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Results tagged with ac.commutative-algebra
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user 31233
Commutative rings, modules, ideals, homological algebra, computational aspects, invariant theory, connections to algebraic geometry and combinatorics.
3
votes
Does 'finite + finitely presented as an algebra' equal 'finitely presented as a module'?
It appears that in the meantime, full proofs have been added to the Stacks Project.
Tag 0564. Let $R \to S$ be a finite and finitely presented ring map. Let $M$ be an $S$-module. Then $M$ is finit …
- 3,312
13
votes
1
answer
626
views
Constructively correct notion of unique factorization domain
Recall the well-known proof that a unique factorization domain is a GCD domain:
Let $x, y \in R \setminus \{ 0 \}$. Factor $x$ and $y$ into pairwise non-associated irreducible elements: $$\begin{a …
- 3,312
5
votes
Is there a universal property for graded localization?
It seems that Martin has provided the answer which Daniel sought. But the question in the title doesn't appear to be answered yet: Is there a universal property for graded localization? More precisely …
- 3,312
9
votes
When can we prove constructively that a ring with unity has a maximal ideal?
If the ring is countable (or the image of a linear well-ordering), then no choice of any kind (not even countable choice) and in fact not even the law of excluded middle is required: There is an expli …
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10
votes
Alternate proofs of Hilberts Basis Theorem
Sorry, this is not an answer, but rather a too-long elaboration on constructive aspects. I post this here because there was some interest about the constructive content of the theorem in the comments. …
1
vote
Affine communication lemma and finite limits in the category of rings
I think that Section 11 on transfer principles in these notes of mine is what you're looking for. A general machinery abstracts the business of tracking all the $f_i$'s and the required high powers. T …
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3
votes
Can the methods of classical algebraic geometry be made rigorous with a synthetic approach?
Is it possible to do algebraic geometry in a synthetic manner that enables rigorous reasoning but is closer to the style of argument employed by classical algebraic geometers?
I sure hope so. You …
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4
votes
constructive Serre classes
Any subclass $\mathcal{C}$ of an abelian category determines a smallest Serre class containing it, by iteratively adding (the zero object and) the object $Y$ for any exact sequence $X \to Y \to Z$ whe …
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6
votes
Constructively, is the unit of the “free abelian group” monad on sets injective?
Yes! Here is a proof which is slightly different from both your proof and the proof in Mines–Richman–Ruitenberg.
First define the similarity relation on $\mathrm{List}(R \times X)$ as in Mines–Richma …
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41
votes
Accepted
Joyal's construction of the spectrum of a commutative ring
Since I don't know precisely which parts of Lawvere's article you have difficulties with, this answer is a bit a long and tries to give a bit of context. If you want me to be more specific at some poi …
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11
votes
Accepted
Constructive proof that a kernel consists of nilpotent elements
This answer provides a scheme how to construct a constructive proof, though I'm still working to actually explicitly extract the constructive proof, so please don't accept the answer just yet. (Update …
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