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Results tagged with ac.commutative-algebra
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user 4194
Commutative rings, modules, ideals, homological algebra, computational aspects, invariant theory, connections to algebraic geometry and combinatorics.
2
votes
$\inf\{i\in \mathbb N \cup \{0\}\cup\infty\mid Ext^i_R(R/I,R)\neq 0\}=0 ?$
I think the answers to both questions are yes.
Put a grading on $R$ so that it is connected (zero in negative degrees, $k$ in degree 0): for example, put $x_n$ in degree $n$. (It also seems safest to …
- 4,254
4
votes
0
answers
97
views
Compute equalizer of maps of polynomial rings, perhaps using Gröbner bases
Suppose that $k$ is a field and I have two ring homomorphisms
$$
\phi, \psi :k[x_1,...,x_m] \to k[y_1,...,y_n].
$$
How can I use Gröbner bases (or other computational tools) to compute the subring of …
- 4,254
7
votes
Over which (graded) rings are all modules decomposable into indecomposables?
In the book Spectra and the Steenrod Algebra, Margolis proves the following (Theorem 21 in chapter 11): if $A$ is a graded connected algebra over a finite field and if $M$ is an $A$-module which is fi …
- 4,254
8
votes
What are some toy models for the stable homotopy groups of spheres?
As Dave Benson says, the Noetherian condition simplifies a lot of things. The derived category of a commutative ring satisfies many of the properties of the stable homotopy category. The derived categ …
- 4,254
4
votes
Is being graded commutative a necessary condition on $A$ such that $H^*(A)$ is commutative?
I think that Mark Grant's answer is great, but if you want a simpler algebra, consider
$$
k\langle a_2, b_2, c_3 \rangle / (ac-ca, bc-cb, [a,[a,b]], [b, [a,b]]).
$$
(Subscripts indicate the degrees of …
- 4,254