3
votes
2answers
68 views

Definition of intersection multiplicity in Hartshorne VS Fulton for plane curves

In exercise I.5.2 in Harshorne there is the following definition of intersection multiplicity for two curves in $\mathbb{A}^2$: \begin{equation} \mathrm{length}_{\mathcal{O}_P}\mathcal{O}_P / (f,g) ...
6
votes
0answers
66 views

Symmetric Algebra and Extension of Scalars

I am reading Gortz and Wedhorn's Algebraic Geometry. In their section on the symmetric algebra they explain the adjoint situation $$ \mathrm{Sym}_A \dashv i_A\colon \mathrm{Alg}(A)\to ...
2
votes
1answer
42 views

Kahler forms of a smooth affine algebra vanish eventually?

If $k$ is a Noetherian ring, then do the Kahler forms of a smooth affine $k$-algebra of dimension $d$ vanish above $d$? I mean is: $\Omega^{d+1}_{A|k}\cong 0$?
2
votes
2answers
61 views

Smoothness and field of fractions

If $k$ is an integral domain and $A$ is a Noetherian finitely presented $k$-algebra for which $A \otimes_k Q(k)$ is a smooth $Q(k)$ algebra, then can it be deduced that $A$ was initially smooth over ...
0
votes
0answers
57 views

Alternative Proof why algebraic groups are smooth

Definitions: Let $k$ be a field, by an affine algebraic group we will mean an affine group scheme $G$ over $Spec(k)$ with the property that the coordinate ring of the underlying affine scheme ...
2
votes
0answers
79 views

Length of a composition series of a module

If $A=\mathbb{C}[x,y]_{(x,y)}$, then what is the length of $A$-module $$A/(x^3-x^2y^2+y^{100},x^3-y^{999})\ ?$$ Any suggestion ?
3
votes
1answer
71 views

Kernel of a morphism of regular rings.

Let $k$ be a field and $f: A \rightarrow B$ be a surjective ring morphism between smooth Noetherian $k$-algebras. By smooth I mean that the module of Kahler Differentials $\Omega_{A|k}$ is a ...
1
vote
0answers
60 views

Prove this morphism is a derivation

Let $A$ a ring, $B$ a $A$-algebra, $I=\ker (B\otimes_A B \to B)$ given by multiplication, i wanna see why is $$d:B \to I/I^2$$ $$b \mapsto b\otimes 1 -1 \otimes b$$ well defined (in fact, why mod ...
1
vote
1answer
64 views

help in an example.

In page 226 of David Eisenbud's book Commutative Algebra with a View Toward Algebraic Geometry there is an example which I need help in some parts of it: why $codim I= 1$? why $dim M = dim R = ...
1
vote
1answer
23 views

pushforward of affines

Suppose $A\rightarrow B$ is a morphism of rings (commutative with unit, not necc. Noetherian) and $f: \rm{Spec}\; B\rightarrow \rm{Spec} \;A$ the canonical morphism of schemes. Is it true that ...
1
vote
1answer
42 views

Tensor of quotients question

Let $R = \mathbb{Z}[x]$ be the 1-variable polynomial ring over the the integers, $q$ and $p$ arbitrary prime numbers, $R_1 = \mathbb{Z}[[x]]/(p)$, and $R_2 = \mathbb{Z}[x]/(q, x)$. Question: what ...
1
vote
1answer
63 views

Hartshorne II Prop. 6.9

Prop. 6.9: Let $X \to Y$ be a finite morphism of non-singular curves, then for any divisor $D$ on $Y$ we have $\deg f^*D=\deg f\deg D$. I can not understand two points in the proof: (1) (Line 9) ...
2
votes
1answer
29 views

Smooth algebras and quasi-freeness

Let A be a unital associative algebra over a field k. Then A is smooth if and only if X:=Spec(A) is smooth. That is $\Omega_{X|Spec(k)}$ is locally-free. The later module is isomorphic to ...
0
votes
1answer
67 views

Flatness question

In reading on the stacks project I came across a result I don't quite follow: "Assume M is finitely presented and flat, i.e., (1) holds. We will prove that (7) holds. Pick any prime p and x1,…,xr∈M ...
3
votes
1answer
73 views

Question on Nakayama?

In reading a certain proof on the stacks project "http://stacks.math.columbia.edu/tag/00NV", I can't see how Nakayama's lemma is used to make the following conclusion: "Assume M is finitely presented ...
1
vote
1answer
55 views

Associated sheaf to a $\mathbb{C}$-module

In section II.5 Hartshorne describes the notion of a sheaf $\tilde{M}$ associated to a module M over a ring A. Now in the case $A:= \mathbb{C}$, we have $\operatorname{Spec}(\mathbb{C}) = {0}$. ...
4
votes
2answers
108 views

Tate module of product of abelian varieties

Is there a way to relate the Tate module $T_{l}(A \times B)$ of the product of two abelian varieties $A$ and $B$ over a field $k$ (where $l \neq \text{char}(k)$), to the Tate modules $T_{l}(A)$ and ...
3
votes
0answers
90 views

Question from Liu, Chapter 5 Ex 1.16

I have a question from Liu's book on Algebraic Geometry, doing Chapter 5 Question 1.16: Let $f:X\rightarrow S$ be a morphism of schemes, with the following base change. $$\require{AMScd} \begin{CD} ...
4
votes
2answers
78 views

Quasi coherent module over affine schemes

I am stuck by this question from Liu's algebraic geometry textbook on quasi-coherent modules. Let X be an affine scheme $\mbox {Spec} A $. Let $\mathcal {F} $ be a quasi-coherent $\mathcal{O}_{X} $ ...
0
votes
0answers
51 views

composition series of modules

Let $R$ be a noetherian ring and $M$ an $R$-module of type $R/I$ for some ideal $I$ of $R$. Then we can define the composition series $$0=M_0\subset M_1\subset \cdots\subset M_k=M$$ such that $M_i$ is ...
4
votes
0answers
102 views

The importance of being Cohen-Macaulay

I am starting to study Cohen-Macaulay rings, mainly from Bruns-Herzog book. In that book there are many examples and sentences of the type "If something satisfies this properties, then it is ...
1
vote
1answer
41 views

Given a sheaf $\mathcal{N}$ of $\mathcal{O}_Y$-modules defined by a finitely generated $\mathbb{C}[x]$-module, what is $f^*\mathcal{N}$?

Say $f : \mathbb{A}^2 \rightarrow \mathbb{A}$ is projection on the first coordinate. Given a sheaf $\mathcal{N}$ of $\mathcal{O}_Y$-modules defined by a finitely generated $\mathbb{C}[x]$-module, what ...
3
votes
1answer
147 views

vector bundles on the affine line over a PID

Let $R$ be a PID. Is every finitely generated projective $R[T]$-module free? In other words, is every vector bundle on $\mathbb{A}^1_R$ trivial? For $R=k[X]$ this is true by the Theorem of ...
4
votes
3answers
245 views

(geometric/intuitive) interpretation of ext

In my current work I have to deal a lot with ext-groups (of modules). I feel kind of familar with the formalism, e.g. the connection between n-th extensions and ext. But I don't have a feeling about ...
1
vote
0answers
83 views

Characterization of Hilbert schemes of points

Let $X=\operatorname{Proj}(R)$ be a projective scheme with a $k$-algebra $R$. Let $\mathcal{M} \in \operatorname{Coh}(X)$ be a coherent sheaf on $X$ whose Hilbert polynomial $\chi(\mathcal{M})$ is a ...
4
votes
1answer
138 views

Flat sheaves over a non-flat base $X \rightarrow Y$

I have a relatively naive question. Suppose that $f: X \rightarrow Y$ is a map of schemes. Then, we get a map of local rings $\mathcal{O}_{Y,f(x)} \rightarrow \mathcal{O}_{X,x}$ and thus for any sheaf ...
0
votes
1answer
54 views

The variety associated to a polynomial ring with a particular grading

We impose various grading on an algebra or a module to understand its homological properties. So I made up the following problem and I would like to understand its correspondence as a variety. ...
9
votes
1answer
436 views

is the pushforward of a flat sheaf flat?

Let $f:X \to Y$ be a morphism of schemes and let $F$ be an $\mathcal{O}_X$-module flat over $Y$. Is $f_*F$ flat over $Y$? What's wrong with this argument? [EDIT: as Parsa points out, the (underived) ...
19
votes
4answers
963 views

Why isn't $\mathbb{C}[x,y,z]/(xz-y)$ a flat $\mathbb{C}[x,y]$-module

Why isn't $M = \mathbb{C}[x,y,z]/(xz-y)$ a flat $R = \mathbb{C}[x,y]$-module? The reason given on the book is "the surface defined by $y-xz$ doesn't lie flat on the $(x,y)$-plane". But I don't ...
1
vote
0answers
82 views

Torsion free flat module over a discrete valuation ring is Cohen-Macaulay

Why a torsion free flat module over a discrete valuation ring is Cohen-Macaulay?