Questions tagged [ac.commutative-algebra]
Commutative rings, modules, ideals, homological algebra, computational aspects, invariant theory, connections to algebraic geometry and combinatorics.
5,337
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local cohomology of Buchsbaum ring
Let $(R,m)$ be a Buchsbaum ring of dimension d. Can we say that $d$-th local cohomology $H_{m}^d(R)$ has finite length?
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Polynomial analogue of "prime independence"
In number theory a well-known fact is that congruence modulo distinct primes are 'independent'. That is, to know that $n \equiv a \pmod{p}$ does not change the probability as to what $n \equiv x \pmod{...
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Decomposition of a quotient module
Let $R=k[v,x,y,z]/I$, with $I=\langle v^2,z^2,xy,vx+xz,vy+yz,vx+y^2,vy-x^2\rangle$,and let
$f:R^2 \rightarrow R^2$ denote the map given by the matrix
$$M=\begin{pmatrix}
v & y \\
x & z
\end{...
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Morphsim from F_p[[X_1,...,X_d]].
Let f: F_p[[X_1,...,X_d]] --->> R be a surjection from a power series ring. Also assume that there is another surjection g: F_p[[Y_1,...,Y_d]] --->> R to the same local ring R.
Question: Is it ...
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analytically isomorphic singularities
Two plane curves $X,Y$, defined by polynomials $f(x,y)=0$ and $g(x,y)=0$,are analytically isomorphic(at the origin). i.e., the complete local rings $k[[x,y]]/(f)$ and $k[[x,y]]/(g)$ are isomorphic.
...
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local complete intersection
The following is an Exercise 1.1.11 of Hartshorne's Algebraic Geometry.
Let $Y\subset \mathbb{A}^3$ be the curve given parametrically by $x=t^3, y=t^4, z=t^5$. Show that $I(Y)$ is a prime ideal of ...
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An example of a ring $R$ with the property that for each $P=Ann_R(r)\in {\rm Min}(R)$ we have $Ann_R(P)=Rr$.
I'm looking for an example of a commutative (preferably local) ring $R$ such that ${\rm dim}R>0$ and $R$ has the property that for each $P=Ann_R(r)\in {\rm Min}(R)$ we have $Ann_R(P)=Rr$.
This ...
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Representations over $\mathbb{Z}_p$
Hi,
I would like to work with indecomposable representations over a commutative ring and start with $R=\mathbb{Z}_p$
Notations: $G$ a finite group, $S$ a subgroup, $R=\mathbb{Z}_p$ the p-adic ring, ...
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Finitely Generated Commutative Z-algebra
Let $R$ be a commutative finitely generated $\mathbb{Z}$-algebra. Then the nilradical is equal to the Jacobson radical.
I am not able to make much traction on this, nor can I find this result in any ...
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minimal number of generators for the square of an ideal
If $R$ is a local Noetherian regular ring and $I$ is an ideal contained in the maximal ideal.
Can we compare the number of minimal set of generator of $I$ and $I^2$?
Thanks a lot for helping me.
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Detecting and counting free direct summands
If $M$ is a finitely generated module over a local ring $(R, \mathfrak{m})$, we can detect whether $M$ has a nonzero free direct summand as follows: Consider the natural map
$$\phi_M\colon \mathrm{Hom}...
- 5,776
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Does the global dimension gldim R equal the projective dimension of R as bimodule over its enveloping algebra?
I know that generally the answer is no, for example the weyl algebra。
But is this true for commutative algebra？ or we may restrict to affine commutative algebras。
Maybe ，it is a classical result. So,...
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fixed point scheme in caracteristic p
Let X\rightarrow A^{n} a smooth affine scheme over an affine space. Everything is defined over a field k.
Let G a finite group acting on X and suppose that his order is divisible by the caracteristic ...
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Pullbacks and Inclusions of Smooth function algebras of manifolds.
Let $M$ and $N$ be two smooth finite dimensional manifolds and
$C^\infty(M)$ as well as $C^\infty(N)$ their smooth function algebras.
Is the following true:
Let $\imath: M \to N$ be an embedding. ...
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tensor of powers of an ideal
Let $I$ be an ideal of a Noetherian ring $R$. $M$ is a finitely generated $R$-module. I question is:
Does there exist $n_0$ such that for all $n \geq n_0$, the short exact sequence
$$I^n/I^{n+1} \...
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Prime ideals in coordinate rings
Is there a way to characterise prime ideals in affine coordinate rings (i.e. quotients of polynomial rings). To be more specific, how can I say if principal ideals in such rings are prime or not in an ...
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Chain of ideals in a complex algebra
Suppose $\mathfrak{A}$ is an unital algebra over complex numbers and $\mathfrak{J}$ is chain of left-ideals in $\mathfrak{A}$ ordered by inclusion such that none of its elements is countably generated....
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A small question on commutative algebra
Recently I read the book Intersection Theory by Fulton. I think one property in his book relies on this commutative algebra conclusion. I'm not sure whether it is right.
Assume all rings are of finite ...
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$\Phi: Hom_R(A,B) \to Hom_R(A,R)\otimes_R B$
Let $R$ be a commutative ring and $A$ and $B$ two $R$-module. Suppose that $A$ is free of rank $n$ with basis $a_1,\dots,a_n$. Then there is an isomorphism $\Phi: Hom_R(A,B) \to Hom_R(A,R)\otimes_R B$ ...
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Cohen Macaulay, free and finitely generated module
Here is an unsolved problem for me in Kaplansky's "Commutative rings" p. 103, no. 18.
Let $R$ be a Cohen-Macaulay ring. Let $T$ be a ring containing $R$ and suppose that as an $R$-module it is free ...
- 1,651
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Almost clean module
Please give me an example of an almost clean module $M$ over a ring $S$ so that if $x$ is a $M$ regular element then $M/xM$ is not almost clean.
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A problem on Moebius transformations
We have the following result:
Let $R=\mathbb{C}[t]_f$, with $f=(t-a_1)(t-a_2)\cdots (t-a_n)$. Then the automorphism group of $R$ is isomorphic to the group of all Moebius transformations which fix (...
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Are pullbacks from a factor of a product scheme flat over the other factor?
Given two smooth projective surfaces $X$ and $Y$ over some algebraically closed field.
Given a torsion free coherent sheaf $M$ on $X$. One has the projections $\pi_X$ and $\pi_Y$ from the product $X\...
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Why is multiplication with a scalar no global morphism?
Given a smooth projective surface $S$ over an algebraically closed field, a sheaf rings or algebras $R$ on $S$ and a simple left $R$-module $M$, i.e. $Hom_R(M,M)=k$.Then we have $Hom_R(M,M(-i))=H^{0}(...
- 1,391
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Morphisms of a simple sheaf over an algebra to its double dual
Given a smooth and projective surface $S$ over an algebraically closed field $k$ and a sheaf of Azumaya algebras $R$, i.e. $R$ is a locally free $O_S$-module of finite rank. Let $M$ be a coherent and ...
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Non-representability by a binary quadratic form
Let $k$ be an arbitrary field, $d\in k$, and $d$ is not a square in $k$.
Consider the binary quadratic form $f(x,y)=x^2-d y^2$
(it is the norm from $k(\sqrt{d})$ to $k$).
I am looking for a reference ...
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The structure of the module of Kähler differentials of R[[x]] over R
It seems that $\Omega_{R[[x]]/R}^1$ is rather big. For example, take $R$ to be the rational numbers.
We can see that $\Omega_{R((x))/R}^1$ is a $R((x))$-vector space of infinite rank. As by results ...
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Unique factorization in polynomial rings
Everybody knows that polynomial rings over fields have unique factorization, and that if $R$ has unique factorization, then so does $R[X]$. And everybody knows who proved these results first.
Well, ...
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Particular example of a quadratic extension of a nonunital ring
I want to construct a concrete non-unital ring $R$ with the following properties:
$R$ is a noncommutative non-unital ring with a right unite $r$ i.e $t.r=t$ for any $t\in R$.
$S\subset R$ is a ...
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Perturbing pole of Laurent polynomial/series in a single summand
I am working with the ring of Laurent polynomials $\mathbb{F}[X,X^{-1}]$ over $\mathbb{F}$ for some algebraically closed field $\mathbb{F}$ of any characteristic. I encountered a problem emerging from ...
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Zero divisors in the boolean polynomial ring $\mathbb{F}_2[x_1,x_2,...,x_n]/(x_1^2-x_1,x_2^2-x_2,...x_n^2-x_n)$
Related to this question.
Let $n$ be positive integer and let $K$ be the boolean polynomial ring
$\mathbb{F}_2[x_1,x_2,...,x_n]/(x_1^2-x_1,x_2^2-x_2,...x_n^2-x_n)$.
For all $a$ in $K$ we have $a= -a$ ...
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Degree three, codimension one subvarieties lying on a quadratic hypersurface
Let $H$ be an irreducible hypersurface in $\mathbb P^n$ of large-ish degree, say 14. This question is about subvarieties $V$ of $H$ such that
$V$ has codimension 1 in $H$ (i.e. $V$ has dimension $n-2$...
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Embedding noetherian domains in a PID with finite index
The starting point of this post is the following question:
Embedding number fields in fields with class number 1
It is shown that in the answers that , given an number field $K$, we cannot necessarily ...
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On "minimal presentation" of local rings essentially of finite type over a field
Let $k$ be a field of characteristic $0$. Let $(R,\mathfrak m)$ be a local ring essentially of finite type over $k$ (https://stacks.math.columbia.edu/tag/07DR). Then, $R$ is the homomorphic image of ...
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The map from the ring of integers to the residue field of a valuation subring is surjective
Let $L$ be a number field and let ${\mathcal{O}}_L$ be its ring of integers. Let $B$ be a valuation subring of $L$ and let $k_B$ be the residue field of $B$. Then the map from ${\mathcal{O}}_L$ to $...
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Finitely generated module, which is a virtually small complex, embeds into a module of finite projective dimension?
Let $R$ be a commutative Noetherian ring, and let $\text{mod } R$ denote the abelian category of finitely generated $R$-module. Consider the bounded derived category $D^b(\text{mod } R) $ which is a ...
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A question on curves on effective divisors
Let $X$ be a smooth projective variety over $\mathbb{C}$ with $\dim X=3$ and $\mathrm{Pic}(X)=\mathbb{Z}\cdot D$, where $D$ is a very ample effective Cartier divisor on $X$. Let $Z$ and $C$ be two ...
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Duogenic quartic rings
Recall that a commutative, unital ring $R$ of finite rank which is isomorphic to $\mathbb{Z}^n$ for some $n \geq 1$ as $\mathbb{Z}$-module is said to be monogenic if there exists an element $\gamma \...
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Flatness criterion for $I$-adic ring: $I$-torsion free
Let $R$ be an $I$-adically separated and complete valuation ring, with $I$ finitely generated.
It is used a few times in Bosch, Lectures on Formal and Rigid Geometry e.g. first lines of pg. 164, Cor. ...
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Symbolic power of an ideal associated to non-singular algebraic set
Let $Z\subset \mathbb P^n$ be a reduced non-singular algebraic set and $I$ denote the saturated homogeneous ideal of $Z$. I have seen the following result without proof:
For all $ n\geq 1$, $I^{(n)}=(...
- 1,703
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Open affine subscheme of a direct limit of smooth algebras
Let $R$ be (assumed to be commutative, Noetherian) a regular local ring. Let $A$ be a direct limit of $R$-smooth algebras, such that the transition maps are $R$-étale.
Let $U= Spec(B)$ be an affine ...
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Height of truncated system of parameter
Let $R$ be a Noetherian local ring of dimension $d$, and $a_1,\dots,a_d$ is a system of parameters. I am wondering whether the following statement is true:
$\mathrm{ht}(a_1,\dots,a_i)=i$ for all $i$, ...
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Are there algebras over reals besides complex numbers, where identities, analoguous to $(-1)^i=e^{-\pi}$ and $i^i=e^{-\pi/2}$ hold?
Are there algebras over real numbers (with exponentiation), such that there is such $z$ that does not include components in $\mathbb{C}$ (or in a subset isomorphic to $\mathbb{C}$), for which $(-1)^z\...
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Creating prime ideals in rings
Are there different ways to create prime ideals in a ring other than taking quotients? I recently came across a construction of a prime ideal in a Noetherian ring $A$ given in the book on Algebraic ...
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Automorphisms of the ring of Laurent polynomials
Is the group of automorphisms of the ring $\mathbb{F}[t,t^{-1}]$ of Laurent polynomials known? Here, $\mathbb{F}$ is an algebraically closed field of characteristic $0$ and I am considering not ...
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Krull dimension and elimination theory over the integers
Let $K:=\mathbb{C}$, and let $R:=K[x_1,\dots , x_n]$.
Then, a system of polynomial equations $p_1=0, p_2=0, \dots , p_r = 0$, where the $p_i$ are polynomials in the $x_j$, has finitely many solutions $...
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Finiteness of the integral closure of an integral domain in its field of fractions
I've just started reading John Milne's book ''Etale Cohomology". Prop. 1.1 of Sec 1 in Ch1 reads as
follows: If $X$ is a normal scheme and $X'\to X$ is its normalization in a certain finite ...
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A question about Dedekind schemes and proper morphisms
The following is Exercise 15.3 of Görtz-Wedhorn Algebraic Geometry I:
Let $S$ be a Dedekind scheme with function field $K$ and let $f: X\to S$ be a proper morphism of schemes. Then the canonical map $...
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143
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Geometric meaning of colocalization of modules?
Let $A$ be a commutative ring and $S\subset A$ a subset. A localization of $A$ at $S$ is defined as a ring morphsim $A\to A[S^{-1}]$ which is initial with respect to inverting $S$. Similarly, a ...
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Positive integer solutions of linear equations under the constraint of Frobenius number
Let $a,b,c$ be three pairwise coprime positive integers, and $\Gamma=\langle a,b,c\rangle$ be the corresponding numerical semigroup. Consider the linear equations:
$n_1a=m_{12}b+m_{13}c$
$n_2b=m_{...
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