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Simplicial commutative rings are very easy to describe. They're just commutative monoids in the monoidal category of simplicial abelian groups. However, I just realized that a priori, it's not clear that even some of the simplest facts we can prove for ordinary commutative rings (in particular those that depend integrally on the axiom of choice, or even those that depend on the law of the excluded middle) will hold for simplicial commutative rings. However, we have at least one saving grace. That is, the interesting parts of simplicial commutative algebra come from considering things up to homotopy.

So, for example, as far as it make makes sense, can we prove that every simplicial ideal of a simplicial commutative ring is weakly equivalent to one contained in a maximal simplicial ideal? Perhaps a better way to state this would be something like, "every noncontractible simplicial commutative ring admits at least one surjective map to a simplicial commutative ring that's weakly equivalent to a simplicial field", or some variation on where the homotopy equivalence appears. Given that the axiom of choice does not necessarily hold in $sSet$, it doesn't seem reasonable to think that the ordinary theorem will hold.

Is there a version of the Hilbert basis theorem that holds up to isomorphism? How about weak equivalence?

What other well-known theorems will fail, even up to homotopy?

Simplicial commutative rings are very easy to describe. They're just commutative monoids in the monoidal category of simplicial abelian groups. However, I just realized that a priori, it's not clear that even some of the simplest facts we can prove for ordinary commutative rings (in particular those that depend integrally on the axiom of choice, or even those that depend on the law of the excluded middle) will hold for simplicial commutative rings. However, we have at least one saving grace. That is, the interesting parts of simplicial commutative algebra come from considering things up to homotopy.

So, for example, as far as it make sense, can we prove that every simplicial ideal of a simplicial commutative ring is weakly equivalent to one contained in a maximal simplicial ideal? Perhaps a better way to state this would be something like, "every noncontractible simplicial commutative ring admits at least one surjective map to a simplicial commutative ring that's weakly equivalent to a simplicial field", or some variation on where the homotopy equivalence appears. Given that the axiom of choice does not necessarily hold in $sSet$, it doesn't seem reasonable to think that the ordinary theorem will hold.

Is there a version of the Hilbert basis theorem that holds up to isomorphism? How about weak equivalence?

What other well-known theorems will fail, even up to homotopy?

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# What facts in commutative algebra fail miserably for simplicial commutative rings, even up to homotopy?

Simplicial commutative rings are very easy to describe. They're just commutative monoids in the monoidal category of simplicial abelian groups. However, I just realized that a priori, it's not clear that even some of the simplest facts we can prove for ordinary commutative rings (in particular those that depend integrally on the axiom of choice, or even those that depend on the law of the excluded middle) will hold for simplicial commutative rings. However, we have at least one saving grace. That is, the interesting parts of simplicial commutative algebra come from considering things up to homotopy.

So, for example, as far as it make sense, can we prove that every simplicial ideal of a simplicial commutative ring is weakly equivalent to one contained in a maximal simplicial ideal? Given that the axiom of choice does not necessarily hold in $sSet$, it doesn't seem reasonable to think that the ordinary theorem will hold.

Is there a version of the Hilbert basis theorem that holds up to isomorphism? How about weak equivalence?

What other well-known theorems will fail, even up to homotopy?