# Tagged Questions

**2**

votes

**1**answer

58 views

### Does the category PCM (partial commutative monoids) have a closed symmetric monoidal product?

A partial commutative monoid (PCM) is, roughly speaking, a set with a partially defined binary operation that is as associative as it can be (given that not all products are defined) and commutative. ...

**0**

votes

**0**answers

130 views

### Reference: A nowhere vanishing section of a vector bundle is locally split

Well-known fact:
If $(A, \mathfrak{m})$ is a local Noetherian ring, $E$ is a finitely generated free $A$-module, and $e\in E$ is an element not contained in $\mathfrak{m}E$, then $E/eA$ is also a ...

**2**

votes

**1**answer

124 views

### Vertices of a polytope as algebra generators

I am wondering if the following kind of objects has some name, or are there any studied examples. I apologize for perhaps too specific definition, this is an adoptation of a situation that arises in ...

**2**

votes

**0**answers

52 views

### Is this duality operation on simplicial complexes/Stanley-Reisner rings previously known?

Let $K$ be an abstract simplicial complex on vertices $x_1,\ldots,x_n$, then there is the familiar construction of the face ideal $I_K=\langle x_{i_1}\cdots x_{i_r} | ...

**1**

vote

**1**answer

93 views

### Original sources for two theorems by Bass, Matlis and Papp

It is an interesting fact that a commutative ring $R$ is noetherian if and only if direct sums of injective $R$-modules are injective, and if and only if every injective $R$-module is a direct sum of ...

**5**

votes

**1**answer

218 views

### Can one prove that toric varieties are Cohen-Macaulay by finding a regular sequence?

It's well-known that if $P$ is a finitely generated saturated submonoid of $\mathbb{Z}^n$, then the monoid algebra $k[P]$ is Cohen-Macaulay (at the maximal ideal $m = (P-0)k[P]$). However, all proofs ...

**1**

vote

**1**answer

347 views

### Study of convex polytopes via commutative algebra

Let $P \subset \mathbb{R}^d$ be any convex polytope with integral vertices, and let $M$ be the additive submonoid of $\mathbb{R}^{d+1}$ which is generated by $\{ (v,1) : v \in P \cap \mathbb{Z}^d \}$. ...

**2**

votes

**1**answer

152 views

### Name and references for a “twisted” addition in a ring

This question may seem a little unmotivated, but it actually isn't something that just occurred to me out of nowhere. It stems from a discussion I had the other day with a friend and is directly ...

**5**

votes

**2**answers

319 views

### Formal completion of the normal bundle

Let me for simplicity start with affine case. If $X=\operatorname{Spec}(A)$ is an affine variety $Z \subset X$ is a closed affine subvariety $Z=\operatorname{Spec}(A/I)$. What conditions are ...

**5**

votes

**0**answers

59 views

### K-Theory and completion [duplicate]

I vaguely remember the existence of a statement that relates the $K$-theory (in the sense of Quillen) of a noetherian local ring $A$ with maximal ideal $\mathfrak{m}$ with the $K$-theory of the ...

**3**

votes

**2**answers

427 views

### An identity in an arbitrary commutative ring

This fact might be either trivial, wrong, or well known.
Let $R$ be a commutative ring. Let $u_1,\dots,u_{s-1},u_s\in R$ and $m,M\in R$. Let us assume that $m,M$ satisfy
$$(m-u_1) \dots ...

**3**

votes

**1**answer

192 views

### Tor dimension in polynomial rings over Artin rings

I found this tricky problem in trying to understand some properties of local rings at non-smooth points of embedded curves. But this would be a very long story. So I make it short and I try to go ...

**7**

votes

**0**answers

278 views

### name for a degree-like invariant of a power series over a commutative ring

Let $R$ be a commutative ring, and let $f \in R[\![X]\!]$ be a formal power series. Sometimes (and for example, this will always be possible if $R$ is Noetherian), one may write $f$ in the form $$
f ...

**3**

votes

**1**answer

329 views

### What sort of ring-theoretic properties does the representation ring of a compact Lie group possess?

Recall the definition of the representation ring $R(G)$ of a compact Lie group $G$. I'd like a reference that gives me basic ring-theoretic properties that $R(G)$ always has, or enough info that I can ...

**3**

votes

**2**answers

254 views

### Counterexample to Openness of Flat Locus

Let $A$ be a commutative Noetherian ring and $B$ a finitely generated $A$-algebra. Then the set $$U\colon=\{P\in\operatorname{Spec}B\mid B_P\ \mathrm{is\ flat\ over}\ A\}$$ is open in ...

**1**

vote

**1**answer

246 views

### Cohomology after completion

I've been scouring google and asking friend about something I was certain must be absolutely the easiest thing to people who do homological algebra, and none seem to know the answer to this, so if ...

**8**

votes

**1**answer

282 views

### Presenting $\mathbb{Q}[[t]]$ as an explicit colimit of smooth $\mathbb{Q}$-algebras: an explicit example for the Popescu's theorem

By the seminal Popescu's theorem, $R=\mathbb{Q}[[t]]$ is a filtered colimit of smooth $\mathbb{Q}$-algebras. Could you give me a hint: which $\mathbb{Q}$-algebras can yield such a colimit? My problem ...

**2**

votes

**1**answer

125 views

### Which algebras can be presented as filtered colimits of f.g. regular ones with smooth connecting morphisms?

Let $R$ be a regular (commutative associative unitial) algebra over a prime field $F$ (i.e. $F=F_p$ or $F=\mathbb{Q}$); assume that it is noetherian excellent (and even of Krull dimension $1$). What ...

**0**

votes

**0**answers

119 views

### Fitting ideal sheaves and determinant bundles

I am working on a problem in algebraic geometry which comes down to a fact in commutative algebra that I am hoping is well-known.
Suppose $F$ is a coherent sheaf on a smooth variety $S$, and that the ...

**1**

vote

**1**answer

142 views

### formally étale morphisms which are also universally closed

A morphism of schemes which is formally unramified, universally closed, and a monomorphism is a closed immersion. Is it possible to characterize morphisms which are formally etale and universally ...

**4**

votes

**1**answer

123 views

### Which power of $2$ kills $W(k)$?

Is the following fact "well-known": if $-1$ is a sum of squares in a field $k$, then the Witt group $W(k)$ of quadratic forms is killed by multiplication by $2^N$ for some $N\ge 0$? What can one say ...

**1**

vote

**1**answer

192 views

### Resolution of singularity of polynomials

Let $f$ be polynomial on a vector space $V$. Let $Z$ be the zero set of $f$ in $V$. Let $Z_{sing}$ be the singular part of $Z$.
By Hironaka's desingularization theorem, there exists a birational map ...

**5**

votes

**1**answer

441 views

### Resolution of a module as an $A_\infty$ module over resolution of an algebra

The following question looks like a basic question on resolutions over commutative rings, unfortunately I was not able to find a reference.
Let $A$ be a regular commutative noetherian ring (and ...

**1**

vote

**1**answer

144 views

### Obstruction map for local singularities via tangent (Andre-Quillen) cohomology

Let $R$ be a local singularity (for example $R=\mathbb{C}[[x_1, \ldots , x_n]]/I$) ring over $\mathbb{C}$. Let $\mathbb{L}_{R}$ be a cotangent complex of $R$, then one can define tangent ...

**1**

vote

**0**answers

161 views

### Composite families of formal power series over $\mathbb C$ as algebraic variety

I was led to prove that the set of composite families $(f_j)_{j \leq k}$ of germs at $0\in \mathbb C^m$ of a holomorphic function (composite = sharing a common divisor belonging to the maximal ideal) ...

**7**

votes

**1**answer

235 views

### Explicit formula for associator of commutative power series

Perhaps this question is too elementary, but if it's written down anywhere, I'd love to know about it. Suppose I have a power series $f\in R[[x,y]]$ for some commutative, unital ring. I've recently ...

**2**

votes

**0**answers

52 views

### Upper semicontinuity of Betti numbers of submodules

Theorem 8.29 in "Combinatorial commutative algebra" by Miller and Sturmfels states the upper-semicontinuity property for Groebner deformations (say, over an algebraically closed field with ...

**4**

votes

**2**answers

179 views

### Germs at infinity of sequence of integers

Consider the $\mathbb Z$-module $\mathcal Z$ obtained as the set of sequences of integers $\mathbb Z ^ \mathbb N$ modulo the relation that two sequences are deemed equivalent when their difference is ...

**3**

votes

**0**answers

88 views

### Flatness over Jacboson ring

This is an elementary question which did not get answered on math.stackexchange. I would like to know the answer for expository purposes.
I need either a reference or a counter-example to the ...

**6**

votes

**0**answers

166 views

### I. Kaplansky, Going up in polynomial rings, unpublished manuscript, 1972

Anyone got a copy of this article?

**1**

vote

**0**answers

40 views

### Integral Leray Number?

The Leray number of a finite simplicial complex $K$ relative to a field $\Bbbk$ is defined to be the least $d\geq 0$ such that $\widetilde H^n(C,\Bbbk)=0$ for all $n\geq d$ and all induced ...

**3**

votes

**0**answers

80 views

### Singularity locus in terms of ideals.

Let $X$ be a smooth affine variety over a field, $Z\subset Y\subset X$ are closed (reduced) subvarieties. What are the possible ways to verify whether $Y$ is singular at $Z$ i.e. whether $Z$ is ...

**3**

votes

**1**answer

135 views

### Flatness and intersections of Cohen-Macaulay subvarieties

There's a commutative algebra fact that I would very much like to be true but could, for all I know, be completely false. One version that would be sufficient is:
Say $A$ is a smooth projective ...

**3**

votes

**2**answers

447 views

### Higher dimensional Bezout via Hilbert polynomials: a reference

For the purposes of teaching my elementary course in algebraic geometry I am looking for a reference (or notes) that contains a complete proof of a higher-dimensional weak Bezout theorem. I only want ...

**4**

votes

**3**answers

287 views

### Spectrum and scheme of the commutative group-algebra of an abelian group.

The group-algebra of an abelian group is commutative, so we can consider the spectrum of this algebra. Are there any information about the abelian group that we can obtain from such considerations? ...

**2**

votes

**0**answers

67 views

### maximal degree of generators of graded ideals

Let $A$ be a commutative Noetherian ring, $\mathfrak{a}$ an ideal. Let $R(\mathfrak{a}) = \oplus_{n\geq 0} \mathfrak{a}^n$ be the Rees algebra of $\mathfrak{a}$. Let $I$ and $J$ be two graded ideals ...

**5**

votes

**1**answer

321 views

### Base change of trace for Gorenstein or Cohen-Macaulay morphisms

This is basically a question of functoriality for base change of CM morphisms.
EDIT: $\text{ }$ Brian Conrad sent me an email explaining the that this is indeed true, and follows from his book. ...

**5**

votes

**0**answers

331 views

### Jacobian ideals reference

Suppose that $f : X \to V$ is a flat equidimensional (of dimension $h$) morphism of schemes of finite type and $V$ is excellent (or a variety) For this one can formulate something called the Jacobian ...

**8**

votes

**0**answers

423 views

### Conceptual proofs for the computation of the structure sheaf

The following lemma in commutative algebra is important for the foundations of algebraic geometry:
If $A$ is a commutative ring, $U \subseteq A$ is a finite subset generating the unit ideal, then ...

**0**

votes

**1**answer

159 views

### Kernel elements for the Grothendieck group map of a commutative monoid

This is just a nomenclature question. Let $T$ be a commutative monoid, and let $T^*$ be its Grothendieck group. That is, $T^* \cong T \times T \ / \sim$, where $(s,s') \sim (t, t')$ if $s+t'+e = ...

**5**

votes

**3**answers

282 views

### Groebner bases for power series rings (reference request)

Hello,
Could you help me with a reference to elementary properties of Groebner bases in rings of formal power series over a field? I am especially interested in generic initial ideals.
Thank you in ...

**5**

votes

**1**answer

231 views

### Is this height-transcendence-degree inequality true without AC ?

Let $R$ be a $k$-algebra ($k$ a field) and a domain of finite Krull dimension. In
$\quad$ Krull dimension <= transcendence degree?
it is shown that
$$\text{Krull-dim}(R) \le \text{trans.deg}_k ...

**0**

votes

**0**answers

125 views

### Is there any unified theory among “ localization” and “quotient” of a ring?

Now I'm reading Introduction to Commutative Algebra,I found that the property of localization of a ring is always in accord with the quotient's.That's amazing!
e.g. Some commutative properties, ...

**2**

votes

**1**answer

217 views

### Level of a commutative ring and its quotient field.

Reading Lam's Introduction to Real Algebra, he remarks that:
For a Dedekind domain $A$ with quotient field $F$, then $s(A)$ is either $s(F)$ or $s(F) + 1$. Furthermore, $s(A)$ is either $\infty$, ...

**3**

votes

**0**answers

155 views

### A non-matroidal notion of dependence on a set of ideals

Assume we are given a set of ideals $I_1, \dots, I_s$ in a commutative polynomial ring. Let's define a subset indexed by $A\subseteq [s] = \{ 1,2,\dots, s\}$ as dependent if there exists an $a\in A$ ...

**2**

votes

**2**answers

302 views

### Free Resolution of this determinantal variety.

I am looking for a free resolution of the ideal generated by $2\times 2$-minors of a $3\times 3$ -matrix. More precisely let $M$ be a matrix (sorry but I cannot write a matrix for some TeX technical ...

**4**

votes

**1**answer

352 views

### Why are minimal resolutions of polynomial ideals important?

Background: Let $k$ be a field and denote by $P = k[x_1,\ldots,x_n]$ the polynomial ring in $n$ (commuting) variables over $k$. A resolution of an ideal $I \lhd P$ is an exact sequence of $P$-modules ...

**4**

votes

**1**answer

1k views

### Why is this theorem attributed to Serre?

Page $117$ of Atiyah, MacDonald's Introduction to Commutative Algebra text has the following theorem. Let $P(M,t)$ denote the Poincare- series of $M$.
$\textbf{Theorem.}$ ...

**16**

votes

**3**answers

1k views

### Commutative Algebra with a View Toward Algebraic Number Theory

Someone asked me this today, and I don't know what the standard answer is:
Is there an analogue of David Eisenbud's rather amazing Commutative Algebra With a View Toward Algebraic Geometry but with a ...

**3**

votes

**2**answers

249 views

### Reference Request: Smith Normal Form for maps between free _graded_ modules

I feel like this should be easy, but I cannot quite find a literature reference for this:
We know (i.a. from the Kaplansky reference in Does Smith normal form imply PID?) that sufficient for Smith ...