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Results tagged with homological-algebra
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user 2083
(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.
5
votes
Accepted
How to construct a ring with global dimension m and weak dimension n?
If $R$ is Noetherian then they are equal.
For $n=0$ one can use the fact that any Boolean ring has weak dimension $0$ (any module is flat), but a free Boolean ring on $\aleph_n$ generators have global …
- 30.6k
4
votes
Why depth, dimension, etc?
A problem with the first question is that the definition of grade function in McAdam involves some ring-theoretic concepts. For example, I do not see how to express condition (iii) of Theorem 2.4 (abo …
- 30.6k
6
votes
Accepted
In what degrees does Ext(S/(f),S) vanish?
It is not true, $Ext^1(S/(f), S)\neq 0$ as Ben pointed out. However, to prove what you want $Ext^m(S/I,S)\cong Ext^m(S/fI,S)(deg(f))$ for $m\geq 2$, just note that
$$Ext^m(S/I,S) = Ext^{m-1}(I,S) $$ …
- 30.6k
12
votes
Accepted
Differential graded structures on free resolution?
It is true if the projective dimension of $M$ over $A$ is at most $3$, and counter examples exist when the projective dimension is $4$.
The first counter example was given in Lucho Avramov's paper
" …
- 30.6k
4
votes
Accepted
Extensions of torsion modules
Actually, it would be easier to look at the other end of the exact sequence. Namely, your map $f^*$ is trivial implies the map $g^*: Hom_S(T,Q) \to Hom_S(M,Q)$ is an isomorphism.
Now, since $M$ is p …
- 30.6k
6
votes
Accepted
Are any finitely generated reflexive module a 2nd syzygy?
Over a normal domain (in fact, you only need Gorenstein in codimension 1, being second syzygy and reflexive are equivalent). This is Theorem 3.6 of Evans-Griffith "Syzygies" book.
- 30.6k
8
votes
Betti sequence of finite dimensional commutative algebras
I will first show that the general answer is no (in fact one should expect the opposite even in the graded case) by the Pigeonhole principle. Then we shall construct some concrete examples via an old …
- 30.6k
11
votes
Accepted
Commutative algebras with modules of small complexity
There are no other examples. This property is equivalent to $A$ being a hypersurface (see Avramov's note "Infinite Free Resolutions"). By Cohen Structure Theorem, an Artinian local hypersurface (which …
- 30.6k
6
votes
When are MCM ideals principal?
On question 1, for what rings all MCM ideals are principal, we can say quite a bit more if one knows that $R$ is parafactorial (that is, the Picard group of the punctured spectrum $Spec^o(R):=Spec(R)- …
- 30.6k
5
votes
Accepted
Modules over a Gorenstein ring
Here is a proof which may not be the best but demonstrates some standard techniques:
Since $R$ has finite inj. dim. one can replace $M$ by a high syzygy, so one can assume $M$ has full depth. Thus on …
- 30.6k
11
votes
Tor and projective dimension
Since your question is really about projective dimension of flat modules, it is worth noting the following result (see Raynaud-Gruson MR0308104, Cor 3.3.2 or Jensen MR0407091, Thm 5.8) which compleme …
- 30.6k
3
votes
1
answer
564
views
When can one localize Ext?
Let $R\to S$ be a ring map such that $S$ is projective over $R$ (I am willing to assume $S=R[X_1,...,X_n]$). Let $M,N$ be finite $S$-modules. Let $P\in Spec R$ such that $M_P$ is $R_P$-flat. Under wha …
- 30.6k
21
votes
Serre's theorem about regularity and homological dimension
ADDED: There is an account written by Buchsbaum (see page 1 and 2 of number 23 here) which described in more details what they wrote in [1]. So the localization problem for regular rings was definitel …
- 30.6k
6
votes
Projective resolution of modules over rings which are regular in codimension n
Dear Liu,
I like your updated question a lot. To make things easier to discuss, let me define the following properties for a Noetherian local ring $R$ and $n>0$:
($A_n$) every ideal with height le …
- 30.6k
6
votes
Generalizing miracle flatness (Matsumura 23.1) via finite Tor-dimension
Hi Bryden,
I agree with Graham that it would be hard to have a generalization in the sense you want.
As Graham pointed out you already have finite Tor-dimension if $A$ is regular. In general, finite …
- 30.6k