All Questions
Tagged with reference-request ac.commutative-algebra
402 questions
4
votes
0
answers
182
views
$\mathcal{C}$-filtering of modules inherited by submodules
I'll state the question about modules, but I'm open to examples in other contexts. I am not an algebraist, so please forgive any non-conventional terminology.
DEFINITION: Let $\mathcal{C}$ be a ...
- 2,231
1
vote
0
answers
224
views
The dimension of the Zariski tangent space is bounded for a finitely generated algebra
Can anyone suggest a published reference for the following fact:
For a given finitely generated algebra over an algebraically closed field, the dimension of the Zariski tangent space at maximal ...
- 11
0
votes
1
answer
215
views
Does $\sum_ia\cap b_i=a\cap(\sum_ib_i)$ and $a(\bigcap_i b_i)=\bigcap_iab_i$ for infinite sums and intersections in arithmetic rings (Prufer domains)?
Note: Please let me know if this question is too basic for MathOverflow. It is about a subject commonly taught in graduate school (commutative algebra), and is based in large part on a (very ...
8
votes
0
answers
548
views
Foundational Questions on Adic Spaces
There are some foundational questions on adic spaces that I can't find in the literature. It seems that these questions are pretty natural, so I guess that an answer should be known to the experts in ...
- 2,923
9
votes
2
answers
566
views
Involutions in $\mathbb{F}_p[[x]]$
Question: For a prime $p$, is every involution in $\mathbb{F}_p[[x]]$ with a zero constant term a reduction modulo $p$ of some involution in $\mathbb{Z}[[x]]$?
Here involution in $A[[x]]$ means $f\in ...
- 1,190
8
votes
1
answer
359
views
Global to local principle for f.g. $\mathbb{Z}[x]$ modules
In graduate school, while I was working on the maximal subgroup growth of certain metabelian groups, I discovered and proved a lemma which gave me the impression that it was already known. Do you know ...
1
vote
1
answer
229
views
Koszul -regular sequences (reference request)
Let $R$ be a ring and let $f:P\rightarrow P'$ be a surjective morphism of smooth $R$-algebras. Let $J$ be the kernel of this map. If $R$ is Noetherian, one can show that $J$ is locally generated by a ...
- 309
9
votes
1
answer
692
views
Equivalence of definitions of Cohen-Macaulay type
I know that the Cohen-Macaulay type has these two definitions:
Let $(R,\mathfrak{m},k)$ be a Cohen-Macaulay (noetherian) local ring and $M$ a finite $R$-module of depth $t$. The number $r(M) = \dim_k ...
- 113
1
vote
0
answers
399
views
Infinite-dimensional representation theory of $K[x]$
Let $K$ be an algebraically closed field. The finite-dimensional representation theory of the polynomial algebra $K[x]$ is tame and completely understood, which I shall first summarise. It's ...
- 482
0
votes
0
answers
448
views
Behavior of Ext under base change
Let $x$ be a nonzerodivisor on a local Noetherian ring $(R,m).$ Let $M,N$ be finitely generated $R/xR$-modules.
How to show the existence of the following exact sequence
$\cdots\longrightarrow ...
- 1,713
2
votes
0
answers
159
views
Graded Betti numbers $\beta_{n,j}$ for points in $\mathbb{P}^n$
Let $S = \mathbb{C}[z_0, \dots, z_n]$, and let $X$ be a set of points in $\mathbb{P}^n$. I'm looking for references concerning results for the graded Betti numbers $\beta_{n,j}(S/I(X))$, i.e., the ...
- 294
18
votes
2
answers
1k
views
Is a matrix similar to its transpose over $\mathbb{Z}_p$?
Is every $n \times n$ matrix with entries in $\mathbb{Z}_p$ (or even $\mathbb{Z}$) conjugate to its transpose via a matrix in $GL_n(\mathbb{Z}_p)$?
On the one hand, I know the analogous fact is false ...
- 2,242
7
votes
1
answer
553
views
Relationship between Hilbert-Samuel multiplicity and polar multiplicity
Let $f \in \mathbb{C}[[x,y]]$ be the germ of an isolated plane curve singularity. Then the Hilbert-Samuel multiplicity $e_f$ of $f$ is given as follows:
$$e_f = \lim_{s \to \infty}\frac{1}{s} \cdot \...
1
vote
0
answers
180
views
Skew-symmetric multi-derivations of $k[x_1,…,x_n]/I$
Let $I = \langle f_1, \ldots f_r \rangle$ be an ideal in $R=k[x_1,\ldots,x_n]$ where $k$ is a field, and put $A = R/I$.
(If $I$ is prime then $A$ is the coordinate ring of an irreducible affine ...
- 832
3
votes
0
answers
97
views
Infinite-dimensional wild commutative algebras with non-trivial units
Let $A$ be an arbitrary (not necessarily finite-dimensional) associative algebra over an algebraically closed field $K$, and let $\mathrm{fin\,}A$ denote the category of finite-dimensional $A$-modules....
- 482
9
votes
1
answer
211
views
Reference for Kakutani result on power sum bases of symmetric functions
Numerical semigroups are additive submonoids $A$ of the natural numbers such that the greatest common divisor of all elements of $A$ is 1. The complement of a numerical semigroup in $\mathbb{N}$ is ...
- 528
7
votes
1
answer
212
views
The Image of a Derivation is Contained in the Jacobson Radical
Let $A$ be a finite-dimensional unital commutative associative algebra over a field $K$ of characteristic $0$. Is it true that for any derivation $D$ of $A$ we have $D(A) \subseteq J(A)$ where $J(A)$ ...
- 71
7
votes
1
answer
294
views
A commutative variant of the exterior algebra
Consider $A = \mathbb{R}[t_1,\ldots,t_{k}]$, the ring of real $k$-variate polynomials in the indeterminates $t_1,\ldots,t_k$. For $y\in\mathbb{R}$, define the element $p(y) \in A$ through
$$
p(y) = ...
- 213
8
votes
1
answer
257
views
Minimal resolution of local cohomology module
Let $(R,\mathfrak m)$ be a Noetherian local ring of dimension $d\geq 1.$ Suppose $\lambda(H_{\mathfrak m}^i(R))<\infty$ for some $0\leq i\leq d-1.$
Question Can we say anything about Betti numbers ...
- 1,713
7
votes
1
answer
837
views
Intersection of free/affine submodules, comparison with vector spaces
If $W_1,W_2 \subset V$ are finite-dimensional $k$-vector spaces of dimensions $d_1, d_2 \leq d$, respectively, then $d_1 + d_2 > d$ suffices to guarantee $W_1 \cap W_2 \neq \{0\}$. There are ...
- 1,142
3
votes
0
answers
133
views
Hilbert's irreducibility theorem for prime ideals
A typical formulation of Hilbert's irreducibility theorem is like this (see [1]):
Let $k=\mathbb{Q}$ and $f\in k[x_1,\ldots,x_n,y_1,\ldots,y_m]$ be an irreducible polynomial. There exists a Zariski ...
- 3,041
9
votes
1
answer
712
views
Curious anti-commutative ring
Has anyone seen the ring $\Lambda[x_0, x_1, x_2, \ldots]/(x_i x_j - (i+1) x_0 x_{i+j})$ in some natural context?
Here $\Lambda[x_0, x_1, x_2, \ldots]$ is the (graded-)commutative algebra (either over ...
- 3,526
1
vote
0
answers
67
views
Classification of Maximal Rank Skew-Symmetric Matrices with Linear forms as entries
I am interested in canonical forms/simplifying assumptions of $(2n+1)\times (2n+1)$ skew symmetric matrices with homogeneous degree 1 polynomials in $k[x,y,z]$ as entries, and whose rank is $2n$.
I ...
- 553
5
votes
0
answers
92
views
Question concerning the representation dimension of a special algebra
I would like to know, if the following problem is still open:
Let $k$ denote an algebraically closed field of characteristic 3.
Determine the representation dimension of $k(C_3\times C_3)$, where $...
- 1,755
1
vote
1
answer
154
views
A property similar to arithmetical property
By an arithmetical ring is understood a commutative ring $R$ with
identity for which the ideals form a distributive lattice, i.e., for which
$(I+J)\cap K=(I\cap K)+(J\cap K)$ for all ideals $I, J$ ...
- 21
1
vote
2
answers
134
views
What would be a standard reference for the formula of the discriminant of $f(t^d)$?
I've posted this to Math.SE about a month ago:
Seems like
$$
\Delta(a_0+a_1t^d+a_2t^{2d}+...+a_nt^{nd})=(-1)^{n\frac{d(d-1)}2}d^{nd}(a_0a_n)^{d-1}[\Delta(a_0+a_1t+a_2t^2+...+a_nt^n)]^d,
$$
where $\...
- 18.4k
4
votes
0
answers
213
views
Reference request: Formal Existence for stacks
Is there a formal existence Theorem for coherent sheaves on locally Noetherian Artin Stacks, in the spirit of Grothendieck's Formal GAGA?
Is it available for more general stacks?
user122630
2
votes
1
answer
168
views
Approximating finite type algebras over a formal power series ring
Let $k$ be a ring, let $A := k[x_{1},\dotsc,x_{d}]$ be the polynomial ring and let $A^{\wedge} := k[[x_{1},\dotsc,x_{d}]]$ be the formal power series ring. For a $d$-tuple $\mathbf{e} = (e_{1},\dotsc,...
- 2,017
8
votes
1
answer
683
views
Is the strict henselization isomorphic to the filtered colimit of finite etale algebras?
Let $(A,\mathfrak{m})$ be a local ring, and let $A^{\mathrm{sh}}$ be the strict henselization of $A$ at $\mathfrak{m}$. Let me denote $A^{\mathrm{sh},\mathrm{fin}}$ for the filtered colimit of finite ...
- 2,017
1
vote
1
answer
338
views
when a family of curve is an affine morphism
Let $f: X\to B$ be a family of curves, i.e. $f$ is flat, surjective and of relative dimension 1. If each fiber is an affine curve, can we conclude that $f$ is an affine morphism? If it is not true, ...
- 1,457
14
votes
1
answer
419
views
Hilbert series of graded Cohen-Macaulay domains, 28 years later?
I am reading through Richard Stanley's 1990 paper "On the Hilbert Function of a Graded Cohen-Macaulay Domain" to present in a seminar. I am trying to provide a reasonable conclusion for this talk, and ...
1
vote
0
answers
75
views
Formula for the index of regularity of a generic Hilbert function
Is there an explicit formula for the index of regularity of a generic Hilbert function in two variables? (i.e., the Hilbert function of an ideal of $k[X,Y]$ generated by $r$ generic forms $f_{i}$ of ...
- 11
4
votes
0
answers
472
views
formal completion of smooth morphism
Let $f: X\to S$ be a smooth morphism of schemes. Let $p$ be a section. Let $\hat{\mathscr{O}}_{X, p(S)}$ be the completion $X$ along $p(S)$. Then I think for every point $s\in S$, there exists an ...
- 1,457
3
votes
0
answers
123
views
Frobenius stratification of imperfect fields
Suppose $k$ is a (non-perfect) field of characteristic $p$, and $Fr$, its Frobenius map. I’d appreciate comments and references on the structure of the fitration/stratification of $k$ given by the ...
user122285
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 ...
- 41
3
votes
0
answers
96
views
Luroth's theorem for Discrete valuation rings?
Luroth's theorem states that if $k$ is a field and $L$ is a field extension of $k$ such that $k \subset L \subseteq k(X)$, then $L=k(f(X))$ for some $f(X) \in k(X) $ . My question is ; is there any ...
user111524
3
votes
1
answer
451
views
Commutative algebra for the Conway games
I was reading the book On Number And Games and I have some question. In this book Conway constructed the set of "games" with a addition and a multiplication. I understand that the surreal numbers are ...
- 527
3
votes
1
answer
436
views
Stone topological Boolean algebras
I am looking for an initial reference for a theorem which is known, namely:
Theorem: A Boolean algebra $A$ admits a Stone space topology (i.e. is the underlying algebra of a Stone topological ...
- 774
3
votes
0
answers
112
views
Indecomposablity in purely inseparable extensions
Let $k$ be a field of characteristic $p$ (e.g. the separable closure of $\mathbb{F}_p(t)$) and consider the extension $k(x)/k$ where $x^p\in k$ but $x$ does not. Consider a (finitely generated) ...
- 31
11
votes
2
answers
2k
views
Idea behind Grothendieck's proof that formally smooth implies flat?
From this answer I learned that Grothendieck proved the following result.
Theorem. Every formally smooth morphism between locally noetherian schemes is flat.
The book Smoothness, Regularity, and ...
- 10.5k
16
votes
1
answer
733
views
Where was $I_x/I_x^2$ first introduced? (DG or AG)
Cotangent space appears in both differential geometry (DG) and algebraic geometry (AG).
In DG, given a smooth manifold $M$ and $x\in M$ one has an isomorphism $I_x/I_x^2 \cong T^*_xM$, where $I_x$ is ...
- 1,615
1
vote
0
answers
161
views
Projective dimension of a principal ideal
Let $R$ be a polynomial ring, $I$ a homogeneous ideal, and let $\operatorname{pd}(R/I)$ denote its projective dimension. Is there a characterization of homogeneous elements $a\in R\setminus I$ for ...
- 868
12
votes
2
answers
831
views
What is known about ideal and divisibility lattices of GCD domains and their generalizations?
The divisibility relation "$a$ divides $b$", or concisely, $a \vert b$ defined over a commutative integral domain $R$ with identity induces a partial order on the multiplicative semigroup $R/R^{\times}...
- 439
3
votes
1
answer
193
views
Definability of nilradical in the model theory of rings
I am looking for a reference dealing with the first-order definability of the nilradical of a commutative ring. The only thing I have found so far is an exercise in Wilfrid Hodges' book Model Theory (...
1
vote
1
answer
382
views
singular locus of semi-normal variety
Let X be a seminormal variety and S be the singular locus of X. Is Blow$_S X=$ the normalisation of X?
Is the singular locus given by the conductor ideal?
user111251
11
votes
1
answer
949
views
Detailed modern references for basic properties of Pfaffians over commutative rings
Pfaffians are important to algebraic combinatorics, at least.
This is to propose the making of a 'wiki' list, more modern, precise and compressed than e.g. the relevant Wikipedia page (nothing against ...
- 6,051
2
votes
0
answers
329
views
Endomorphism algebra of a coherent sheaf is locally free
What is an example of a Noetherian ring $A$ and a finitely generated $A$-module $M$ such that the endomorphism algebra $\mathrm{Hom}_{A}(M,M)$ is flat as an $A$-module but $M$ is not flat?
- 428
3
votes
0
answers
432
views
When is every submodule of a module a direct sum of indecomposable submodules?
Is there any reference for modules over a commutative ring with identity such that every submodule of them is a direct sum of indecomposable submodules? Or is there any characterization of such ...
- 41
1
vote
0
answers
113
views
Reference request. The adjunction $\hom_{CDGA}(\Omega^\bullet(A),B^\bullet)\cong\hom_{CA}(A,B^0)$
We have the adjunction
$$\hom_{CDGA}(\Omega^\bullet(A),B^\bullet)\cong\hom_{CA}(A,B^0)$$
where $CDGA$ is the category of commutative diffferential graded algebras and $CA$ is the category of ...
- 1,615
3
votes
0
answers
119
views
Finite generation of the module of invariant vector fields
Let $G$ be a linear algebraic group (not necessarily reductive) and let
$X$ be an affine variety with a regular $G$-action (everything defined over the field of complex numbers $\mathbb{C}$). Denote ...
- 413