All Questions
Tagged with picard-group ac.commutative-algebra
14 questions
5
votes
0
answers
288
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Picard group of almost module category
I am very new to the world of almost mathematics and I am curious about the following:
Fix an almost mathematics situation $(R,I)$ throughout. Very generally, the almost module category comes with a ...
- 51
2
votes
1
answer
185
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Finite étale cover of factorial ring
Let $A$ be a regular factorial ring.
Consider $B=A[X]/(P)$ such that $B$ is finite étale over $A$. When do we have that $B$ is also factorial?
- 3,472
2
votes
1
answer
165
views
$K_0((k[x]/(x^2))[y])$
Let $K_0(R):= K_0(P(R))$ where $P(R)$ is the category of finitely generated projective $R$-modules, where $R$ is a commutative ring with unity. Now if $R = k[x]/(x^2)$, $R$ is a local ring thus all ...
- 167
4
votes
1
answer
194
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When $R $ is a cusp then $K_0(R) \ncong K_0(R[s])$
Quillen's classical result shows that if $R$ is a regular ring then $K_0(R) \cong K_0(R[t_1,...,t_m])$ for all $m \in \mathbb{N}$. So I wanted to construct some elementary examples where $K_0(R)$ ...
- 167
3
votes
0
answers
72
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Automorphisms of matrix algebras and Picard group
This is a repost of https://math.stackexchange.com/questions/4692364/automorphisms-of-matrix-algebras-and-picard-group (asked on MSE).
Notation. In what follows, $R$ is a commutative ring with $1$, $n\...
- 1,766
3
votes
0
answers
124
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Picard group of a cusp [duplicate]
$\DeclareMathOperator\Pic{Pic}$For a commutative ring with unity $R$, if $R_{\mathrm{red}}$ is semi-normal then we have an equivalent criterion which states that $\Pic(R) \cong \Pic(R[t])$. Here $\Pic(...
- 167
7
votes
2
answers
2k
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Picard group vs class group
The question.
Let $R$ be a commutative ring. Let $M$ be an $R$-module with the property that there exists an $R$-module $N$ such that $M\otimes_R N\cong R$. Does there always exist an ideal $I$ of $R$ ...
- 41.4k
9
votes
1
answer
607
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Bézout ring with non-trivial Picard group?
[I asked this on stackexchange here a few weeks ago to no response]
A ring is called Bézout when its finitely generated ideals are principal.
Q: Is there a nice example of a Bézout ring $R$ with ...
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9
votes
2
answers
695
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On a morphism from the Brauer group to the Picard group
Suppose that $k$ is a commutative ring and that $A$ is an Azumaya
$k$-algebra. Then there is a well-known morphism from $Aut_{Alg_k}(A)$, the group of algebra automorphisms, to the Picard group $Pic(k)...
- 52.7k
9
votes
1
answer
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Picard group and reduced schemes
$\DeclareMathOperator\Pic{Pic}$If $A$ is a ring, then we know that $\Pic(A)=\Pic(A_\text{red})$, but for a scheme $X$ it is false in general.
On the other hand, we have that $\Pic(X)=H^{1}_{et}(X,\...
- 3,472
6
votes
0
answers
129
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Center Picard group non-commutative algebra
I am wondering if there is a way to describe the center of the Picard group of a non-commutative algebra.
Namely, let $A$ be a finitely generated algebra over a field $k$. Denote by $\mathrm{Pic}(A)$...
- 7,320
22
votes
4
answers
2k
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Two questions about finiteness of ideal classes in abstract number rings
Let us say that an abstract number ring is an integral domain $R$ which is not a field, and which has the "finite norms" property: for any nonzero ideal $I$ of $R$, the quotient $R/I$ is finite.
(I ...
- 65.4k
15
votes
6
answers
2k
views
Seeking Noetherian normal domain with vanishing Picard group but not a UFD
Once again, the question says it all.
My motivation is the article on factorization I am writing. I want to explain (as well as to understand!) why for normal Noetherian domains of dimension greater ...
- 65.4k
34
votes
4
answers
5k
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What is the right definition of the Picard group of a commutative ring?
This is a rather technical question with no particular importance in any case of actual interest to me, but I've been writing up some notes on commutative algebra and flailing on this point for some ...
- 65.4k