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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
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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 $\...
SARTHAK GUPTA's user avatar
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. (*) ...
Dattier's user avatar
  • 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. ...
Jason McCullough's user avatar
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{...
Sebastian Pozo's user avatar
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 ...
Dima Roytenberg's user avatar
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 &...
Junyan Xu's user avatar
  • 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[...
Justin Bloom's user avatar
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 ...
Sasha Kucherenko's user avatar
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 ...
Zuka's user avatar
  • 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{...
Zuka's user avatar
  • 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\...
Zuka's user avatar
  • 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 \...
user47660's user avatar
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 $a_i$. ...
A. Gupta's user avatar
  • 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 ...
DVL-WakeUp's user avatar
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 \...
user267839's user avatar
  • 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 $s \in \mathbb{N}$, ...
QZ2025's user avatar
  • 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 ...
Rustam T's user avatar
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 ...
Jz Pan's user avatar
  • 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 ...
Serge the Toaster's user avatar