# Questions tagged [projective-resolution]

The projective-resolution tag has no usage guidance.

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### Projective dimension and subrings

$\DeclareMathOperator\pd{pd}$Suppose that $R$ is a commutative ring and $R'$ is a subring of $R$ such that $R$ is a free $R'$-module of finite rank. Assume that both $R$ and $R'$ are regular local ...

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### Is the projective dimension of finite torsion-free modules over regular ring of dimension $n$ smaller that $n$?

Let $R$ be a Noetherian regular integral domain of Krull dimension $n$. Let $M$ be a finite torsion-free $R$-module. Is this true that $M$ has projective dimension $<n$ ?
This would be a ...

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### Exact sequences with two FL-modules

Let $R$ be a ring. An $R$-module $M$ is called FL (FP) if it has a finite resolution consisiting of finitely generated free (projective) modules.
Given an exact sequence of $R$-modules, $0\to M_1\to ...

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### Finitely generated module, which is a virtually small complex, embeds into a module of finite projective dimension?

Let $R$ be a commutative Noetherian ring, and let $\text{mod } R$ denote the abelian category of finitely generated $R$-module. Consider the bounded derived category $D^b(\text{mod } R) $ which is a ...

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### When does $FP_n(R)$ imply $F_n$?

It is known that if a group $G$ is of type $F_2$ (finitely presented) and of type $FP_n(\mathbb{Z})$, then $G$ is of type $F_n$.
However, is this true also for other rings which are not $\mathbb{Z}$? ...

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### Non-finitely presented FP groups with cohomological dimension $2$

In this recent preprint, the authors construct a certain uncountable family of non-finitely presented FP groups. Recall that group is an FP group if the trivial $\mathbb Z[G]$-module $\mathbb Z$ has a ...

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### Conditions for a minimal derived $A_\infty$-algebra to be bounded

I was looking for some examples of derived $A_\infty$-algebras (or $dA_\infty$-algebras) in the original reference by Sagave, DG-algebras and derived A-infinity algebras, where some examples obtained ...

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### On infinite global dimensions of "slightly non-commutative" rings

Assume $R$ is a commutative Noetherian ring of finite Krull dimension; $R'$ is a not commutative ring that contains $R$ in its center and also finitely generated as an $R$-module.
If the (left) global ...

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### Existence of a finite resolution

I have tried to formulate a question in which I was very curious, any hints suggestions are also welcomed. Thanks in advance.
Let $M$ be an $R$ module ($R$ commutative ring with unity). It is given ...

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### Cohomological dimension of torsion-free groups and its subgroups

In this thesis by Martin Hamilton on
Finiteness Conditions in Group Cohomology there is on page 11 a reference to following result:
Theorem 1.2.14. If $G$ is a torsion-free group and $H$ is a
subgroup ...

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### derived tensor product and finite projective dimension

Let $(R, \mathfrak m)$ be a Noetherian local ring. Let $M,N$ be non-zero finitely generated $R$-modules.
Is it known that $M\otimes_R^{\mathbf L} N$ has finite projective dimension if and only if $M$ ...

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### Vector bundles admitting resolution by ample line bundles

Let's assume we are working a smooth projective variety. Let $C$ be the category of vector bundles constructed by taking successive extensions of line bundles of the form $\mathcal{O}(n)$ for $n\in \...

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### contracting homotopies of Koszul resolution of $\mathbb{C}[x_1, \ldots, x_n]$ and $\mathbb{C}_{q}[x_1, \ldots, x_n]$

Let $A : = \mathbb{C}[x_1, \ldots, x_n],$ $A_q : =\mathbb{C}_q[x_1, \ldots, x_n] = \mathbb{C} \langle x_1, \ldots, x_n \rangle / (x_ix_j = q x_jx_i)$.
By Koszul resolution I mean
$$\ldots \to A \...

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### Any exact faithful functor is represented by a unique projective generator

In the book 'Tensor Categories' by Pavel Etingof, Shlomo Gelaki, Dmitri Nikshych and Victor Ostrik on page 10 it says:
'Conversely, it is well known (and easy to show) that any exact faithful functor ...

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### Explicit locally free resolution of a perfect complex $E\oplus F\to (E\oplus F)\otimes \mathcal{O}_X(D)\to (E\otimes \mathcal{O}_X(D))|_D$

Let $X$ be a smooth projective variety over $\mathbb{C}$. Let $E,F\to X$ be 2 holomorphic vector bundles and $D\hookrightarrow X$ be a smooth divisor. Denote by $\mathcal{O}_X(D)$ the line bundle ...

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### What are the projective dimensions of big fraction fields?

Let $A$ be an integral domain, $B$ is its fraction field. Can the projective dimension of the $A$-module $B$ be greater than $1$? This surely cannot happen if the spectrum of $A$ is countable (since ...

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### Equivalence of definitions of Cohen-Macaulay type

I know that the Cohen-Macaulay type has these two definitions:
Let $(R,\mathfrak{m},k)$ be a Cohen-Macaulay (noetherian) local ring and $M$ a finite $R$-module of depth $t$. The number $r(M) = \dim_k ...

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### Image of a quiver variety under natural morphism

We know that the natural morphism $\pi:\mathfrak{M}_{\theta}(Q,\mathbf{v},\mathbf{w})\rightarrow \mathfrak{M}_0(Q,\mathbf{v},\mathbf{w})$ between a smooth and affine quiver variety is not necessarily ...

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### Resolution of a torsion sheaf

Let $J$ be the hyperplane divisor in $\mathbb{C}P^2$, and let $i:C \hookrightarrow \mathbb{C}P^2$ be the closed immersion of a smooth generic curve of degree 2. We know that $C\simeq \mathbb{C}P^1$, ...

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### Projective dimension of graded modules

Short version:
Why is the projective dimension of a graded module the same as the projective dimension of its underlying ungraded module?
Longer version:
Let $G$ be a commutative group, let $R$ ...

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### Intuition behind the canonical projective resolution of a quiver representation

Let $Q$ be a finite acyclic quiver, and $X$ some representation of $Q$. For $i \in Q_0$ define the $kQ$-modules $P_i = kQe_i$, and $X(i) = e_i X$. The representation $X$ has a canonical projective ...

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### resolution tautological sheaf projective dual $G(3,6)$

I consider the Plücker embedding of $G(3,6)$ in $\mathbb{P}^{19}$. I denote by $X \subset {\mathbb{P}^{19}}^*$ the projective dual to $G(3,6) \subset \mathbb{P}^{19}$. The variety $X$ is a quartic ...

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### Homological dimension of pure coherent sheaves and specialization

Let $X$ be a projective variety, not necessarily smooth, $R$ a DVR with residue field $k$ (assume char$(k)=0$). I am looking for examples of a pure coherent sheaf, say $F$, on $X_R:=X \times_k \mathrm{...

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### on the ``generic" modules of finite length (skyscrapers)

Let $R$ be a local or graded ring. (If it helps, can assume the ring is "good", e.g. $R=k[[x_1,..,x_p]]$, where $k$ is a field of zero characteristic.)
Let $M$ be a finitely generated $R$-module ...

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### 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 ...

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### 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 $\mathscr{C}$-...

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### 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 ...

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### 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 ...

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### 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
$$
pdim_{R}(S)=pdim_{R/(f)}(S)+1?
$$

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### 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
$$
pdim_{R}(M)=pdim_{R_{\mathfrak{m}}}(...

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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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### 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 ...