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2. Commutative algebra aspects


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      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.


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          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.


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              Problem 2.15.

              How do we intersect tropical ideals? How do we add tropical ideals?


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                  Problem 2.2.

                  How can we describe a tropical ideal with a finite amount of information?


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                      Problem 2.25.

                      Is there a primary decomposition for tropical ideals?


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                          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.


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                              Problem 2.35.

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


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                                  Problem 2.4.

                                  How does the matroidal condition defining tropical ideals interact with Berkovich skeleta?

                                    • End of workshop problems


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                                      Problem 2.45.

                                      Generate (more) examples of interesting tropical ideals, especially non-realizable tropical ideals.


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                                          Problem 2.5.

                                          Find a new example of a tropical ideal which is prime in the naive sense.


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                                              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?


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                                                  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 $1$.


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                                                      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.