2
votes
2answers
56 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
52 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 ...
3
votes
0answers
58 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
69 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
57 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
22 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 ...
0
votes
1answer
57 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
23 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
66 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
72 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
52 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 $\operatorname{Spec}(\mathbb{C}) = {0}$. ...
3
votes
2answers
104 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
88 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
70 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
50 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
93 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
137 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
226 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
80 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
128 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
53 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. ...
8
votes
1answer
412 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) ...
20
votes
4answers
914 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 ...
2
votes
0answers
73 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?