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4
votes
1answer
155 views

Different ways of having infinite global dimension

Is there any ring $R$ of infinite global dimension such that any $R$-module is a retract (i.e. direct summand) of some $\oplus_{i\in I}M_i$ where each $M_i$ has finite projective dimension? I ask ...
2
votes
1answer
295 views

Semi-free resolutions

Let $\mathscr{C}$ be a DG category (not much will be lost if you assume that $\mathscr{C}$ has one object, i.e. is a DG algebra). One way to construct the unbounded derived category of ...
4
votes
0answers
190 views

Coordinate free Koszul-Tate resolution

Tate's original construction of the Koszul-Tate resolution involved choosing cocycles representing the cohomology to be killed. Where is it written in a coordinate free treatment, perhaps via a ...
2
votes
2answers
332 views

Free Resolution of this determinantal variety.

I am looking for a free resolution of the ideal generated by $2\times 2$-minors of a $3\times 3$ -matrix. More precisely let $M$ be a matrix (sorry but I cannot write a matrix for some TeX technical ...
1
vote
1answer
106 views

Projective dimension over hypersurface

Let $R$ be (not necessarily commutative) ring and $S$ a simple right $R$-module. Let $f\in Ann(S)$ be normalizng and a non-zero divisor. Is it always true that $$ pdim_{R}(S)=pdim_{R/(f)}(S)+1? $$
4
votes
1answer
305 views

Projective dimension of simple module

Let $R$ be a (not necessarily commutative) ring and $M$ a simple right $R$-module. Then $\mathfrak{m}=Ann(M)$ is a maximal ideal of $R$. It is seems known that $$ ...
4
votes
0answers
159 views

Minimal Koszul-Tate resolutions

In what generality of commutative associative algebras does there exist a minimal Koszul-Tate resolution? Or what is the most general condition known?
3
votes
1answer
263 views

Is there a notion of 'local ample/Kähler cone' for resolved singularities?

Google searches for "local ample cone" and "local Kähler cone" yield no results, but maybe there is a different term. Let $\pi : \hat X \to X$ be a resolution of an isolated singularity on the ...
10
votes
1answer
632 views

Higher “Cartan-Eilenberg” Resolutions

I am a number theory graduate student learning a bit of homological algebra, and I am curious about higher complexes in abelian categories. I apologize if my post is slightly vague as I am not an ...
0
votes
0answers
209 views

Ideal of real points in $\mathbb{C}[x_0,x_1,\dotsc,x_k]$

I fell a little uncomfortable with this real stuff. The question here is more general, but we can suppose $\mathbb{K}=\mathbb{R}$. Take a set of (distinct) points in $\mathbb{P}^n$, the complex ...
1
vote
1answer
341 views

Rank of a module

I have seen the definition of a module,not neccessary free, the alternatin sum of free modules in a free resolution of that module. it's clear that when the module is free our definition Coincide the ...
2
votes
1answer
269 views

Is resolution of singularities effective?

Suppose I have a singular projective variety defined by some homogeneous equations in complex projective space. Is the resolution of singularities effective? That is, do I actually know which smooth ...
3
votes
4answers
2k views

projective module

Is it true that if $\mbox{Ext}^{1}_{A}(P,A/I)=0 \ \forall I$ then P is projective? Similar statements are true for flat and injective modules, but I'm beginning to suspect that projective modules ...
-3
votes
1answer
285 views

Length of a resolution [closed]

Is a (say, projective) resolution (of a module) consisting entirely of zero modules considered to have a length (of zero) at all? I think this possibility causes problems in some books.
2
votes
3answers
859 views

Projective dimension

Is it true that if a module has a free resolution of length $d$ then any of its submodule has a free resolution of length $\leq d$?
7
votes
1answer
530 views

Is there a good computer package for working with complexes over non-commutative rings?

I'm interested in doing computations with certain non-commutative rings, most of which involve taking derived tensor products. Does anyone know of a computer algebra package which will find ...