Questions tagged [ac.commutative-algebra]
Commutative rings, modules, ideals, homological algebra, computational aspects, invariant theory, connections to algebraic geometry and combinatorics.
5,496 questions
3
votes
1
answer
263
views
On the exactness of some completed tensor products
Let $X$ be an affinoid variety over a discretely valued non-archimedean field $k$ with valuation ring $\mathcal{R}$. Fix a uniformizer $\omega$. On the section 3.2 of the paper https://arxiv.org/abs/...
- 541
16
votes
5
answers
2k
views
Why is the Hochschild homology of k[t] just k[t] in degrees 0 and 1?
Background: the Hochschild homology of an associative algebra is the homology of the complex
$$ \ldots \longrightarrow A \otimes A \otimes A \longrightarrow A \otimes A \longrightarrow A$$
where ...
- 35.5k
24
votes
1
answer
4k
views
Minimal number of generators of a homogeneous ideal (exercise in Hartshorne)
In the very first chapter Hartshorne proposes the following seemingly trivial exercise (ex. I.2.17(ii)):
Show that a strict complete intersection is a set theoretic complete intersection.
Here are ...
- 1,313
10
votes
0
answers
463
views
Does the book "Algebra III" exist (within the Encyclopaedia of Mathematical Sciences series from Springer)?
Within the series "Encyclopaedia of Mathematical Sciences", as published by Springer, one finds the 8 volumes, namely,
the volumes I, II, IV, V, VI, VII, VIII, IX but zbMath has no listing ...
- 4,713
4
votes
1
answer
242
views
Comparing the first-order theories of different kinds of local rings of a complex variety
Let $X$ be a complex variety containing some point $x$. Then $X$ is naturally a complex-analytic space, and we have an inclusion of rings $\mathbb{C}[X]_x\hookrightarrow\mathbb{C}\{X\}_x\...
- 1,969
5
votes
1
answer
419
views
Multiple root of resultant
Let us suppose that we have two polynomials $F_1(x,y)$ and $F_2(x,y)$. Generally speaking, each of them defines a curve on the plane and the system of polynomial equations defined by them computes the ...
- 66.3k
4
votes
0
answers
211
views
Diagonalization over valuation rings
Let $\mathcal{R}$ be a valuation ring, and consider an $\mathcal{R}$-linear endomorphism $L:\mathcal{R}^{n}\rightarrow \mathcal{R}^{n}$. Is there any criterion for telling when $L$ can be diagonalized?...
- 541
4
votes
0
answers
108
views
Gröbner deformations
Let $I \subseteq S=\mathbb{k}\left[x_{1}, \ldots, x_{d}\right]$ be an ideal, $<$ be a monomial order on $S$ and let $T=S[t]=\mathbb{k}\left[x_{1}, \ldots, x_{d}, t\right]$. There exists $\omega \in ...
- 177
4
votes
1
answer
714
views
Jacobian criterion for algebraic independence over a perfect field in positive characteristics
It is well known that the Jacobian criterion for algebraic independence does not hold in general for fields of positive characteristics. However, the following partial statement seems promising:
...
- 156k
2
votes
0
answers
90
views
Monomial order and prime ideals
Let $S=K[x_1,\ldots, x_n]$ polynomial ring. Let $I \subseteq S$ an ideal and $<$ be a monomial order in $S$. Is it possible to describe the minimal primes of in$_<(I)$ from the minimal primes of ...
- 177
3
votes
1
answer
255
views
How to determine the degree of a rational function field over a relatively algebraic subfield?
Let $K$ be a field and $K(x_1,\cdots,x_n)$ be the degree-$n$ purely transcendental extension of $K$. Given homogeneous polynomials $f_1,\cdots,f_n\in K[x_1,\cdots,x_n]\setminus K$ with $\deg f_i=d_i$,...
- 5,339
3
votes
1
answer
164
views
Explicit computation of D-modules pullback
Consider the $D_{\mathbb{A}^1}$-module $M:=D_{\mathbb{A}^1}/(x)$, and the map $f:z\mapsto z^k$. I want to know $f^*(M)$. I believe it only has a single non-zero cohomology, namely in degree $0$, which ...
- 148k
62
votes
5
answers
10k
views
Does "finitely presented" mean "always finitely presented"? (Answered: Yes!)
Precisely, if an R-module M has a finite presentation, and Rk → M is some unrelated surjection (k finite), is the kernel necessarily also finitely generated?
Basically I want to believe I can ...
- 64k
3
votes
1
answer
459
views
Frobenius functor and length of local cohomology
Let $(R,\mathfrak{m})$ be a Noetherian local ring of positive prime characteristic $p$ and let $F$ be the Frobenius functor. Write $d$ for dimension of $R$. Assume that for some $0\leq i< d $ the ...
- 2,821
2
votes
0
answers
80
views
Prime elements in integrally closed extension of domains
Let $R \subseteq S$ be an extension of integral domains such that $R$ is integrally closed in $S$. Let $P$ be a prime ideal of $R$.
Is $PS$ always a prime ideal of $S$?
For classical examples of ...
- 365
10
votes
0
answers
454
views
What does Hilbert's 90 theorem tell us about Galois fixed points in projective space?
Consider the following statement:
If $K\subseteq L$ is a Galois extension of fields with Galois group $G$ and $x \in \mathbb{P}^n(L)$ is such that $\sigma(x)=x$ for all $\sigma\in G$, then $x \in \...
- 32.5k
4
votes
0
answers
213
views
Reference request: Radicial morphisms & Jacobson-Bourbaki correspondence
My name is Chemy (Przemysław). I am a PhD student at UvA (Amsterdam), and I work on projects related to foliations in algebraic geometry in positive characteristic. Therefore I am avidely reading two ...
- 485
16
votes
2
answers
4k
views
A geometric reference for (affine) Gorenstein varieties and singularities
I would like to ask for a reference to some text that explains in relatively down to earth (if possible geometric) terms (for dummies) what is a Gorenstein singularity and Gorenstein variety (for a ...
- 35.5k
6
votes
0
answers
515
views
Quasi-syntomic descent and prismatic F-crystals
I am reading Bhatt and Scholze's paper on F-crystals, and they seem to be using the following result in the proof of Theorem 5.6:
let $X \to Y$ be a quasisyntomic cover of formal schemes over $\...
- 461
6
votes
1
answer
260
views
Vanishing linear combinations of minors
Let $V$ be the set of $k$ by $n$ matrices ($k<n$) with entries in $\mathbb{C}$, and let $\mathbb{C}[V]$ denote the set of polynomial functions on $V$. For any subset $I \subseteq [n] = \{1,2,\dotsc,...
- 7,300
6
votes
1
answer
272
views
Groebner Bases for submodule over polynomial ring with integer coefficients
It is well-known that there exist Groebner bases for ideals in polynomial ring $\mathbb Q[x]$ which can be found algorithmically Moreover, I don't think it is hard to show that there exist Groebner ...
- 1,446
1
vote
0
answers
76
views
Determine whether a set generates a residue field of an invariant ring
Fix two positive integers $m>n$.
Let $(A|Y)$ be an $m\times (n+1)$ augmented matrix consisted of $m\times (n+1)$ indeterminates, where $Y$ is a column symbolic vector of length $m$.
Denote $R=\...
- 331
1
vote
0
answers
165
views
spectral sequence Ext(R/I,H^g(M)) => Ext^{p+q}(R/I,M)
I am reading papers of Local cohomology and came across some spectral sequences. I then started reading about spectral sequences from Rotman's book. I havent finished reading the chapters on spectrals ...
- 183
1
vote
1
answer
181
views
Non-separable $\mathbb{A}^2$-form is trivial
Suppose $A$ is a finitely generated $\mathbb{Q}$-algebra and $A \otimes_{\mathbb{Q}} \mathbb{R} \cong \mathbb{R}[X, Y]$. Then is $A \cong \mathbb{Q}[X, Y]$? Here $\mathbb{R}[X, Y]$ is a two variable ...
- 4,111
1
vote
2
answers
193
views
Annihilators of sum of two ideals
Let $R$ be a commutative Noetherian ring and $I$, $J$ be two ideal of $R$.
If $x\in R$, then is $((I+J):x)=(I:x)+(J:x)$?
I would be very grateful if someone comment me.
- 113
4
votes
0
answers
352
views
What does the cotangent complex tell you when it takes animated inputs?
These two links: What is the cotangent complex good for? and Intuition about the cotangent complex? are quite helpful in giving intution for the cotangent complex in terms of deformations but I don't ...
- 301
4
votes
1
answer
564
views
Nondegenerate pairings versus perfect pairings for finitely generated projective modules
Let $R$ be a (not necessarily commutative) ring, $M$ a left $R$-module, and $N$ a right $R$-module. We say that a pairing
$$
\langle -,-\rangle:M \otimes_R N \to R
$$
is non-degenerate if, for all $n \...
- 145
3
votes
1
answer
172
views
On the linearizability of the action of a finite group on a formal polydisc
$\DeclareMathOperator\Aut{Aut}\DeclareMathOperator\Gl{Gl}$Let $\mathcal{D}=\mathbb{C}[[t_{1},\dotsc , t_{n}]]$ be a formal polydisc over $\mathbb{C}$, and $G$ be a finite group. On Lemma 7.8 of ...
- 148k
6
votes
2
answers
388
views
Abelian groups such that $A \cong \mathrm{End}(A)$ and "complete rings"
Motivation: for any ring $R$ there is the natural monomorphism $\mathrm{in} \colon R \to \mathrm{End}(R_{add}): r \mapsto (x \mapsto rx)$, where $R_{add}$ is an additive abelian group ( rings are ...
- 3,041
3
votes
0
answers
345
views
On a conjecture of Hartshorne
Hartshorne has a conjecture in his book Ample Subvarieties of Algebraic Varieties. It's in page 126 Conjecture 5.16 where he writes that if $B$ is a finitely generated flat module over a regular local ...
- 2,275
3
votes
0
answers
91
views
On the descent of noetherianess along completion
Let $A$ be a commutative local ring with maximal ideal $m$ and $\hat{A}$ be its $m$-adic completion. Are there any non-trivial conditions on $A$, under which $\hat{A}$ noetherian implies $A$ ...
- 541
5
votes
1
answer
524
views
Is there a non-split algebraic torus (over a finite field) satisfying the following properties?
Is there a non-split algebraic torus $T$ (over a finite field $\mathbb{F}_{\!q}$) satisfying the following properties?
$T$ is not $\mathbb{F}_{\!q}$-isomorphic to the direct product of algebraic tori ...
- 12.9k
1
vote
0
answers
41
views
A bi-variate polynomial interpolation question
Let $R$ be a commutative unital ring, and $R^{m\times k}$ denote the set of $m\times k$ matrices with entries from $R$. A matrix $U\in R^{m\times m}$ is elementary if $U$ is obtained from the identity ...
CommunityBot
- 1
2
votes
0
answers
108
views
Left-elements of a numerical semigroup generated by two elements
A numerical semigroup $S$ is a semigroup in $\mathbb{N}$ such that $\mathbb{N}\backslash S$ is finite. It is known that there exists always a set $M$ such that an element in $S$ can be expressed as a ...
- 115
3
votes
0
answers
71
views
Sums of powers in regular local rings
The following theorem is proven in Jensen-Lenzig's "Algebra, Logic, and Applications, vol 2" Sublemma 3.34.1:
Suppose that $R$ is a regular local Henselian ring with maximal ideal $\mathfrak{...
- 1,689
3
votes
0
answers
324
views
Roots of polynomials over $\mathbb{Z}/p^k\mathbb{Z}$
Over a finite field, such as $\mathbb{Z}/p\mathbb{Z}$, the number of roots of a polynomial is no larger than the degree. I'm interested in how does this generalize to $\mathbb{Z}/p^k\mathbb{Z}$.
I'm ...
CommunityBot
- 1
1
vote
0
answers
140
views
The first syzygy module of a binomial ideal
It is known how you compute the first syzygy module of a monomial ideal but it seems an hard work to do the same for binomial ones. I don't know any procedure to aim that, so I would like kindly if ...
- 31
1
vote
2
answers
1k
views
Extension class and cup product
Recall that the group $Ext^1(F'',F')$ parametrizes extensions $$0 \rightarrow F' \rightarrow F \rightarrow F'' \rightarrow 0$$ as follows: given one such extension, consider the long exact cohomology ...
- 4,492
4
votes
1
answer
631
views
On the multiplicities of an ideal on a smooth variety
Let $X$ be a smooth variety, $\xi$ be a point of $X$ and $\mathfrak{a}$ be an ideal sheaf. If we define $mult_{\xi} \mathfrak{a}$ to be the largest integer $p$ such that $\mathfrak{a} \cdot \mathcal{O}...
- 35.5k
2
votes
1
answer
258
views
Possible "algebraic" direction in hyperplane arrangements
I have been studying hyperplane arrangements from R. Stanley's notes and so far, I have read until lecture 5. But, Lecture 6 and end of Lecture 5 seems very combinatorial. I'm more interested in the &...
- 24.2k
15
votes
1
answer
649
views
Primes that must occur in every composition series for a given module
Let $M$ be a finitely generated module over the commutative noetherian ring $R$. Let ${\cal C}(M)$ be the set of all primes $P$ in $R$ such that $R/P$ appears as a quotient in every composition ...
- 2,821
11
votes
2
answers
530
views
Undecidability of irreducibility of infinite families of integer polynomials?
A recent question, Is irreducibility of polynomials $\in\mathbb{Z}[X]$ over $\mathbb{Q}$ an undecidable problem? was quickly answered in the negative. I am wondering if there is a simple example of a ...
- 82.7k
3
votes
0
answers
149
views
What direction does the derivation of an inseparable algebraic variable point in?
I've been thinking about the geometry of inseparable field extensions lately, since I'm studying smoothness in commutative rings in an advanced topics course this semester. I've generally come to the ...
- 1,969
15
votes
2
answers
1k
views
Is irreducibility of polynomials $\in \mathbb{Z} [X]$ over $\mathbb{Q}$ an undecidable problem?
There are a number of criteria for determining whether a polynomial $\in \mathbb{Z} [X]$ is irreducible over $\mathbb{Q}$ (the traditional ones being Eisenstein criterion and irreducibility over a ...
- 12.9k
36
votes
4
answers
12k
views
Flatness and local freeness
The following statement is well-known:
Let $A$ be a commutative Noetherian ring and $M$ a finitely generated $A$-module. Then $M$ is flat if and only if $M_{\mathfrak{p}}$ is a free $A_{\mathfrak{p}}$-...
- 4,219
7
votes
0
answers
365
views
Residue field of a ring does not depend upon the maximal ideal
Let $\mathbb{K}$ be a field and let $A$ be a $\mathbb{K}$-algebra. We will say that $A$ is residually $\mathbb{K}$ if for every maximal ideal $\mathfrak{m}$ we have that the structural morphism $\...
0
votes
0
answers
185
views
Exactness of $I$-adic completion in a certain non-finitely generated case
I would like the functor
$$(-\otimes_{\mathbb Z} F)\hat{}: \mathbb{Z}[x_1,\dots,x_r]\text{-Mod}_{\mathrm{f.g.}}\longrightarrow \mathbb{Z}[x_1,\dots,x_r]\text{-Mod}$$
to be exact, where completion is w....
- 91
8
votes
3
answers
740
views
Is there some example that nicely extends the multiplication of natural numbers?
Motivation: In mathematics, it is natural to decompose a complicated thing into simpler ones. In the system of natural numbers, the process to decompose large number is to factor it. The ...
- 10.5k
2
votes
1
answer
234
views
Algorithm for compact polynomial expressions
Sometimes an ugly polynomial (perhaps in several variables) can be expressed as a small sum of much simpler polynomials. Can this be done algorithmically? More precisely:
Is there a reasonable
...
- 63.9k
3
votes
0
answers
92
views
Solutions of the vector field $D=A\frac{\partial}{\partial X}+B\frac{\partial}{\partial Y}$ with $A,B\in k[[X,Y]]$
I am trying to understand the article Reduction of Singularities of the Differential Equation $Ady=Bdx$ by Arno van den Essen.
Let $A,B\in k[[X,Y]]$ be formal power series and $D$ the vector field $A\...
- 63.9k