All Questions
Tagged with reference-request ac.commutative-algebra
402 questions
2
votes
1
answer
194
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Name and references for a "twisted" addition in a ring
This question may seem a little unmotivated, but it actually isn't something that just occurred to me out of nowhere. It stems from a discussion I had the other day with a friend and is directly ...
- 3,513
10
votes
2
answers
1k
views
Formal completion of the normal bundle
Let me for simplicity start with affine case. If $X=\operatorname{Spec}(A)$ is an affine variety $Z \subset X$ is a closed affine subvariety $Z=\operatorname{Spec}(A/I)$. What conditions are ...
- 1,545
5
votes
0
answers
73
views
K-Theory and completion [duplicate]
I vaguely remember the existence of a statement that relates the $K$-theory (in the sense of Quillen) of a noetherian local ring $A$ with maximal ideal $\mathfrak{m}$ with the $K$-theory of the $\...
- 165
3
votes
2
answers
470
views
An identity in an arbitrary commutative ring
This fact might be either trivial, wrong, or well known.
Let $R$ be a commutative ring. Let $u_1,\dots,u_{s-1},u_s\in R$ and $m,M\in R$. Let us assume that $m,M$ satisfy
$$(m-u_1) \dots (m-u_{s-1})=0,...
- 21.8k
3
votes
1
answer
422
views
Tor dimension in polynomial rings over Artin rings
I found this tricky problem in trying to understand some properties of local rings at non-smooth points of embedded curves. But this would be a very long story. So I make it short and I try to go ...
- 165
5
votes
2
answers
732
views
The Weyl algebra modules which are also rings
Consider the space $C^\infty(\mathbb{R})$. We have on it a natural action of the Weyl algebra generated by $x$ and $d/dx$, where $x$ is the coordinate on $\mathbb{R}$, and there is plenty of Weyl ...
- 389
8
votes
0
answers
430
views
name for a degree-like invariant of a power series over a commutative ring
Let $R$ be a commutative ring, and let $f \in R[\![X]\!]$ be a formal power series. Sometimes (and for example, this will always be possible if $R$ is Noetherian), one may write $f$ in the form $$
f =...
- 1,802
4
votes
1
answer
592
views
What sort of ring-theoretic properties does the representation ring of a compact Lie group possess?
Recall the definition of the representation ring $R(G)$ of a compact Lie group $G$. I'd like a reference that gives me basic ring-theoretic properties that $R(G)$ always has, or enough info that I can ...
- 35.5k
3
votes
2
answers
519
views
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
1
vote
1
answer
359
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Cohomology after completion
I've been scouring google and asking friend about something I was certain must be absolutely the easiest thing to people who do homological algebra, and none seem to know the answer to this, so if it'...
- 1,049
12
votes
1
answer
551
views
Presenting $\mathbb{Q}[[t]]$ as an explicit colimit of smooth $\mathbb{Q}$-algebras: an explicit example for the Popescu's theorem
By the seminal Popescu's theorem, $R=\mathbb{Q}[[t]]$ is a filtered colimit of smooth $\mathbb{Q}$-algebras. Could you give me a hint: which $\mathbb{Q}$-algebras can yield such a colimit? My problem ...
- 16.9k
2
votes
1
answer
235
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Which algebras can be presented as filtered colimits of f.g. regular ones with smooth connecting morphisms?
Let $R$ be a regular (commutative associative unitial) algebra over a prime field $F$ (i.e. $F=F_p$ or $F=\mathbb{Q}$); assume that it is noetherian excellent (and even of Krull dimension $1$). What ...
- 16.9k
0
votes
0
answers
548
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Fitting ideal sheaves and determinant bundles
I am working on a problem in algebraic geometry which comes down to a fact in commutative algebra that I am hoping is well-known.
Suppose $F$ is a coherent sheaf on a smooth variety $S$, and that the ...
- 5,857
1
vote
1
answer
483
views
formally étale morphisms which are also universally closed
A morphism of schemes which is formally unramified, universally closed, and a monomorphism is a closed immersion. Is it possible to characterize morphisms which are formally etale and universally ...
- 761
4
votes
1
answer
141
views
Which power of $2$ kills $W(k)$?
Is the following fact "well-known": if $-1$ is a sum of squares in a field $k$, then the Witt group $W(k)$ of quadratic forms is killed by multiplication by $2^N$ for some $N\ge 0$? What can one say ...
- 16.9k
2
votes
1
answer
450
views
Resolution of singularity of polynomials
Let $f$ be polynomial on a vector space $V$. Let $Z$ be the zero set of $f$ in $V$. Let $Z_{sing}$ be the singular part of $Z$.
By Hironaka's desingularization theorem, there exists a birational map ...
- 1,457
5
votes
1
answer
675
views
Resolution of a module as an $A_\infty$ module over resolution of an algebra
The following question looks like a basic question on resolutions over commutative rings, unfortunately I was not able to find a reference.
Let $A$ be a regular commutative noetherian ring (and ...
- 1,545
1
vote
1
answer
202
views
Obstruction map for local singularities via tangent (Andre-Quillen) cohomology
Let $R$ be a local singularity (for example $R=\mathbb{C}[[x_1, \ldots , x_n]]/I$) ring over $\mathbb{C}$. Let $\mathbb{L}_{R}$ be a cotangent complex of $R$, then one can define tangent (Andre-...
- 1,545
3
votes
1
answer
280
views
Composite families of formal power series over $\mathbb C$ as algebraic variety
I was led to prove that the set of composite families $(f_j)_{j \leq k}$ of germs at $0\in \mathbb C^m$ of a holomorphic function (composite = sharing a common divisor belonging to the maximal ideal) ...
- 5,428
7
votes
1
answer
304
views
Explicit formula for associator of commutative power series
Perhaps this question is too elementary, but if it's written down anywhere, I'd love to know about it. Suppose I have a power series $f\in R[[x,y]]$ for some commutative, unital ring. I've recently ...
- 10.4k
3
votes
0
answers
182
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Upper semicontinuity of Betti numbers of submodules
Theorem 8.29 in "Combinatorial commutative algebra" by Miller and Sturmfels states the upper-semicontinuity property for Groebner deformations (say, over an algebraically closed field with ...
- 902
6
votes
2
answers
265
views
Germs at infinity of sequence of integers
Consider the $\mathbb Z$-module $\mathcal Z$ obtained as the set of sequences of integers $\mathbb Z ^ \mathbb N$ modulo the relation that two sequences are deemed equivalent when their difference is ...
- 5,428
3
votes
0
answers
123
views
Flatness over Jacboson ring
This is an elementary question which did not get answered on math.stackexchange. I would like to know the answer for expository purposes.
I need either a reference or a counter-example to the ...
- 63
6
votes
0
answers
243
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I. Kaplansky, Going up in polynomial rings, unpublished manuscript, 1972
Anyone got a copy of this article?
- 1,388
1
vote
0
answers
71
views
Integral Leray Number?
The Leray number of a finite simplicial complex $K$ relative to a field $\Bbbk$ is defined to be the least $d\geq 0$ such that $\widetilde H^n(C,\Bbbk)=0$ for all $n\geq d$ and all induced ...
- 38.6k
3
votes
0
answers
124
views
Singularity locus in terms of ideals.
Let $X$ be a smooth affine variety over a field, $Z\subset Y\subset X$ are closed (reduced) subvarieties. What are the possible ways to verify whether $Y$ is singular at $Z$ i.e. whether $Z$ is ...
- 16.9k
4
votes
1
answer
203
views
Flatness and intersections of Cohen-Macaulay subvarieties
There's a commutative algebra fact that I would very much like to be true but could, for all I know, be completely false. One version that would be sufficient is:
Say $A$ is a smooth projective ...
- 1,510
7
votes
3
answers
1k
views
Higher dimensional Bezout via Hilbert polynomials: a reference
For the purposes of teaching my elementary course in algebraic geometry I am looking for a reference (or notes) that contains a complete proof of a higher-dimensional weak Bezout theorem. I only want ...
- 14.3k
5
votes
3
answers
677
views
Spectrum and scheme of the commutative group-algebra of an abelian group.
The group-algebra of an abelian group is commutative, so we can consider the spectrum of this algebra. Are there any information about the abelian group that we can obtain from such considerations? ...
- 263
4
votes
0
answers
110
views
maximal degree of generators of graded ideals
Let $A$ be a commutative Noetherian ring, $\mathfrak{a}$ an ideal. Let $R(\mathfrak{a}) = \oplus_{n\geq 0} \mathfrak{a}^n$ be the Rees algebra of $\mathfrak{a}$. Let $I$ and $J$ be two graded ideals ...
- 2,773
6
votes
1
answer
670
views
Base change of trace for Gorenstein or Cohen-Macaulay morphisms
This is basically a question of functoriality for base change of CM morphisms.
EDIT: $\text{ }$ Brian Conrad sent me an email explaining the that this is indeed true, and follows from his book. I'...
9
votes
0
answers
2k
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Jacobian ideals reference
Suppose that $f : X \to V$ is a flat equidimensional (of dimension $h$) morphism of schemes of finite type and $V$ is excellent (or a variety) For this one can formulate something called the Jacobian ...
- 20.5k
9
votes
0
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644
views
Conceptual proofs for the computation of the structure sheaf
The following lemma in commutative algebra is important for the foundations of algebraic geometry:
If $A$ is a commutative ring, $U \subseteq A$ is a finite subset generating the unit ideal, then ...
- 63.1k
0
votes
1
answer
232
views
Kernel elements for the Grothendieck group map of a commutative monoid
This is just a nomenclature question. Let $T$ be a commutative monoid, and let $T^*$ be its Grothendieck group. That is, $T^* \cong T \times T \ / \sim$, where $(s,s') \sim (t, t')$ if $s+t'+e = s'+t+...
- 8,512
7
votes
3
answers
900
views
Groebner bases for power series rings (reference request)
Hello,
Could you help me with a reference to elementary properties of Groebner bases in rings of formal power series over a field? I am especially interested in generic initial ideals.
Thank you in ...
- 1,839
4
votes
1
answer
427
views
Is this height-transcendence-degree inequality true without AC ?
Let $R$ be a $k$-algebra ($k$ a field) and a domain of finite Krull dimension. In
$\quad$ Krull dimension less or equal than transcendence degree?
it is shown that
$$\text{Krull-dim}(R) \le \text{...
- 16.2k
4
votes
2
answers
308
views
Level of a commutative ring and its quotient field
Reading Lam's Introduction to Real Algebra, he remarks that:
For a Dedekind domain $A$ with quotient field $F$, then $s(A)$ is either $s(F)$ or $s(F) + 1$. Furthermore, $s(A)$ is either $\infty$, $2^{...
- 143
6
votes
3
answers
1k
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Radical generation of ideals in Noetherian rings
It is well-known that any ideal in a Dedekind domain can be generated by at most two elements. However, already for Noetherian domains of dimension 2, it is easy to construct examples of ideals that ...
- 15.6k
4
votes
0
answers
188
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A non-matroidal notion of dependence on a set of ideals
Assume we are given a set of ideals $I_1, \dots, I_s$ in a commutative polynomial ring. Let's define a subset indexed by $A\subseteq [s] = \{ 1,2,\dots, s\}$ as dependent if there exists an $a\in A$ ...
- 1,961
3
votes
2
answers
571
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Free Resolution of this determinantal variety.
I am looking for a free resolution of the ideal generated by $2\times 2$-minors of a $3\times 3$ -matrix. More precisely let $M$ be a matrix (sorry but I cannot write a matrix for some TeX technical ...
- 31
7
votes
1
answer
2k
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Why are minimal resolutions of polynomial ideals important?
Background: Let $k$ be a field and denote by $P = k[x_1,\ldots,x_n]$ the polynomial ring in $n$ (commuting) variables over $k$. A resolution of an ideal $I \lhd P$ is an exact sequence of $P$-modules $...
- 15.5k
6
votes
1
answer
2k
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Why is this theorem attributed to J.-P. Serre?
Page $117$ of Atiyah, MacDonald's Introduction to Commutative Algebra text has the following theorem. Let $P(M,t)$ denote the Poincare- series of $M$.
$\textbf{Theorem.}$ $\bigl(\mathsf{Hilbert-Serre}...
- 4,795
24
votes
3
answers
4k
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Commutative algebra with a view toward algebraic _number theory_
Someone asked me this today, and I don't know what the standard answer is:
Is there an analogue of David Eisenbud's rather amazing Commutative Algebra With a View Toward Algebraic Geometry but with a ...
3
votes
2
answers
427
views
Reference Request: Smith Normal Form for maps between free _graded_ modules
I feel like this should be easy, but I cannot quite find a literature reference for this:
We know (i.a. from the Kaplansky reference in Does Smith normal form imply PID?) that sufficient for Smith ...
- 2,537
15
votes
2
answers
870
views
A space of ideals
Definition: Let $R$ be a commutative ring with 1. Endow the power set $2^R$ with the product topology. The ideal space $\mathcal{I}(R)$ is defined to be subset of $2^R$ consisting of ideals, ...
- 25k
6
votes
1
answer
437
views
"Archimedeanising" an ordered field
If $K$ is an ordered field, let $B$ be the subring comprising the $x \in K$ such that $|x| \le n$ for some $n \in N$, and let $I$ be the ideal of $B$ comprising the infinitesimal elements (i.e. the $x ...
- 879
17
votes
1
answer
4k
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A Relative Algebraic Hartogs Lemma
The Algebraic Hartogs Lemma states that in a Noetherian normal scheme, a rational function that is regular outside a closed subset of codimension at least two, is in fact regular everywhere.
In a ...
- 7,338
11
votes
1
answer
675
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When is there a deformation of a given singularity to a normal singularity
Question: Given a variety $X_0$ with a singularity (say Cohen-Macaulay), when does this exist as a special fiber of a flat family $X \to C$ mapping to a smooth curve $C$, such that the generic fiber ...
- 20.5k
12
votes
5
answers
2k
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analysis over non-Archimedean ordered fields
Can anyone suggest any good references for (or any experts on) analysis over non-Archimedean ordered fields, such as the field of rational functions in one variable (ordered at 0, or if you prefer at ...
- 19.7k
4
votes
1
answer
319
views
Reference request on Leray numbers
The Leray number $L_{\Bbbk}(K)$ (relative to a field $\Bbbk$) of a simplicial complex $K$ is the least $d\geq 0$ such that $\widetilde H_n(C,\Bbbk)=0$ for all $n\geq d$ and all induced subcomplexes $C$...
- 38.6k