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## 2. Commutative algebra aspects

1. #### Problem 2.05.

What is a good definition of prime tropical ideals?
The naive definition (that the complement is closed under tropical multiplication) has some nice properties, but also some issues. For example, the prime ideal $(x-1) \subseteq K[x,y]$ tropicalizes to a tropical ideal which is not prime in the naive sense.
• #### Problem 2.1.

Produce examples of tropical ideals whose varieties are non-realizable.
For example, produce a tropical ideal $I$ such that $V(I)$ is the Bergman fan of a non-realizable matroid.
• #### Problem 2.15.

How do we intersect tropical ideals? How do we add tropical ideals?
• #### Problem 2.2.

How can we describe a tropical ideal with a finite amount of information?
• #### Problem 2.25.

Is there a primary decomposition for tropical ideals?
• #### Problem 2.3.

Develop descent theory for modules over a $\mathbb{T}$-algebra $A$. This will likely require finding an appropriate subcategory of the category of $A$-modules.
• #### Problem 2.35.

Are there syzygies in tropical geometry?
Begin by tropicalizing simple examples, e.g. the twisted cubic.
• #### Problem 2.4.

How does the matroidal condition defining tropical ideals interact with Berkovich skeleta?
• ### End of workshop problems

#### Problem 2.45.

Generate (more) examples of interesting tropical ideals, especially non-realizable tropical ideals.
• #### Problem 2.5.

Find a new example of a tropical ideal which is prime in the naive sense.
• #### Problem 2.55.

For a given tropical ideal $I$, describe the set of tropical ideals which agree with $I$ in low degree.
For example, which tropical ideals have $I_1$ equal to the vectors of a fixed valuated matroid?
• #### Problem 2.6.

Develop a notion of prime ideals such that under the reduction map $\mathbb{T}_+[x_1, \ldots, x_n] \to \mathbb{B}[x_1, \ldots, x_n]$ (here $\mathbb{T}_+ = \mathbb{R}_{\geq 0} \cup \{\infty\}$), a prime tropical ideal in $\mathbb{T}_+[x_1, \ldots, x_n]$ drops its dimension by $1$.
• #### Problem 2.65.

What is the semiring analogue of a DVR (or valuation ring more generally)?

Cite this as: AimPL: Foundations of tropical schemes, available at http://aimpl.org/tropschemes.