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Results tagged with homological-algebra
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user 5292
(Co)chain complexes, abelian Categories, (pre)sheaves, (co)homology in various (possibly highly generalized) settings, spectra, derived functors, resolutions, spectral sequences, homotopy categories. Chain complexes in an abelian category form the heart of homological algebra.
2
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3
answers
2k
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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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-1
votes
2
answers
662
views
Do Gorenstein rings necessarily have a finite projective dimension (as a module over itself)? [closed]
Do Gorenstein rings necessarily have finite projective dimensions?
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1
vote
Modules over a Gorenstein ring
I found this proof in Kaplansky's Commutative Rings:
Induction on $\mbox{dim }A$.
$\mbox{dim }A =0 \ $:
Suppose $M\neq 0$. $\mbox{id}(M)=\mbox{depth}(A)=0$ so $M$ is injective, and hence is a direc …
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2
votes
2
answers
853
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Modules over a Gorenstein ring
$A$ a Gorenstein ring, $M\neq 0$ a finite $A$-module with finite injective dimension. According to Bruns, this implies that $M$ has finite projective dimension. How do I see that?
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3
votes
3
answers
4k
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Does Ext commute with direct limit?
Is it true that if $\mbox{Ext}^{1}(P,M)=0$ for every finitely generated module $M$ then $P$ is projective? Or that if $\mbox{Ext}^{1}(M,Q)=0$ for every finitely generated module $M$ then $Q$ is inject …
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6
votes
3
answers
3k
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Tor and projective dimension
Is it possible that $\mbox{Tor }^{r+1}(M,N)=0 \ \ \forall N$ yet $\mbox{proj. dim }M>r$?
What I do know is that if $(A,\mathfrak{m})$ is Noetherian local and $M$ is finitely generated over $A$ then $ …
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0
votes
1
answer
419
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Are maximal Cohen-Macaulay modules supported everywhere?
Let $A$ be a local CM ring, and $M$ a maximal CM $A$-module. Is it true that $\operatorname{Supp}M=\operatorname{Spec}A$ ? This suspicion stems from such statements as:
If $\omega$ is a canonical mo …
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4
votes
2
answers
700
views
Why are canonical modules supported everywhere?
Let $A$ be a local CM ring, and $\omega$ a canonical module of $A$. Here are two properties of $\omega$ from Bruns & Herzog:
$\omega_{\mathfrak{p}}$ is a canonical module of $A_{\mathfrak{p}}$ for e …
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9
votes
3
answers
3k
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Free resolution dimension?
Is there a notion of a dimension associated to free resolutions like projective and injective dimensions associated to projective and injective resolutions? My guess is that it coincides with projecti …
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3
votes
0
answers
755
views
Finite generatation of Ext
If $A$ is a Noetherian ring and $M$, $N$ are finitely generated modules over $A$, it is easy to see that $\mbox{Ext}_{A}(M,N)$ is finitely generated by taking a finitely generated projective resolutio …
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2
votes
2
answers
413
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Homological dimensions of module
$(A,\mathfrak{m})$ a Noetherian local ring, $M\neq 0$ a finitely generated $A$-module. As I understand, $\mbox{Ext }^{j}(A/\mathfrak{m}, M) = 0$ for $j<\mbox{depth }(M)$ and for $j>\mbox{inj. dim }(M) …
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3
votes
2
answers
1k
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Depth and dimension
$A$ a Noetherian local ring, $M\neq 0$ a finite $A$-module. Then is it true that $\mbox{depth }M\le\mbox{depth }A$ just like $\mbox{dim }M\le\mbox{dim }A$? I don't see any relation between an $M$-sequ …
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9
votes
2
answers
3k
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Projective & injective dimensions
$A$ a Noetherian local ring, $M\neq 0$ a finite $A$-module. I'm not quite sure about the relation between finiteness of projective and injective dimensions of $M$. Does the finiteness (or infiniteness …
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0
votes
1
answer
183
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Projectively splitting module
Is there a name for such class of modules $M$ such that $M\rightarrow N\rightarrow 0$ splits for every $N$?
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14
votes
3
answers
2k
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Projective dimension of zero module
Is there any consensus on what the projective dimension of the zero module should be? Here are three statements one commonly encounters in textbooks, sometimes with or without the condition $M\neq 0$: …
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