# Questions tagged [projective-resolution]

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### 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?
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### 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 (...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### Is it true that if $\operatorname{Ext}^{1}_{A}(P,A/I)=0$ for all $I$ then $P$ is projective?

Is it true that if $\operatorname{Ext}^{1}_{A}(P,A/I)=0$ forall $I$ then $P$ is projective? Similar statements are true for flat and injective modules, but I'm beginning to suspect that projective ...
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### 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 $\leq d$?
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