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2 votes
0 answers
124 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 \...
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^{...
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 " ...
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?
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,...
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 $y=f_m(x)$. ($x,y,a,b$ ...
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 ...
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 $...
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 ...
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 $\...
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 ...
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 ...
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 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 ...
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 ...
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 ...
7 votes
1 answer
351 views

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-...
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 ...
8 votes
3 answers
921 views

Generic Noether normalisation

Suppose that $M$ is a finitely generated module over $A=k[X_1,\ldots,X_n]$ of Krull dimension $m$ with $k$ an infinite field. Then one version of Noether normalisation says there is an $m$-dimensional ...
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 ...
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 ...
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 ...
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 ...
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=...
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 ...
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(...
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 ...
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 ...
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 ...
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 ...
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 ...
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 $X$ ...
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-...
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 ...
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-...
21 votes
1 answer
2k views

Two conjectures by Gabber on Brauer and Picard groups

In a paper I need to make reference to two conjectures by Gabber, from Ofer Gabber, On purity for the Brauer group, in: Arithmetic Algebraic Geometry, MFO Report No. 37/2004, doi:10.14760/OWR-2004-37 ...
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 ...
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 ...
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\...
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 ...
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) ...
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 ...
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 ...
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 $\...
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 votes
1 answer
157 views

Projecting onto the span of a generic Veronese variety

Let $\sigma_d:\mathbb{P}^2\to\mathbb{P}^n$ be the d-th Veronese map and let $X=\sigma_d(\mathbb{P}^2)$. Let $W\subset\mathbb{P}^n$ be a 2-plane such that $W\cap X=\emptyset$. For a line $L\subset \...
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). ...
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 ...
2 votes
0 answers
137 views

Weak Lefschetz property Jacobian ring smooth hypersurface

Let $A_{.}$ be a graded commutative ring. We say that $A_{.}$ satisfies the weak Lefschetz property if for generic $L \in A_1$ the multiplication maps $ \times L : A_i \longrightarrow A_{i+1}$ has ...
8 votes
0 answers
548 views

Foundational Questions on Adic Spaces

There are some foundational questions on adic spaces that I can't find in the literature. It seems that these questions are pretty natural, so I guess that an answer should be known to the experts in ...