CRing.sql

INSERT INTO category_property_assignments (
	category_id,
	property_id,
	is_satisfied,
	reason
)
VALUES
(
	'CRing',
	'locally small',
	TRUE,
	'There is a forgetful functor $\mathbf{CRing} \to \mathbf{Set}$ and $\mathbf{Set}$ is locally small.'
),
(
	'CRing',
	'finitary algebraic',
	TRUE,
	'Take the algebraic theory of a commutative ring.'
),
(
	'CRing',
	'strict terminal object',
	TRUE,
	'If $f : 0 \to R$ is a homomorphism, then $R$ satisfies $1=f(1)=f(0)=0$, so that $R=0$.'
),
(
	'CRing',
	'Malcev',
	TRUE,
	'This follows in the same way as for groups, see also Example 2.2.5 in <a href="https://ncatlab.org/nlab/show/Malcev,+protomodular,+homological+and+semi-abelian+categories" target="_blank">Malcev, protomodular, homological and semi-abelian categories</a>.'
),
(
	'CRing',
	'coextensive',
	TRUE,
	'[Sketch] A ring homomorphism $f : A \times B \to R$ yields the idempotent element $e := f(1,0) \in R$, so that $R \cong eR \times (1-e)R$. Then $f$ decomposes into the ring homomorphisms $f_A : A \to eR$, $f_A(a) := f(a,0)$ and $f_B : B \to (1-e)R$, $f_B(b) := f(0,b)$.'
),
(
	'CRing',
	'balanced',
	FALSE,
	'The inclusion $\mathbb{Z} \hookrightarrow \mathbb{Q}$ is a counterexample.'
),
(
	'CRing',
	'cogenerating set',
	FALSE,
	'We apply <a href="/lemma/missing_cogenerating_sets">this lemma</a> to the collection of fields: If $F$ is a field and $R$ is a non-trivial commutative ring, any ring homomorphism $F \to R$ is injective. For every infinite cardinal $\kappa$ the field of rational functions in $\kappa$ variables has cardinality $\geq \kappa$ and a non-trivial automorphism (swap two variables).'
),
(
	'CRing',
	'skeletal',
	FALSE,
	'This is trivial.'
),
(
	'CRing',
	'countably codistributive',
	FALSE,
	'The canonical homomorphism $\mathbb{Q} \otimes \mathbb{Z}^{\mathbb{N}} \to (\mathbb{Q} \otimes \mathbb{Z})^{\mathbb{N}} = \mathbb{Q}^{\mathbb{N}}$ is not an isomorphism: its image consists of those sequences of rational numbers whose denominators can be bounded.'
),
(
	'CRing',
	'semi-strongly connected',
	FALSE,
	'There is no homomorphism between $\mathbb{F}_2$ and $\mathbb{F}_3$.'
),
(
	'CRing',
	'coregular',
	FALSE,
	'See <a href="https://math.stackexchange.com/questions/3745302" target="_blank">MSE/3745302</a>.'
),
(
	'CRing',
	'co-Malcev',
	FALSE,
	'See <a href="https://mathoverflow.net/questions/509552">MO/509552</a>: Consider the forgetful functor $U : \mathbf{CRing} \to \mathbf{Set}$ and the relation $R \subseteq U^2$ defined by $R(A) := \{(a,b) \in U(A)^2 : ab = a^2\}$. Both are representable: $U$ by $\mathbb{Z}[X]$ and $R$ by $\mathbb{Z}[X,Y] / \langle XY-X^2 \rangle$. It is clear that $R$ is reflexive, but not symmetric.'
),
(
	'CRing',
	'regular quotient object classifier',
	FALSE,
	'Assume that $P \to \mathbb{Z}$ is a regular quotient object classifier. If $J$ denotes its kernel, this means that every ideal $I \subseteq A$ of any commutative ring has the form $I = \langle \varphi(J) \rangle$ for a unique homomorphism $\varphi : P \to A$. If $\sigma : A \to A$ is an automorphism with $\sigma(I)=I$, then uniqueness gives us $\sigma \circ \varphi = \varphi$, which means that $\varphi(J)$ lies in $A^{\sigma}$, the fixed ring of $\sigma$. But then $I$ is generated by elements in the fixed ring. This fails for $A = \mathbb{Z}[X]$, $I = \langle X \rangle$, $\sigma(X)=-X$. The fixed ring is $\mathbb{Z}[X^2]$, and if $I$ was generated by elements $f \in \mathbb{Z}[X^2] \cap I$, they would be multiples of $X^2$, but $X$ is not a multiple of $X^2$.'
);