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Stream: learning: questions

Topic: Why are invertible fractional ideals projective?

John Baez (Apr 02 2025 at 02:55):

Here's a proof that invertible fractional ideals are projective, and if you follow the links you can learn what all the words in that statement mean:

• Proof of invertible ideals are projective.

I would like to see a more category-theoretic proof.

John Baez (Apr 02 2025 at 03:05):

The idea seems to be this... I could easily screw up here...

If you have a commutative ring $R$ that's an integral domain, you can form its field of fractions $\mathrm{Frac}(R)$. Since it contains $R$ as a subring, $\mathrm{Frac}(R)$ becomes an $R$-module, and any submodule of this is called a fractional ideal of $R$. These are really important in number theory.

The set of fractional ideals of $R$ becomes a monoid because given two fractional ideals $I,J$ we can define a new one

$IJ = \{i_1 j_1 + \cdots + i_n j_n : i_k \in I, j_k \in J\} \subseteq \mathrm{Frac}(R)$

John Baez (Apr 02 2025 at 03:09):

Q1: Is this isomorphic to the tensor product of $I$ and $J$ as $R$-modules?

We say a fractional ideal $I$ of $R$ is invertible if there's a fractional ideal $J$ with

$I J = 1$

where $1$ is the fractional ideal that's $R$ itself.

John Baez (Apr 02 2025 at 03:09):

Q2: Why are invertible fractional ideals projective as $R$-modules?

John Baez (Apr 02 2025 at 03:13):

I'm hoping there's some explanation like this: for any commutative ring $R$ we can take the tensor product of $R$-modules so $R \mathsf{Mod}$ becomes a symmetric monoidal category. This lets us define an invertible $R$-module $M$ to be one for which there's a module $N$ with

$M \otimes N \cong I$

where $I$ is the unit for the tensor product, namely $R$ itself viewed as an $R$-module. I'm hoping that any invertible $R$-module is projective!

Q3: If $R$ is a commutative ring, is any invertible $R$-module projective?

If so, I'm hoping there's some rather simple category-flavored proof.

John Baez (Apr 02 2025 at 03:39):

Q4: Can we use the hoped-for fact that invertible $R$-modules are projective to show that invertible fractional ideals are projective?

An affirmative answer to Q1 should make this possible.

John Baez (Apr 02 2025 at 05:23):

Oh, I get the answer to Q3. Yes, because any $R$-module with a dual is projective (see the Corollary here for a proof of that), and in any symmetric monoidal category an inverse of an object is a dual of that object.