are finitely generated as rings
Let $K$ be a global field (number field or algebraic function field over a finite field), $\mathcal{V}$ the set of $\mathbb{Z}$-valuations on $K$, $S \subseteq \mathcal{V}$ a finite set. The ring of $...
- 277
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2
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505
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A good reference to the Gauss result on the structure of the multiplicative group of a residue ring
I need a good reference (desirably some textbook in Number Theory) to the following known result, attributed to Gauss in Wikipedia.
Theorem (Gauss). Let $p$ be a prime number, $k\in\mathbb N$ and $\...
- 41.8k
3
votes
1
answer
153
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Artinian Tor modules (Reference request)
I am looking for a reference for the following basic fact:
Let $R$ be a noetherian ring, let $M$ be an artinian $R$-module, let $N$ be a finitely generated $R$-module, and let $i\in\mathbb{N}$. ...
- 6,700
6
votes
1
answer
607
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rationality of weighted projective space
A complex weighted projective is $\mathbb{P}(k_1, \cdots, k_{n+1})=Proj(\mathbb{C}[x_1, \cdots, x_{n+1}])$ with $x_i$ of degree $k_i$ (sometimes people ask for each $n$ of the weights being coprime). ...
- 337
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1
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142
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Valuation theory on semisimple algebras used in the paper of Cohen-Martinet: reference request
I'm currently reading the paper of Henri Cohen & Jacques Martinet "Etude heuristique des groupes de classes des corps de nombres"
On the 2nd section, they recall some facts on valuations, ...
- 1,013
5
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2
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601
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Reference book for understanding Hilbert Series/functions
For my bachelor thesis my goal is to understand the reasoning behind "Hilbert series" and how they connect to the idea of "dimension".
https://en.wikipedia.org/wiki/...
user129588
14
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1k
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Is there a slick proof of the fundamental theorem of dimension theory?
The fundamental theorem of dimension theory in commutative algebra states that given a module $M$ over a noetherian local ring $A$, we have $s(M)=\text{dim}(M)=d(M)$ (where $s(M)$ is the infimum of ...
- 1,514
2
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0
answers
137
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Weak Lefschetz property Jacobian ring smooth hypersurface
Let $A_{.}$ be a graded commutative ring. We say that $A_{.}$ satisfies the weak Lefschetz property if for generic $L \in A_1$ the multiplication maps $ \times L : A_i \longrightarrow A_{i+1}$ has ...
- 7,300