All Questions
145 questions
2
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124
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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 \...
- 21
5
votes
1
answer
151
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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^{...
- 3,230
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 " ...
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,...
- 128k
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 ...
- 961
6
votes
1
answer
1k
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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 ...
- 5,339
5
votes
0
answers
107
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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 $y=f_m(x)$. ($x,y,a,b$ ...
- 10.5k
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?
- 21
2
votes
1
answer
365
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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 $\...
- 613
2
votes
0
answers
92
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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 ...
- 6,962
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 ...
- 7,127
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,...
- 5,428
7
votes
1
answer
351
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Has anyone attempted to generalize the notion of a higher differential of $ A $ and the sheaf of differentials $ \Omega_{A/k} $?
Over a field $ k $ of characteristic zero, actions of $ \widehat{\mathbb{G}_{a}} $ on a variety $ \operatorname{Spec}(A) $ are obtained from $ A $-derivations $ \delta $ of $ A $ over $ k $ via the co-...
- 912
3
votes
1
answer
496
views
Regular ring is smooth when the field is perfect
Take $A$ a (not necessarily local) commutative algebra over a field $k$ which is essentially of finite type (i.e. a localization of a finitely generated algebra). In simple words, I just want to know ...
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 ...
- 156k
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 ...
user497552
2
votes
0
answers
128
views
On the generalization of a Cech-to-sheaf type spectral sequence
Let $(X,O_{X})$ be a ringed space and $\mathcal{F^{\bullet}}$ be a bounded below complex of $O_{X}$-modules on $(X,O_{X})$. On [Stacks Project 20.25.1][1] it is shown that there is a weakly convergent ...
- 541
4
votes
2
answers
519
views
Quasi-compact surjective morphism of smooth k-schemes is flat
I have precedently posted the same question on Math.Stackexchange (https://math.stackexchange.com/questions/4277856/quasi-compact-surjective-morphism-of-smooth-k-schemes-is-flat), but to no avail; I ...
- 55
7
votes
1
answer
206
views
$2$-dimensional complete local normal domain with rational singularity that has exactly one exceptional curve
Let $(R, \mathfrak m)$ be a complete local normal domain of dimension $2$ with residue field $R/\mathfrak m$ algebraically closed and characteristic $0$. Assume Spec$(R)$ has rational singularity, let ...
- 413
3
votes
1
answer
212
views
Flatness of finitely presented algebras
Let $R$ be a commutative (noetherian, if needed) ring, let $f_1，\ldots，f_r\in R[x_1，\ldots，x_n]$ and $A=R[x_1，…，x_n]/(f_1，\ldots，f_r)$, when is $A$ flat over $R$?
I found a nice answer for the case $n=...
- 1,906
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 $...
- 912
2
votes
1
answer
437
views
Extending functors between K-algebras to schemes
Assume we have $K$ and $L$ (comm.) rings, and we have a functor $F$ from the category of $K$-Algebras to the category of $L$-Algebras (I work only with commutative rings). What conditions need to ...
- 608
5
votes
1
answer
299
views
The universal multiset for a finite scheme - reference request
If $X$ is a finite set of size $n$, then by listing the elements of $X$ we get a canonical element of the symmetric power $X^n/\Sigma_n$, which we can call the universal multiset for $X$.
Now let $X$ ...
- 56.9k
8
votes
1
answer
855
views
What is the motivation for excellent rings?
First of all I am not formally educated in mathematics so pardon my ignorance if this is obvious and I am skipping something vital, but I am interested nonetheless in what the original motivation and ...
- 81
35
votes
3
answers
5k
views
Matrix factorizations and physics
I have heard during some seminar talks that there are applications of the theory of
matrix factorizations in string theory. A quick search shows mostly papers written by physicists. Are there any ...
- 30.5k
2
votes
0
answers
115
views
When $k(f(x), g(x)) = k(x)$? In other words, when is a given polynomial parametrization of an affine planar (rational) curve proper?
If $f, g \in k[x]$, where $k$ is a field, then $k(f, g) = k(h)$ for some rational function $h \in k(x)$ (this is a special case of Lüroth's theorem).
Question 1: Under what conditions does the above ...
- 5,339
8
votes
1
answer
264
views
Class group of hypersurfaces of finite representation type
Let $k$ be an algebraically closed field of characteristic different from $2,3$ and $5$, and let $R=k[[x,y,x_2,\dots,x_d]]/(f)$, where $f\in(x,y,x_2,\dots,x_d)^2$, $f\neq0$. By results of Buchweitz-...
- 411
6
votes
1
answer
275
views
Algebraic geometry additionally equipped with field automorphism operation
I am looking for some facts on theory, which is essentially algebraic geometry but with field automorphisms added as 'basic' operations. (Precisely, I mean universal algebraic geometry for (universal) ...
- 63
5
votes
1
answer
383
views
Euler characteristic and rational Poincaré series
$\DeclareMathOperator\len{len}\DeclareMathOperator\Tor{Tor}$Let $(A,\mathfrak{m})$ be a regular local ring, and $x \in \mathfrak{m}^2$ be a non-zero prime element. So $R:=A/(x)$ is a non-regular Cohen-...
- 125
4
votes
0
answers
79
views
Possible number of zeros of a stable perturbation of a germ $(\mathbb{R}^n, 0) \to (\mathbb{R}^n, 0)$
Let $f:(\mathbb{R}^n, 0) \to (\mathbb{R}^n, 0)$ be an analytic germ. Assume that it has isolated zero at 0, that is, $f^{-1}(0)=\{0\}$, what is more, assume that the dimension of the local algebra* $Q(...
- 73
6
votes
1
answer
607
views
rationality of weighted projective space
A complex weighted projective is $\mathbb{P}(k_1, \cdots, k_{n+1})=Proj(\mathbb{C}[x_1, \cdots, x_{n+1}])$ with $x_i$ of degree $k_i$ (sometimes people ask for each $n$ of the weights being coprime). ...
- 337
1
vote
0
answers
74
views
Characterise set of polynomials which are zero over an ideal
This is not a specific question, but rather a question about possible techniques approaching a problem. Although this question came from research, it might not fit this forum; in which case I will ...
- 11
30
votes
6
answers
8k
views
Algebraic stacks from scratch [closed]
I have a pretty good understanding of stacks, sheaves, descent, Grothendieck topologies, and I have a decent understanding of commutative algebra (I know enough about smooth, unramified, étale, and ...
5
votes
0
answers
324
views
Earliest reference for infinitesimal neighborhoods of the diagonal
Where was $I_x/I_x^2$ first introduced? (DG or AG) asks about the algebraic cotangent space. The paper First neighborhood of the diagonal and geometric distributions by Kock claims Grothendieck ...
- 10.5k
1
vote
0
answers
149
views
Cohen-Macaulay coordinate rings defined by regular sequences
Consider the polynomial ring $R = k[x_1, \ldots, x_n]$ in $n$ indeterminates over an algebraically closed field $k$ (my relevant case is the complex numbers). Furthermore, consider an algebraic ...
- 111
16
votes
1
answer
733
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Where was $I_x/I_x^2$ first introduced? (DG or AG)
Cotangent space appears in both differential geometry (DG) and algebraic geometry (AG).
In DG, given a smooth manifold $M$ and $x\in M$ one has an isomorphism $I_x/I_x^2 \cong T^*_xM$, where $I_x$ is ...
- 1,615
16
votes
2
answers
4k
views
A geometric reference for (affine) Gorenstein varieties and singularities
I would like to ask for a reference to some text that explains in relatively down to earth (if possible geometric) terms (for dummies) what is a Gorenstein singularity and Gorenstein variety (for a ...
- 14.3k
0
votes
0
answers
265
views
Algebraic closure of field of fractions of multivariate polynomial ring over $\mathbb{R}$
I am searching for good references on the topic of the behaviour of the elements in the algebraic closed field $(\mathbb{R}[x_{1},\dots,x_{n}])^{\operatorname{alg}}.$ I imagine that, when we try to ...
- 573
39
votes
2
answers
6k
views
What is Serre's condition (S_n) for sheaves?
The Serre's condition $(S_n)$, especially $(S_2)$, has been mentioned in a few MO answers: see here and here for example. I am pretty sure I have seen it in other questions as well, but could not ...
- 30.5k
2
votes
1
answer
656
views
Closed points of a closed subscheme of $\mathbb{P}^n$ over the residue field and the fraction field of a valuation ring $R$
Let $(R, M)$ be a valuation ring with algebraically closed fraction field $k$. Let $L = R/M$ be the residue field of $R$; it follows that $L$ is algebraically closed. I would like to understand the ...
- 3,625
4
votes
1
answer
692
views
Picard group of hypersurfaces in $\mathbb{P}^r\times\mathbb{P}^s$
Let $k$ be an algebraically closed field, say $k=\mathbb{C}$. Let $r,s$ be sufficiently large integers.
Is it true that, for any irreducible hypersurface $X$ of bi-degree $(d,1)$ in $\mathbb{P}^r\...
user39380
3
votes
1
answer
332
views
Algebraic vector bundles on the punctured spectrum: an exact reference for a result
Let $(R, \mathfrak m)$ be a Noetherian local ring of depth at least $2$. Let $X=Spec(R)$ denote the affine- scheme with structure sheaf $\mathcal O_X$ and $U=Spec(R)\setminus \{\mathfrak m\}$ be the ...
- 642
3
votes
1
answer
451
views
Commutative algebra for the Conway games
I was reading the book On Number And Games and I have some question. In this book Conway constructed the set of "games" with a addition and a multiplication. I understand that the surreal numbers are ...
- 527
1
vote
1
answer
229
views
Generic Galois alteration of an arithmetic model with semistable special fiber
Let $R$ be a DVR of char $0$ and $S=Spec(R)$. Let $X\longrightarrow S$ be a proper flat morphism. Assume $X$ is integral. De Jong's Theorem 8.2 in his paper Smoothness, semi-stability and alterations ...
- 43
1
vote
0
answers
72
views
Equivalence between smoothly regular and analytically regular
I think the following statement is true.
Let $M$ be a real analytic manifold. Let $S \subset M$ be an analytic or semianalytic subset. A point $p \in S$ is called smoothly regular resp. analytically ...
- 803
8
votes
1
answer
359
views
Global to local principle for f.g. $\mathbb{Z}[x]$ modules
In graduate school, while I was working on the maximal subgroup growth of certain metabelian groups, I discovered and proved a lemma which gave me the impression that it was already known. Do you know ...
14
votes
0
answers
1k
views
Is there a slick proof of the fundamental theorem of dimension theory?
The fundamental theorem of dimension theory in commutative algebra states that given a module $M$ over a noetherian local ring $A$, we have $s(M)=\text{dim}(M)=d(M)$ (where $s(M)$ is the infimum of ...
- 1,514
11
votes
2
answers
2k
views
Idea behind Grothendieck's proof that formally smooth implies flat?
From this answer I learned that Grothendieck proved the following result.
Theorem. Every formally smooth morphism between locally noetherian schemes is flat.
The book Smoothness, Regularity, and ...
- 10.5k
3
votes
0
answers
186
views
Is cup product of cycle classes on Noetherian regular excellent scheme compatible with intersection
Let $\mathcal{X}$ be a Noetherian regular integral excellent scheme. Let $Y$ and $Z$ be algebraic cycles of codimension $c$ and $d$ on $\mathcal{X}$.
Let $n$ be a positive integer invertible on $\...
- 43
7
votes
1
answer
553
views
Relationship between Hilbert-Samuel multiplicity and polar multiplicity
Let $f \in \mathbb{C}[[x,y]]$ be the germ of an isolated plane curve singularity. Then the Hilbert-Samuel multiplicity $e_f$ of $f$ is given as follows:
$$e_f = \lim_{s \to \infty}\frac{1}{s} \cdot \...
