Questions tagged [ac.commutative-algebra]
Commutative rings, modules, ideals, homological algebra, computational aspects, invariant theory, connections to algebraic geometry and combinatorics.
5,497 questions
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Moduli of finite-dimensional algebras
Let $n\geq 1$ be an integer. There is an obvious family of $n$-dimensional unital algebras parametrized by $\mathbb{C}^{n(n-1)^2}$ such that any $n$-dimensional unital algebra is isomorphic to at ...
CommunityBot
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Cohomology with supports of dualizing sheaf
Let $Z$ be a closed subvariety of $X$ and $T_{Z/X}$ the relative tangent complex. Possibly I want to assume that $Z \hookrightarrow X$ is a regular embedding, so that $T_{Z/X}$ is just a shift of the ...
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Norm forms, slicing, and ideal classes
Let $K$ be a number field, which we may suppose satisfies $n = [K : \mathbb{Q}] \geq 3$. Let $\mathcal{O}_K$ be the ring of integers of $K$, and let $\{\omega_1, \cdots, \omega_{n}\}$ be a basis of $\...
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The isomorphism class of $\mathrm{Ext}^1_\mathbb{Z}(\mathbb{R}/\mathbb{Z},\mathbb{Z})$
In a recent conversation with a colleague, the following question arose:
What is the isomorphism class of $\mathrm{Ext}^1_\mathbb{Z}(\mathbb{R}/\mathbb{Z},\mathbb{Z})$? That is to say, what is $...
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Which abelian groups are $\varprojlim^1$ groups?
Question 1: Let $\mathcal A$ be an abelian group. Does there exist an inverse system $(A^n)_{n \in \mathbb N} = (\cdots \to A^n \to A^{n-1} \to \cdots \to A^0)$ such that $\varprojlim^1 A^\bullet \...
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Computing Gröbner basis elements of some constant degree
I'm wondering if there is any way or any special set of ideals such that there is an efficient way to compute elements of degree at most $d$ in a Gröbner basis for that ideal.
If you have any paper ...
CommunityBot
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An algebra of "integrals"
When discussing divergent integrals with people, I got curious about the following:
Is there an $\mathbb{R}$-algebra $A$ together with a map (could be defined on just a subspace)
$$\int_0^{\infty}: ...
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Algebra with a certain abelian group as the multiplicative group
Let $A$ be an abelian group. Are there an algebra $\mathfrak{X}(A)$ s.t. the multiplication group is isomorphic to A ? i.e.
$$
\mathfrak{X}(A)^{\times} \simeq A.
$$
For example, for $A=\mathbb{Z}/4\...
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Is $X$ closed in $Aut_{\mathbb{C}(t)}(\mathbb{C}(t)[x_1,\ldots, x_n])$?
Consider $\mathbb{C}$-algebras
$$A = \mathbb{C}[t][x_1,\ldots, x_n]\subset\mathbb{C}(t)[x_1,\ldots, x_n] = B$$
Group $\operatorname{Aut}_{k(t)}(k(t)[x_1,\ldots, x_n])$ carry a power series topology (...
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Example of a PID with a residue field of finite characteristic and a residue field of characteristic 0?
I understand that for any nonempty set $S$ of characteristics, there exists a PID $R$ such that the set of characteristics of residue fields of $R$ (i.e. quotients by of $R$ by maximal ideals -- I'm ...
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A problem about an unramified prime in a Galois extension
I asked this question in MathStackExchange, but I didn't receive any answer.
Let $K/\mathbb{Q}$ be a Galois extension of degree $n$, and denote its ring of integers by $\mathcal{O}_K$. Let $\mathfrak{...
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If a PID has no nonzero divisible elements, then is the same true of its finitely-generated modules?
EDIT: The question was originally about general Noetherian rings instead of PID's. Thanks to YCor for pointing out how wrong this was in the comments below (1 2 3).
Question 1: Let $R$ be a PID. ...
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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\...
CommunityBot
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Naive question on tensor product
Let $A, B$ be $\mathbb{C}$-algebras, which are also integral domains. Suppose there is an injective ring homomorphism $f:A \to B$. Assume further than $f$ is a finite morphism in the sense that $f$ ...
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Is every regular local ring a filtered colimit of essentially finitely generated regular local rings?
Is every regular local ring $R$ a filtered colimit of regular local rings which are essentially of finite type over $\mathbb{Z}$ (i.e. localizations of finitely generated rings)?
For comparison, ...
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flatness and reduction
Let $\mathcal J$ be an ideal sheaf on a (Noetherian) $Y$-scheme $X$, and let $\mathcal I$ be the unique primary ideal in a primary decomposition $\mathcal J$ corresponding to a minimal associated ...
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Kernel of the map $\mathbb{C}[G]^U \to \mathbb{C}[U^+]$
$\DeclareMathOperator{\SL}{\operatorname{SL}}$Let $G=\SL_k$ be the special linear group, $U$ the unipotent subgroup consisting of all lower unipotent triangular matrices, $U^+$ the unipotent subgroup ...
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Expressing primes $p\equiv 1 \pmod 3$ in the form $p = x^2 + xy + y^2$
Fermat famously showed that the only primes $p$ of the form $x^2 + y^2$ are the primes such that $p \equiv 1 \mod{4}$. Furthermore, we now know “effective” versions of Fermat's theorem, i.e. given a ...
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Reference request: additive basis of $\mathbb{C}[N]$
Let $N$ be the maximal unipotent subgroup of $SL_k$. I think that the following is an additive basis of $\mathbb{C}[N]$:
$$\{ e_T: T \text{ is a semi-standard Young tableau with at most $k-1$ rows and ...
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Rlim versus tensor product
Let $R$ be a coherent ring, and let $(M_n)_{n\geq 1}$ and $(N_n)_{n\geq 1}$ be two inverse systems of finitely generated flat $R$-modules. If $R^1 \lim M_n=R^1 \lim N_n = 0$, is it true also that $R^1 ...
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An ideal invariant under an automorphism
The following question appears here; hopefully, it is appropriate for MO.
Let $k$ be a field of characteristic zero, and let $\beta: k[x,y] \to k[x,y]$ be the following involution $\beta: (x,y) \...
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SOS polynomials with rational coefficients
Suppose we are given a univariate polynomial with rational coefficients, $p \in \Bbb Q [x]$, and are told that $p$ can be expressed as the sum of $k$ squares of polynomials with rational coefficients. ...
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Maximum number of common zeros of n polynomials in n-1 variables
Given $n$ quadratic polynomials in $n-1$ variables over the complex field, what is the maximum number of common zeros? Can we have $2^{n-1}-1$ common zeros? I assume that a linear combination of the ...
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What are the irreps in this canonical action of $\operatorname{PGL}_2(F_q)$?
Consider the permutation action of $\operatorname{PGL}_2(\mathbb F_q)$ on $\mathbb P^1(\mathbb F_q)$ by fractional linear transformations. We can consider the associated (complex) representation of ...
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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 ...
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$\operatorname{SL}_2(k)$ invariant polynomials in $k[x_1,x_2,y_1,y_2]$
Let $k$ be a field and let $\operatorname{SL}_2(k)$ act on $k[x_1,x_2]$ and $k[y_1,y_2]$ in the usual ways. These actions induce an action on the tensor product $k[x_1,x_2,y_1,y_2]$ that preserves ...
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An example of a special $1$-dimensional non-Noetherian valuation domain
I am looking for a $1$-dimensional non-Noetherian valuation domain $R$ such that there exists a sequence $\{a_i\}_{i=1}^\infty$ of elements of $R$ such that $\langle a_1\rangle \subsetneqq\langle a_2\...
CommunityBot
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Existence theorem for symmetric nondegenerate forms over a ring
There exists a rich theory for inner product spaces (i.e. vector spaces with a symmetric nondegenerate bilinear form) over fields, and it can be discussed in the context of local rings and free ...
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Solutions to nonhomogeneous quadratic equation mod $N$
Is there any way to find non-trivial solutions to the equation $x^2 + y^2 - x \equiv 0 \mod{N}$? There are clearly several trivial solutions, for example $(x, y) = (0, 0), (1, 0), (2^{-1}, 2^{-1}), (2^...
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Cohomology of the complement of the resonance hyperplane arrangement
Here was a question about resonance arrangement. It is defined as follows.
Let $x_i$ be the standard coordinates on $\mathbb{C}^n$. For each nonempty $I\subseteq\{1,\dots,n\}$, define the hyperplane $...
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Properties preserved in addition of ideals
If I and J are prime (radical) ideals then what are the conditions under which we can define a prime (radical) ideal from I+J?
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Separable non-flat simple ring extension
Let $R \subseteq S$ be two commutative $\mathbb{C}$-algebras such that:
(1) $R$ and $S$ are integral domains.
(2) $Q(R)=Q(S)$, namely, their fields of fractions are equal.
(3) $S=R[w]$, for some $w \...
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non constant regular function derivative is zero
Let $R$ be a Noetherian regular $k$-algebra (where $k$ any field of char = 0) of dimension greater than 0. Is it true that $H^{0}_{dR}(R \lvert k) = k$?.
More generally we could ask, Is it true that $...
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When is the cotangent complex perfect?
Let $X\rightarrow S$ be a proper flat morphism of schemes.
When is the cotangent complex $L_{X/S}$ perfect ?
It is well known, that for local complete intersections the cotangent complex is perfect, ...
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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 $Ext_R^i(I,R)$...
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$k[h(x),y] \subseteq k[h(x),y] + \langle h(x),y \rangle \subseteq k[x,y]$
Let $k$ be a field of characteristic zero.
Let $h=h(T) \in k[T]$ with $\deg(h)=d \geq 2$ and $h(0)=0$ (namely, $h$ has zero constant term).
Consider the following chain of $k$-algebras:
$$k \subseteq ...
- 12.9k
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Are quotients of polynomial rings almost UFDs?
If $K$ is a field then the polynomial ring $K[x_1,\ldots, x_n]$ is a UFD. On the other hand, quotients of such a polynomial ring usually don't enjoy unique factorization: consider, for instance, $\...
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Scheme-theoretic image and delta-invariants
Let $(X,o)$ be an affine, isolated, normal, Gorenstein singularity. Let $f$ and $g$ be two morphisms from $\mbox{Spec}(\mathbb{C}[[t]])$ to $X$ (also known as formal arcs) such that the closed point ...
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Question about Zariski cancelation problem
If $\mathbb{Q}[t_1,\ldots, t_n] = A[x_1,\ldots, x_{n-1}]$ has it been proven that $A\cong\mathbb{Q}[t]$?
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A question on $SK_1$ of rings
Let $B$ be a commutative ring with unity and $B/nil(B):=B_{red}$, where $nil(B)$ is the nilradical of $B$. Is $SK_1(B)=SK_1(B_{red}) ?$ In particular, is it true when $B$ is an affine algebra over an ...
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A certain property of integral domains $A \subseteq B$ with $Q(A) \cap B= A$
I have asked the following question in MSE:
Let $k$ be a field of characteristic zero. Let $A \subseteq B$ be $k$-algebras which are also (commutative) integral domains with fields of fractions $Q(A) \...
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What would be the quotient groups $U_{\mathrm{gen}}/U_{\mathrm{gen}}^{(n)}$ and $U_{\mathrm{gen}}^{(n)}/U_{\mathrm{gen}}^{(n+1)}$?
Let $K \supseteq \mathbb{Q}_p$ be a $p$-adic field with ring of integer $O$ and maximal ideal $m$. Let $O^*$ be the group of units in $O$.
Consider the group of units $U^{(0)}=U=O^*$ and $U^{(n)}=1+m^...
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Irreducibility of a palindromic polynomial
I have strong reasons to believe that the palindromic polynomial $p_n(x)$ defined by
$$p_n(x) = x^{2n}+2x^{2n-1}+3x^{2n-2}+ \cdots+ nx^{n+1}+(n+3)x^{n}+nx^{n-1}+\cdots+2x+1$$
is irreducible in $\...
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0
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A composition of a simple extension and a separable extension is simple
Let $K/L/M$ be a tower of finite field extensions with $K/L$ separable and $L/M$ simple (in the sense of being generated by a single element). How does one show that $K/M$ is also simple?
I know that ...
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formal smoothness and cotangent complex
If $k$ is a field and $A$ is a formally smooth $k$-algebra, then we know that $\Omega^{1}_{A/k}$ is projective. What about its cotangent complex $L_{A/k}$? When is it quasi-isomorphic to $\Omega^{1}_{...
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Self-map of short exact sequences
Consider the commutative diagram of finite abelian groups
$\require{AMScd}$
\begin{CD}
0@>>> A @>i>> B@>\pi>> C@>>> 0\\
\ @VV 0 V@VVfV@VV 0 V\\
0@>>>A @&...
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Nonvanishing criterion for a polynomial of polynomials
Let $P \in \mathbb{F}_q[x_1, \dots, x_n][y]$ be some fixed polynomial. Substituting polynomials for the variables in $P$ gives a map $\mathbb{F}_q[x]^{n+1} \to \mathbb{F}_q[x]$ defined by
$$(Q_1(x), \...
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1
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Necessary and sufficient condition for $can : A^X\otimes_A A^Y\rightarrow A^{X\times Y}$ to be an embedding
The two sets are, of course, supposed infinite.
This question is related to that one
Commutation of tensor products with inverse limits in a specific case
where it received a (partial) answer ($A$ ...
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Which theorems in commutative algebra describe the closed property of curves (i.e. algebraic varieties) in algebraic geometry?
In $R^2$, we have those solutions of $x^2+y^2-1=0$ describe a closed unit disk and $y^2 = x^3 − x + 1$ describes an unclosed elliptic curve. When we consider corresponding algebras e.g. ${R^2}/ \...
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In char zero $ \operatorname{Cox}(\operatorname{Bl}_{[1:1:1]}(\mathbb{P}(a:b:c))) $ is finitely generated, but not in char p. How?
Let $ X(a_{1}:a_{2}:a_{3}) $ be the blow-up of $ \mathbb{P}(a_{1}:a_{2}:a_{3}) $ at $ [1:1:1] $, the identity of the torus. In Steven Dale Cutkosky's paper Symbolic Algebras of Monomial Primes ...
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