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Results tagged with ac.commutative-algebra
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user 310
Commutative rings, modules, ideals, homological algebra, computational aspects, invariant theory, connections to algebraic geometry and combinatorics.
8
votes
2
answers
757
views
Can any countably generated k-algebra occur as the ring of global sections of some variety?
In the answer to this question we saw that there exists a nonsingular quasi-projective threefold over a field with non-finitely generated global sections.
I was talking about this previous questi …
- 35.5k
2
votes
When are dual modules free?
In the case of regular local rings a criterion for a reflexive module to be free is given in
M. I. Jinnah, Reflexive modules over regular local rings, Archiv der Mathematik volume 26, (1975) pages 36 …
- 35.5k
20
votes
Accepted
Noetherian local ring and the growth of $\dim_k \operatorname{Ext}^i(k,k)$
The commutative noetherian rings such that the Betti numbers of $k$ eventually grow polynomially are precisely the complete intersections. This is a theorem of Gulliksen, see Theorem 2.3 here .
The …
- 8,681
2
votes
Integrally closed factor rings and projective modules
Would it be particularly surprising if this were true? If I have understood what you mean correctly (so that since you want to consider integrality in full generality a version of essentialness relati …
- 25.8k
1
vote
Is projectiveness a Zariski-local property of modules? (Answered: Yes!)
This is not an answer to your question about Zariski-local projectivity, but it is relevant to being locally free and you might be interested.
One can get away with finitely generated rather than fin …
- 8,681
2
votes
When is a localization of a commutative ring finitely generated as an algebra?
In the case that $A$ is noetherian and we replace finitely generated as an algebra with finitely generated as a module we can argue as follows. We have for any choice of minimal prime ideal $P$ of $\m …
- 8,681
2
votes
Dense section of sheaves of modules
The answer is no - the point is that finitely generated projective modules are locally free but not necessarily globally so.
For instance take a Dedekind domain $A$ which does not have unique factori …
- 8,681
7
votes
Are quotients of polynomial rings almost UFDs?
I would say that it is not true that quotients of polynomials rings are "almost UFDs".
For starters, being a quotient of $k[x_1,\ldots,x_n]$ for some $n$ just says that the ring is finitely generated …
- 8,681
4
votes
Primary decomposition for modules
The second definition is the correct one (at least in my opinion). It is similar to the correct notion of defining torsion. For instance one does not in general want to define an abelian group A to be …
- 8,681
5
votes
Different definitions of the dimension of an algebra
I agree with Greg Muller that homological dimension is very nice. In fact various flavours of it turn out to be the same as Krull dimension. For instance a local commutative ring with unit is regular …
- 8,681
3
votes
Are submersions of differentiable manifolds flat morphisms?
I have an idea for the case of a submersion - maybe it is nonsense but maybe not.
In the algebraic case of nonsingular varieties X,Y over an algebraically closed field one can check smoothness (which …
- 8,681
4
votes
Exactness of filtered colimits
A counterexample which is non-trivial is given in Chapter 6 of Neeman's book Triangulated Categories. The category in question is the full subcategory of additive functors Cat(S^{op}, Ab) where S sati …
- 8,681
8
votes
Accepted
Graded local rings versus local rings
One small thing I know of which changes is that if one has a Z-graded-commutative noetherian ring (where Z is the integers) Matlis' classification of indecomposable injective modules goes through but …
- 8,681
8
votes
What is interesting/useful about big Witt Vectors?
From my point of view (which is I should say basically my, probably incorrect, interpretation of Borger's) part of the interest in the big Witt ring is that, at least in the flat case where there are …
- 8,681