Questions tagged [ac.commutative-algebra]
Commutative rings, modules, ideals, homological algebra, computational aspects, invariant theory, connections to algebraic geometry and combinatorics.
19 questions from the last 30 days
6
votes
1
answer
633
views
Is decomposability of polynomials over a field an undecidable problem?
By a decomposition of a polynomial $F(x)$ over a field $K$ we mean writing $F(x)$ as
$$
F(x)=G(H(x)) \quad(G(x), H(x) \in K[x]),
$$
which is nontrivial if $\operatorname{deg} G(x)>1$ and $\...
- 310
5
votes
1
answer
322
views
Non-negative coefficients polynomials
Let $n \in \mathbb N$ and $P,Q \in \mathbb R_+[x]$.
Is it true that $(x+1)^n\neq (x-2)^2 \times P(x)+(x-4)^2 \times Q(x)$ ?
I have asked, this question here (*), two weeks ago, but no answers.
(*) ...
- 4,074
5
votes
1
answer
267
views
UFD property for power series in characteristic 0
Samuel famously produced an example of a UFD, namely $S = R_{(x,y,z)}$, where $R = K[x,y,z]/(x^2+y^3+z^7)$ and $K$ has characteristic 2, such that the power series extension $S[[ x ]]$ is not a UFD. ...
4
votes
1
answer
227
views
Literature Request: The derived category is Krull-Schmidt
I am looking for literature where it is proven that the derived category of bounded complexes over a finite-dimensional algebra is Krull-Schmidt. I found this question
Literature request: $K^b(\text{...
9
votes
1
answer
300
views
What are the points of the algebra of polynomial functions on an arbitrary vector space?
Let $V$ be an arbitrary vector space over some field $\mathbb{K}$ (UPD: of characteristic 0), $V^*=\mathrm{Hom}(V,\mathbb{K})$ its linear dual. Let $\mathrm{Sym}_\mathbb{K}(V^*)$ be the free ...
- 480
4
votes
1
answer
357
views
When do algebraic elements form a subalgebra?
If $R$ is a commutative ring and $A$ is a commutative $R$-algebra, we say that an element $x\in A$ is algebraic over $R$ if $x$ is a root of a nonzero polynomial $f \in R[X]$, or equivalently, if the &...
- 844
2
votes
1
answer
144
views
Exotic Hopf algebra structures on the $p$-fold direct product in characteristic $p > 0$
Let $k$ be an algebraically closed field of characteristic $p > 0 $ and let $A$ be an algebra over $k$, which is a local ring.
There is an isomorphism of algebras $\prod_{i=1}^p A \cong A \otimes k[...
- 542
3
votes
1
answer
222
views
Finite generativity of algebra with valuation
Let $C$ be a commutative finitely generated algebra with no zero divisors. If necessary, we can assume it to be graded and a unique factorization domain. Let $a\in C$ be a prime element.
Let's also ...
5
votes
1
answer
131
views
Relation between Tor amplitude and $p$-complete Tor amplitude for a ring of characteristic $p$
Fix a prime number $p$. Let $A$ be a commutative ring, and consider an $A$-algbera $B$ of characteristic $p$. So we have a sequence of ring homomorphisms
$$
A \to A/pA \to B.
$$
Assume that we want to ...
- 125
1
vote
1
answer
214
views
Derived completeness of the inverse perfection
Fix a prime number $p$, and let $R$ be a ring of positive characteristic $p$. Consider the inverse perfection of $R$, which is defined as the inverse limit
$$
R^\flat = \varprojlim(\cdots \xrightarrow{...
- 125
2
votes
0
answers
122
views
Derived tensor products and regular sequences
Let $R \to A$ be a homomorphism of commutative rings, and let $x\in R$ be an element (or a sequence of elements in $R$, if you prefer) that is both $R$-regular and $A$-regular. Then we have
$$
A\...
- 125
2
votes
0
answers
125
views
Generalization of Koszul cohomology for wedge product with a fixed q-form: literature and references?
Let $A$ be a commutative ring. For an odd positive integer $q$ and an integer $p$ in the range $1 \leq p \leq q$, consider the cochain complex:
$0 \to \Lambda^p(A^N) \to \Lambda^{p+q}(A^N) \to \cdots \...
- 21
2
votes
0
answers
88
views
Action of torus on Laurent polynomials
Let $F$ be a field and suppose that the torus $(F^*)^n$ acts on the Laurent polynomial ring $L$ in $n$ variables $X_1, \dots, X_n$ defined by $X_i \dashrightarrow a_iX_i$ for suitable scalars $a_i$.
...
- 376
2
votes
0
answers
75
views
Connectedness of equivariant Hilbert schemes of points of affine spaces (or as orbifolds)?
Let $G$ be an abelian finite group act on $\mathbb C^n$, when the equivariant Hilbert scheme $\mathrm{Hilb}^{R}(\mathbb C^n)^G=\mathrm{Hilb}^{R}([\mathbb C^n/G])$ is connected? Now $R$ is a ...
- 249
1
vote
0
answers
69
views
Descent of $G$-invariant formal system of parameters using GAGF
Let $R=(R,\mathfrak{m})$ be a comm local regular ring of char $\neq 2$ (ie $2 \neq 0$ in $R$) with maximal ideal $\mathfrak{m}$ of (Krull) dimension $2$, ie $R$ admits system of parameters $x,y \in \...
- 6,038
1
vote
0
answers
98
views
Cohen-Macaulayness of the homogeneous coordinate ring of projective monomial curves
Let $A = \{a_0, a_1, \ldots, a_{n-1}\} \subset \mathbb{N}$ be a set of non-negative integers where we assume that $a_0 < \cdots < a_{n-1}$ and set $d := a_{n-1}$. For every $s \in \mathbb{N}$, ...
- 21
1
vote
0
answers
28
views
Primary invariants on MAGMA for a graded ring
I have asked this question on mathstacks, but a collegue of mine recommended me to post it here.
I am trying to find an optimal system of parameters for a graded ring using Magma. Specifically, I want ...
- 11
2
votes
0
answers
59
views
Tensor product of two transcendental flat algebras is not a field?
I'm considering the correctness of the following assertion, which is related to linear disjointness (I'm trying to generalize it to subalgebras), What does "linearly disjoint" mean for ...
- 173
0
votes
0
answers
94
views
Length of generic intersection in local ring
Let $(R, \mathfrak{m})$ be a regular local ring and let $I\subset R$ be an ideal of coheight 1. Let $a \in \mathfrak{m}\setminus \mathfrak{m}^2$.
If $a$ that is not a zero divisor of $R/I$ we have ...
