15
votes
Accepted
Intuition behind the canonical projective resolution of a quiver representation
I suggest to think about it in terms of an analogy. If $G$ is a group, and $M$ is a module over $G$, you have a resolution of $M$ known as the bar complex. The way you produce this resolution can be ...
- 4,189
13
votes
Accepted
Cohomological dimension of torsion-free groups and its subgroups
This is Theorem 3.1, p. 190, in Brown, "Cohomology of groups". He also attributes it to Serre.
As a remark, this is the reason that the virtual cohomological dimension (vcd) is well-defined.
- 2,050
11
votes
Accepted
Non-finitely presented FP groups with cohomological dimension $2$
The Bestvina-Brady construction of non-finitely presented groups of type FP produces groups of cohomological dimension two. Bestvina-Brady groups are parametrized by finite flag simplicial complexes. ...
- 3,451
8
votes
Accepted
Resolution of a torsion sheaf
Write the equation of $C$ as $\ XY-Z^2=0$, and consider the homomorphism $u: \mathcal{O}_{\mathbb{P}^2}(-1)^2\rightarrow \mathcal{O}_{\mathbb{P}^2}^2$ given by the matrix $\begin{pmatrix}
X & Z\\Z&...
- 38k
8
votes
Accepted
Equivalence of definitions of Cohen-Macaulay type
For the equivalence you need two more assumptions: (a) $M$ to have finite projective dimension and (b) $R$ to be Gorenstein. (a) is implicitly required in the second condition, otherwise you don't ...
- 30.5k
7
votes
Accepted
Minimality of the Koszul resolution
In the category of ungraded bimodules, the multiplication map $R\otimes_\mathbb{C}R\to R$ is not a projective cover. For example, the proper sub-bimodule of $R\otimes_\mathbb{C}R$ generated by $1\...
- 35.2k
7
votes
Accepted
derived tensor product and finite projective dimension
Let me preface this by saying that I don't know a reference - so if that's what you're really looking for, someone else will have to answer.
Let $k:=R/m$ denote the residue field (I'm assuming "...
- 15.8k
6
votes
Intuition behind the canonical projective resolution of a quiver representation
The answer to all your questions is "yes". I'm not sure I could do a better job than the intuition in Derksen's notes, though: think of the projective modules in terms of paths as described there, and ...
- 4,217
5
votes
Accepted
Any exact faithful functor is represented by a unique projective generator
Let $\mathcal C$ be the category of finite dimensional left modules over a finite dimensional ring $R$. Let $G: \mathcal C \to \mathrm{Vec}$ be an exact and faithful functor to finite dimensional ...
- 3,948
5
votes
Accepted
Explicit locally free resolution of a perfect complex $E\oplus F\to (E\oplus F)\otimes \mathcal{O}_X(D)\to (E\otimes \mathcal{O}_X(D))|_D$
Projections to the second summands define a morphism from that complex to the complex
$$
F \stackrel{s}\to F(D)\tag{*}
$$
of locally free sheaves. The cone of this morphism is the complex
$$
0 \to E \...
- 39.3k
5
votes
Accepted
What are the projective dimensions of big fraction fields?
Yes, this can happen. See for example https://projecteuclid.org/download/pdf_1/euclid.nmj/1118801622
- 27k
4
votes
Accepted
Projective dimension of graded modules
The other inequality follows from Schanuel's lemma.
For $M$ as is in your question, consider a truncated resolution of projective $G$-graded modules
$$Q_{n-1}\xrightarrow{f}Q_{n-2}\to\dots\to Q_1\to ...
- 2,928
4
votes
Accepted
Is the projective dimension of finite torsion-free modules over regular ring of dimension $n$ smaller that $n$?
Yes. If $\operatorname{pd}_A(M)=n $, there exists a maximal ideal $\mathfrak{m}$ of $A$ such that $\operatorname{pd}_{A_{\mathfrak{m}}}(M_{\mathfrak{m}})=n $. By the Auslander-Buchsbaum theorem, this ...
- 38k
4
votes
When does $FP_n(R)$ imply $F_n$?
The paper https://link.springer.com/article/10.1007/BF02804017 shows there are $\mathbb Q$-acyclic $k$-dimensional simplicial complexes with a complete $k-1$-skeleton and $\binom{n-1}{k}$ $k$-...
- 38.6k
4
votes
Extensions of $G$-modules parametrized by $H^1$
No. If $G$ is the trivial group and $q =p$ is a prime, $\operatorname{Ext}^1_G(V, W) = \operatorname{Ext}^1_{\mathbb{Z}}(\mathbb{Z}/ p, \mathbb{Z} / p) \cong \mathbb{Z}/p$ but $H^1(G, V^\vee\otimes_\...
- 542
3
votes
Accepted
Extensions of $G$-modules parametrized by $H^1$
The correct version of your statement is the following. Let $K$ be a commutative ring and suppose that $V,W$ are (left) $KG$-modules with $V$ finitely generated projective over $K$. Then $$\mathrm{...
- 38.6k
3
votes
Accepted
Projective dimension and subrings
This follows from the General Change of Rings Theorem 4.3.1 in Weibel's book: Let $f\colon R'\rightarrow R$ be a ring map, and let $M$ be an $R$-module. Then as an $R'$ module
$$\operatorname{pd}_{R'...
- 878
3
votes
Vector bundles admitting resolution by ample line bundles
This holds true for projective spaces. Indeed, if $E$ is any vector bundle (even any coherent sheaf), the Beilinson spectral sequence for $E(n)$ with $n \gg 0$ gives the required resolution for $E(n)$....
- 39.3k
3
votes
Intuition behind the canonical projective resolution of a quiver representation
There's some exposition along these lines in Theorem 2.15 in Schiffler's "Quiver Representations".
Using your notation, let $Q = (Q_0,Q_1)$ be a finite acyclic quiver and $X$ be some representation ...
- 31
2
votes
Accepted
Homological dimension of pure coherent sheaves and specialization
Take for $X$ the plane curve $X^3=Y^2T$, and for $F$ the ideal sheaf $(X-\pi ^2T,Y-\pi ^3T)$. Its restriction to the generic fiber is the ideal of a smooth point, hence is an invertible sheaf, while ...
- 38k
2
votes
Accepted
Finitely generated module, which is a virtually small complex, embeds into a module of finite projective dimension?
For every $M$, $M\oplus R$ is virtually small, so your question is equivalent to the question: Does every finitely generated $R$-module embed in a finitely generated module of finite projective ...
- 35.2k
1
vote
Accepted
Exact sequences with two FL-modules
Yes, this is true. See Bourbaki, N. Éléments de mathématique. Algèbre. Chapitre 10. Algèbre homologique. (French) [Elements of mathematics. Algebra. Chapter 10. Homological algebra], Theorem 3.9.1
...
- 1,418
1
vote
Image of a quiver variety under natural morphism
The answer is completely known for ADE quiver varieties:
Holds for general quiver varieties, as $\pi$ is a projective morphism so its image is a closed Poisson subvariety of $\mathfrak{M}_0$ so it ...
- 1,677
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