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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
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
0 answers
274 views

What is known about vector subspaces of polynomial rings closed under factors?

Let $R$ be a commutative ring. Call a nonempty subset $F$ of $R$ a factroid if it is closed under sums and factors. That is: If $a,b \in F$, then $a+b \in F$, and If $a,b \in R$ with $a\in R$ ...
Neil Epstein's user avatar
  • 1,802
2 votes
0 answers
66 views

Projective cover (minimal) for (derived)complete modules over Noetherian local rings exist?

Let $(R,\mathfrak m)$ be a commutative Noetherian local ring. Let $M$ be an $R$-module which is $\mathfrak m$-adically derived complete. Then, does there exist a free $R$-module $F$ and a surjective $...
uno's user avatar
  • 412
5 votes
1 answer
151 views

Dimension from Hilbert series with variable-weighted grading?

Suppose $R = F[x_1,...,x_n]/I$ is a finitely generated ring over the field $F$, and suppose there is a function $d:[n] \to \mathbb{Z}$ such that defining the degree of a monomial $x_1^{e_1} ... x_n^{...
Joshua Grochow's user avatar
2 votes
1 answer
232 views

Reference for surjectivity of the canonical map $R^{G_1} \otimes R^{G_2} \to R^{G_1 \cap G_2}$

Let $R$ be a commutative ring, $G$ a finite group with an action over $R$. Let $G_1, G_2 \subset G$ be two subgroups. Then the canonical map $R^{G_1} \otimes R^{G_2} \to R^{G_1 \cap G_2}$ is ...
Zheming Xu's user avatar
1 vote
1 answer
92 views

On analytic transcendence degree and Krull dimension for homomorphic images of power series rings

Let $k$ be a field of characteristic zero and $I$ be a radical ideal of $k[[x_1,\ldots, x_n]]$. Let $P$ be a minimal prime ideal of the reduced ring $R:=k[[x_1,\ldots, x_n]]/I$. Then, $R_P$ is a field ...
Alex's user avatar
  • 480
1 vote
1 answer
88 views

Transcendence degree and Krull dimension for homomorphic images of power series rings

Let $k$ be a field of characteristic zero and $I$ be a radical ideal of $k[[x_1,\ldots, x_n]]$. Let $P$ be a minimal prime ideal of the reduced ring $R:=k[[x_1,\ldots, x_n]]/I$. Then, $R_P$ is a field ...
Alex's user avatar
  • 480
2 votes
1 answer
112 views

Example of non injective module over Noetherian local ring with trivial vanishing against residue field?

Is there an example of a module $M$ over a commutative Noetherian local ring $(R,\mathfrak m, k)$ such that $\text{Ext}_R^{>0}(k, M)=0$ but $M$ is not an injective $R$-module? I know that for such ...
Alex's user avatar
  • 480
2 votes
0 answers
93 views

Minimal injective resolution and change of rings

Let $R$ be a commutative Noetherian ring. For an $R$-module $M$, let $0\to E^0_R(M)\to E^1_R(M)\to \ldots $ denote the minimal injective resolution of $M$. I have two questions: (1) If $I$ is an ...
Alex's user avatar
  • 480
2 votes
1 answer
205 views

Localization of quasi-excellent rings are quasi-excellent?

If $R$ is a quasi-excellent ring, then is $R_{\mathfrak p}$ also quasi-excellent for every prime ideal $\mathfrak p$ of $R$ ? I think Matsumura's commutative ring theory book says that localization of ...
Alex's user avatar
  • 480
4 votes
1 answer
685 views

Who and when proved Artin's Theorem on alternative rings?

I am interested in the history of the proof of Artin's Theorem (on the diassociativity of alternative rings). Question. When has Artin proved this theorem and where was it published for the first ...
Taras Banakh's user avatar
  • 41.8k
1 vote
0 answers
142 views

Explicit computation of Čech-cohomology of coherent sheaves on $\mathbb{P}^n_A$

$\newcommand{\proj}[1]{\operatorname{proj}(#1)} \newcommand{\PSP}{\mathbb{P}}$These days I noticed the following result of (constructive) commutative algebra, which I think is probably well known ...
Jürgen Böhm's user avatar
1 vote
2 answers
830 views

Books one can read for 2nd course in Commutative Algebra ( Self Study)

I am a student who has completed master's but couldn't take admission to a PhD program due to some unfortunate reasons. I have done 1 course in Commutative Algebra where I followed the book " ...
2 votes
0 answers
75 views

When is a finitely generated commutative algebra a projective module over its invariant subalgebra?

For the sake of simplicity, I will work over the complex numbers. Let $A$ be a finitely generated algebra and $G$ any finite group of algebra automorphisms. Then, by Noether's Theorem, $A^G$ is also a ...
jg1896's user avatar
  • 3,318
4 votes
0 answers
119 views

Adjoining new factors for primes in UFDs

It is well-known that if we pass from a UFD to a new ring where we have factored one of the primes, it does not need to stay a UFD. The classic example is passing from $\mathbb{Z}$ to $\mathbb{Z}[\...
Pace Nielsen's user avatar
  • 18.7k
0 votes
0 answers
57 views

Reference for packing property and König property

Can someone please suggest reference material to study about the packing property and König property of ideals and some examples?
Sowbarnika R's user avatar
5 votes
2 answers
754 views

A version of Hilbert's Nullstellensatz for real zeros

$\newcommand\R{\Bbb R}$Let $Q(x_1,\dots,x_n)\in\R[x_1,\dots,x_n]$ be an irreducible polynomial such that the dimension of the set $Z:=\{(x_1,\dots,x_n)\in\R^n\colon Q(x_1,\dots,x_n)=0\}$ (defined, say,...
Iosif Pinelis's user avatar
5 votes
0 answers
107 views

Generalized Puiseux series for diagonal reflections of the curves $y = \frac{x}{(1-ax)(1-bx)^m}$

Reflection of the curve $y = f_m(x) = \frac{x}{(1-ax)(1-bx)^m}$ through the diagonal line $y=x$ in the $xy$-plane can be regarded as local compositional inversion of the curve $y=f_m(x)$. ($x,y,a,b$ ...
Tom Copeland's user avatar
  • 10.5k
1 vote
1 answer
286 views

Unpacking the plethystic substitution $h_n[n\mathbf{z}]$ in a paper by Aval, Bergeron and Garsia

I'm not familiar with the formalism of plethysm, so I need some help in unpacking the plethystic substitution $h_n[n\mathbf{z}]$ found in eqns. 5.6 and 5.9 of "Combinatorics of labelled ...
Tom Copeland's user avatar
  • 10.5k
4 votes
0 answers
302 views

What is known about the number of elements needed to generate a given ideal in $k[X_1,\dots,X_n]$?

In Algebraic Geometry by J.S. Milne, after he proves Hilbert's Basis Theorem, he makes the following aside: One may ask how many elements are needed to generate a given a given ideal $\mathfrak a$ in ...
Joe Lamond's user avatar
0 votes
0 answers
109 views

Affine scheme over ring of meromorphic functions with finite poles on unit circle

I am looking into the set $S$ of meromorphic functions with a finite number of poles on the unit circle (i.e., rational functions with poles on the unit circle). I assume that any $h\in S$ has the ...
Jens Fischer's user avatar
1 vote
0 answers
63 views

Factorization of the symmetric function identity $E(t)=1/H(t)$ with the refined Euler characteristic polynomials of associahedra / Lagrange inversion

I've come across two matrix identities, flagged with daggers below, relating the two sets of elementary and complete homogeneous symmetric polynomials/functions via the two sets of refined Lah and ...
Tom Copeland's user avatar
  • 10.5k
1 vote
0 answers
103 views

Reference for a clear version of multigraded Serre-Grothendieck-Deligne correspondence local cohomology

The Grothendieck-Serre-Deligne correspondence states the following. Let $ R $ be a Noetherian, graded ring and let $ T $ be $ \operatorname{Proj}(R) $. If $ \mathcal{F} $ is a coherent sheaf on $ T $...
Schemer1's user avatar
  • 912
2 votes
0 answers
92 views

Are covering families of localizations stable under pushouts?

For a commutative ring $A$, we call a finite family of localizations $A \to A_{S_i}$ (where $S_i$ are some subsets of $A$) a covering if the canonical morphism $A \to \prod A_{S_i}$ is an effective ...
Arshak Aivazian's user avatar
4 votes
0 answers
95 views

List of equivalent conditions for the invariant subalgebra to be polynomial

Let $k$ be a field, $P_n$ the polynomial algebra in $n$ indeterminates, and $G<\operatorname{GL}_n$ a finite group whose order is coprime to the characteristic of $k$, and that acts on $P_n$ by ...
jg1896's user avatar
  • 3,318
3 votes
0 answers
74 views

Locally compact rings with reciprocals

A topological field is defined to be a topological ring $F$ with reciprocals such that the reciprocal function $F\setminus\{0\} \to F\setminus\{0\}$ is continuous. Locally compact topological fields ...
Andre Kornell's user avatar
4 votes
2 answers
193 views

Intersection of commutative factorial domains: completely integrally closed and Krull domain

Let $A=\bigcap_{t\in T}D_t$ be an integral domain such that $D_t$ is a commutative factorial domain for every $t\in T$. It is quite natural to see that $A$ is a completely integrally closed domain. ...
Mara Pompili's user avatar
2 votes
1 answer
365 views

Correspondence between fundamental group and geometric properties of $X$

At the time of studing some algebraic topology I was wondering about the following. Let $X$ be a topological space and $\pi_1(X)$ be its fundamental group. If we assume some algebraic property of $\...
KAK's user avatar
  • 613
1 vote
0 answers
106 views

Uniqueness of indecomposable decomposition (Krull–Schmidt) for finitely generated modules over commutative Noetherian standard graded rings

Let $R_0=\mathbb C$ and $R=\bigoplus_{i\geq 0} R_i$ be a commutative Noetherian graded ring such that the grading is standard, i.e., $R=R_0[R_1]$. Let $M$ be a finitely generated $R$-module. Evidently,...
uno's user avatar
  • 412
2 votes
0 answers
123 views

An alternative proof that Buchsbaum rings are generalized Cohen-Macaulay

Let $(R,\mathfrak{m})$ be a Noetherian local ring. $R$ is said to be Buchsbaum if, for each ideal $\mathfrak{q}$ generated by a full system of parameters, the number $\lambda_R(R/\mathfrak{q})-e_{\...
walkar's user avatar
  • 253
1 vote
1 answer
131 views

Completion of $\mathbb F_q(T)$

It is easy to prove that for a an irreducible polynomial $P$ of degree $d$ of $\mathbb F_q[T]$, one can embed $\mathbb F_{q^d}$ in $\mathbb F_q(T)_P$ (the completion of $\mathbb F_q(T)$ at $P$) and ...
joaopa's user avatar
  • 3,998
8 votes
1 answer
356 views

Homological conjectures for finite dimensional commutative algebras

$\DeclareMathOperator\Ext{Ext}\DeclareMathOperator\Hom{Hom}$>Question: What are some (open) homological conjectures that are also relevent for finite dimensional commutative algebras over a field $...
Mare's user avatar
  • 26.5k
4 votes
0 answers
92 views

Lie bracket of general unipotent matrices

Let $k$ be a field (of characteristic $0$). Let $$ X:=\begin{pmatrix}1&x_{1,2}&x_{1,3}&\cdots&x_{1,n}\\ &1&x_{2,3}&\cdots&x_{2,n}\\ &&1&\cdots&x_{3,n}\\ ...
Li Guanyu's user avatar
  • 449
6 votes
1 answer
1k views

Discovery of Hilbert polynomial

Presumably it was Hilbert who discovered Hilbert polynomials - where did they first appear? The basic theorem is that for a finitely generated graded module $M = \bigoplus_k M_k$ over the ring of ...
pinaki's user avatar
  • 5,339
5 votes
1 answer
308 views

Reference request for the group of units of a power series ring in one variable

Let $\mathbb F_p$ denote the finite prime field of $p$ elements. What reference can be recommended for an analysis of the structure of the group of units of the power series ring $\mathbb F_p[[x]]$? ...
Peter Kropholler's user avatar
2 votes
0 answers
73 views

An iterative formula for the Kreweras-Voiculescu polynomials (reference request)

Let $$N(x) = 1 + \sum_{k \ge 1} N_k(h_1,h_2,...,h_k) \;x^k$$ $$ = 1 + h_1 x + (h_1^2 + h_2) x^2 + (h_1^3 + 3h_1h_2 + h_3)x^3 + (h_1^4 + 6 h_2 h_1^2 + 4 h_3 h_1 + 2 h_2^2 + h_4) x^4 + \cdots$$ be the ...
Tom Copeland's user avatar
  • 10.5k
4 votes
0 answers
101 views

Theta functions in acyclic cluster algebras

Setup: Let $B$ be a skew-symmetrisable integer matrix and $\mathcal{A}$ be the cluster algebra with principal coefficients at a seed with mutation matrix $\tilde{B}$ whose principal part is equal to $...
Antoine de Saint Germain's user avatar
1 vote
0 answers
116 views

List of automorphism groups of low-dimensional complex commutative algebras?

Let $\mathcal{A}$ be a finite-dimensional commutative associative unital $\mathbb{C}$-algebra. I am looking for a list (of further examples of) $\operatorname{Aut}_\mathbb{C}(\mathcal{A})$, the group ...
M.G.'s user avatar
  • 7,127
4 votes
1 answer
287 views

The $1$-dimensional Jacobian Conjecture over $\mathbb{Z}$-torsion free rings

I am looking for further proofs, preferably in the literature, of the following result: Proposition: Let $R$ be a unital commutative $\mathbb{Z}$-torsion free ring. If $f(x) \in R[x]$ with $f'(x) \in ...
M.G.'s user avatar
  • 7,127
12 votes
2 answers
775 views

Hilbert polynomials of graded algebras evaluated at negative numbers

Let $k$ be a field and let $R$ be a commutative (standard) graded $k$-algebra, that is, $R=\bigoplus_{n=0}^\infty R_n$ with $R_0=k$ (and $R=k[R_1]$). The Hilbert function $h_R:\mathbb{N}\rightarrow \...
walkar's user avatar
  • 253
4 votes
1 answer
181 views

Effective bound on "Jacobian rank" for (regular) planar algebraic curves

Let an irreducible, square-free complex polynomial $f\in \mathbb C[x,y]$ be given. It is well known that the curve $\mathcal C:=\{f=0\}$ is nonsingular if and only if $\mathbb C[x,y]=<f,\partial_xf,...
Loïc Teyssier's user avatar
0 votes
1 answer
349 views

Log associahedra and log noncrossing partitions--raising ops and symmetric function theory for $A_n$ (references)

Where do the following three sets $[LA]$, $[ILA]$, and $[LN]$ of partition polynomials appear in the literature? There are two sets of partition polynomials, not in the OEIS, that serve as the ...
Tom Copeland's user avatar
  • 10.5k
3 votes
0 answers
151 views

Extension of work by Novelli and Thibon on noncommutative symmetric functions and Lagrange inversion

(Edit May 12, 2023: I just put up a brief summary of some of my notes on the partition polynomials described below in my WordPress mini-arXiv at "As Above, So Below". It contains multinomial ...
Tom Copeland's user avatar
  • 10.5k
5 votes
0 answers
160 views

Cohen-Macaulayness of rings of polynomials vanishing at points

Let $V$ be a finite dimensional vector space, let $L_1$, $L_2$, ..., $L_r$ be subspaces and let $w_1$, $w_2$, ..., $w_r$ be positive rational numbers. Define a graded ring $R$ where $R_d$ is those ...
David E Speyer's user avatar
3 votes
0 answers
70 views

Degree of an even/odd part of a formal power series over a polynomial ring

Let $K$ be a field with $\operatorname{char}K\ne 2$ (say, $K=\mathbb{R}$ or $\mathbb{C}$) and consider a formal power series $f=f(x)\in K[[x]]$ such that $[K[x,f]:K[x]\,]=d$. Suppose $f_e,f_o\in K[[x]]...
Alexander Burstein's user avatar
1 vote
0 answers
108 views

On Serre's condition and singular locus of determinantal rings

Let $R$ be a Commutative Noetherian ring. Let $\mathbf X:=[X_{ij}]_{1\le i \le r, 1 \le s \le t}$ be a matrix of indeterminates. Let $t>1$ be an integer, and $I_t(\mathbf X)$ denote the ideal in $...
Snake Eyes's user avatar
3 votes
1 answer
173 views

$\Omega$ for noetherian semiperfect rings

Let $A$ be a a two-sided noetherian semiperfect ring and assume that the injective dimension of the left and right regular modules are equal to $n \geq 1$. Let $\Omega^n(mod A)$ be the category of $n$-...
Mare's user avatar
  • 26.5k
3 votes
0 answers
173 views

Intersection theory on schemes with Gorenstein singularities

Is there a good reference/book on Intersection theory on schemes with Gorenstein singularities? Does the construction of the intersection of cycles discussed in Fulton's book also hold for schemes ...
user avatar
1 vote
2 answers
333 views

Condition for equality of modules generated by columns of matrices

Let $R$ be a commutative ring with unit. Let $M_A$ denote the submodule of $R^m$ generated by columns of a matrix $A$ with entries in $R$. Suppose we are given two matrices $A,B \in R^{m \times k}$. I ...
Rahul Sarkar's user avatar

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