3
votes
1answer
52 views

Tor sheaves on schemes

I was trying to understand the definition of "Tor sheaves", but since it is defined in the derived category of sheaves of $\mathcal{O}_X$-modules and since I am not acquainted with derived categories ...
2
votes
0answers
39 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 ...
1
vote
1answer
44 views

$K(A)\cong \mathbb Z$ for a PID $A$

In Atiyah and Macdonald, chapter 7, exercise 26, iii), it's required to show the Grothendieck group $K(A)\cong \mathbb Z$ for a PID $A$. By ii) of this problem, it's easy to show that $K(A)$ is ...
2
votes
2answers
64 views

Represent localization as a direct limit

Let $A$ be a commutative ring with identity, $S\subset A$ a multiplicatively closed subset and $1\in S$. Does the equation $$S^{-1}A=\varinjlim_{s\in S}A_s$$ make sense? Here $A_s$ is the ...
4
votes
0answers
47 views

The functor $\underline{\mathbf{R}}^if_*$

Let $f: X \to Y$ be a proper morhpism of varieties, and $\mathcal{F}$ be a sheaf on $X$. Then we have $f_* \mathcal{F}$ as a sheaf on Y and we also have a higher derived functor $\mathbf{R}^i ...
17
votes
1answer
357 views

Homological methods in algebraic geometry

This question will probably seem quite silly to those well-versed in algebraic geometry (about which I admittedly hardly know anything); in the preface of Atiyah-Macdonald's book on commutative ...
5
votes
0answers
54 views

Maps between spectral sequences

I am trying to understand a subtle point about how Theorem 2.2.5 is used in Kedlaya, Abbott, and Roe's "Bounding Picard numbers of surfaces using p-adic cohomology". Below I've tried to pose the ...
8
votes
0answers
82 views

Is there anything to be learned from the spectrum of a cohomology ring?

Given some topological space, $X$, is there any benefit to studying $Spec(H^*(X))$, or is everything we care about already available "in the algebra"? As $H^*$ is a graded ring, does this question ...
3
votes
3answers
170 views

Existence of an isomorphism $\mathbb{P}^n\times\mathbb{P}^m \rightarrow \mathbb{P}^{n+m}$ [duplicate]

There exist an isomorphism of varieties? $$\mathbb{P}^n\times\mathbb{P}^m \rightarrow \mathbb{P}^{n+m}$$ I am considering $\mathbb{P}^n\times\mathbb{P}^m$ as the product in the category of ...
5
votes
0answers
63 views

Leray spectral sequence for complexes

Let $f:X\rightarrow S$ be a morphism of schemes. Let $0\rightarrow C_1 \rightarrow C_2 \rightarrow C_3 \rightarrow 0$ be an exact sequence of Abelian sheaves on $X$. Is there a general procedure to ...
2
votes
0answers
56 views

How to define the natural map on the second page of a spectral sequence?

I'm learning about spectral sequences in Ravi Vakil's notes, and can't quite figure out how to define the map ($d_2$) on the bottom of page 59 (he describes it as a worthwhile exercise). It should be ...
0
votes
0answers
61 views

What are V(f) and D(f) in real practice of EGA

https://skydrive.live.com/redir?resid=E0ED7271C68BE47C!355 would like to do and understand what is V(f) and D(f) where D(f) = SpecA - V(f) in the following diagram, it said V(f) is subset of p ...
2
votes
0answers
51 views

Support of a direct sum of local cohomology modules

Let $R$ be a Noetherian ring with unit, $I$ be an ideal of $R$. Let $M$ be a finitely generated $R$ module. How can we show the following: $$\operatorname{Supp}(\bigoplus_{j\ge ...
3
votes
1answer
57 views

All local cohomology modules being zero

Let $R$ be a Noetherian ring with unit, $I$ be an ideal of $R$ and let $M$ be a finitely generated $R$-module. Suppose $H_{I}^j(M)=0$ for all $j$, then how can one show that $M=IM$? The converse of ...
3
votes
0answers
108 views

A question of extension of vector bundles.

Fix $p \in \mathbb{P}^1$. Let $X=\mathbb{P}^1\times \mathbb{P}^1$, $C_1=\mathbb{P}^1\times \{p\}$ and $C_2=\{p\}\times \mathbb{P}^1$. Since $\mathrm{Ext}^1(\mathcal{O}_{C_2},\mathcal{O}_{C_1})\cong ...
-4
votes
1answer
126 views

What is the Grothendieck group and how calculate the $K_0$ functor?

$\ker(f)$ direct sum with $\operatorname{im}(g)$, does it mean that number of polynomials in $\ker(f)$ must be the same as the number of polynomials in $\operatorname{im}(g)$? In other words, for ...
7
votes
2answers
264 views

Derived Category of Coherent Sheaves on Elliptic Curves

I know little about algebraic geometry, however while studying noncommutative geometry some results showed that a category I understand well (holomorphic vector bundles over noncommutative tori) was ...
9
votes
1answer
143 views

Ext between two coherent sheaves

Let $X$ be a smooth projective variety over a field $k = \overline k$. From Hartshorne we know, that $\textrm{dim} \, H^i (X,F)<\infty$ for any coherent sheaf $F$. How to show, that all $Ext^i ...
5
votes
1answer
124 views

Sheaves on $\mathbb{P}^n \times \mathbb{P}^m$, and a commutation relation for derived functors of global sections and tensor products on it.

I'll state my questions first and then provide some background. Question 3 is by far my most important one. We work over $k=\mathbb{C}$ whenever necessary. Is it true that $\text{Pic}(\mathbb{P}^n ...
0
votes
1answer
47 views

how to make factorization by a group action

any algebra and numerical example for Projectivization http://en.wikipedia.org/wiki/Projectivization which book or paper teaching this
4
votes
3answers
219 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 ...
3
votes
1answer
83 views

Calculation of dimension of Socle

Let $S=k[[t^3,t^5,t^7]]$ be a formal power series over field $k$.I wanna know why $$\dim_k \operatorname{Soc}(S/t^3S)=2?$$.($\dim_k$ means dimension as $k$-vector space.) background: ...
3
votes
3answers
113 views

$\mathrm{Spec}(R)\!=\!\mathrm{Max}(R)\!\cup\!\{0\}$ $\Rightarrow$ $R$ is a PID

Is the following true: If $R$ is a commutative unital ring with $\mathrm{Spec}(R)\!=\!\mathrm{Max}(R)\!\cup\!\{0\}$, then $R$ is a PID. If yes, how can one prove it? Since $0$ is a prime ...
0
votes
2answers
118 views

Extention of vector bundles on projective line: $Ext^1({\mathcal O_{\mathbb{P}^1}}(n),{\mathcal O_{\mathbb{P}^1}}(m))=$??

I want to know the value $Ext^1({\mathcal O_{\mathbb{P}^1}}(n),{\mathcal O_{\mathbb{P}^1}}(m))$ for integer m, n.
9
votes
2answers
146 views

What does $Tor_{R}^n(M,N)$ represent?

Let $R$ be a commutative ring and $M$ and $N$ be $R$-modules (I am not sure if one really needs commutativity in the following). It is well-known that $Ext_{R}^n(M,N)$ for $n>1$ parametrizes ...
1
vote
0answers
35 views

Inequality of numerical invariants of complex algebraic surfaces?

Let $S, T$ be algebraic surfaces over $k=\mathbb{C}$, and $\phi: S \longrightarrow T$ a surjective morphism. Furthermore we have the numerical invariants: \begin{align*} q(S) &:= \dim H^1(S, ...
2
votes
1answer
72 views

Exact sequence of four sheaves in Beauville: associated l.e.s.?

This question is about an exact sequence of four sheaves on a smooth projective surface $S$ over $k=\mathbb{C}$, to be found in Beauville: complex algebraic surfaces, theorem I.4, page 3 (second ...
1
vote
0answers
109 views

some question of calculating betti number

For a graded finitely generated $k[x_1, \cdots, x_n]$ module $V$, I know that $$ b_{i,p}(V)=\operatorname{dim}_k H_i(K\otimes V)_p$$ where $K$ be the Koszul complex of $k$. I also know that $K$ is ...
5
votes
0answers
165 views

Computing the hypercohomology of a complex of acyclic sheaves

Let $K^{\bullet}$ be a cochain complex of sheaves of finite-dimensional vector spaces, I wanted to compute $\mathbb{H}^{\bullet}(X,K^{\bullet})$ = the hypercohomology of the complex $K^{\bullet}$, the ...
3
votes
1answer
276 views

Motivation for studying quadratic algebras, Koszul algebras, Koszul duality

I'm trying to gain a practical understanding of Koszul duality in different areas of mathematics. Searching the internet, there's lots of homological characterisations and explanations one finds, but ...
18
votes
4answers
844 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 ...
7
votes
2answers
200 views

English translation or summary of “Relevements modulo $p^2$ et decomposition du complexe de de Rham. ”

I'm looking for either an English translation or summary of the article "Relevements modulo $p^2$ et decomposition du complexe de de Rham." by Deligne. I'm attempting to read this for background ...
0
votes
1answer
86 views

Right derived functor of diagonal morphism equals direct image on line bundles?

Let $X$ be a smooth projective variety. The map $i:X\to X\times_k X$ induced by the identity is a closed immersion. Denote its image by $\bigtriangleup$. We have ...
3
votes
2answers
148 views

Vanishing of $ H^1(\mathcal{M})$ implies vanishing of $H^1(U\otimes\mathcal{M}) $ on a curve.

Let $C$ be a smooth projective curve of genus $g\geq 1$ over an algebraically closed field. Let $\mathcal{M}$ be a line bundle with $deg \mathcal{M}\geq 2g -1$. Let $T$ be torsion and denote by $U$ ...
9
votes
1answer
253 views

Tangent space in a point and First Ext group

Let $X$ be an abelian variety over an algebraically closed field $k$. I have read that one has for an arbitrary closed point $x$ on $X$ a canonical identification $$T_x(X)\simeq ...
5
votes
2answers
613 views

Arbitrary products of quasi-coherent sheaves?

I have a short question: Does the category of quasi-coherent sheaves on a scheme have arbitrary products? I know that it does if the scheme is affine and I know that they will not be isomorphic to ...
5
votes
1answer
132 views

Grothendieck spectral sequence

given functors $F,G$, left exact, with as good properties as you want we have a spectral sequence $R^p F\circ R^q G$ abutting to $R^{p+q}(F\circ G)$. I am looking for an analogous for a "mixed ...
4
votes
5answers
302 views

Derived category and so on

I am looking for an introductive reference to the theory of derived categories. Especially I need to start from the very beginning and I need to know how to use this in examples which comes from ...
19
votes
1answer
430 views

cones in the derived category

If I have two exact triangles $X \to Y \to Z \to X[1]$ and $X' \to Y' \to Z' \to X'[1]$ in a triangulated category, and I have morphisms $X \to X'$, $Y \to Y'$ which 'commute' ...
6
votes
2answers
259 views

Status of mixed motives

From the wikipedia page: http://en.wikipedia.org/wiki/Motive_(algebraic_geometry) it appears that the category of Mixed motives $MM(k)$ over a field $k$ is still conjectural; but there is a good ...