2. Commutative algebra aspects

Problem 2.05.
What is a good definition of prime tropical ideals? 
Problem 2.1.
Produce examples of tropical ideals whose varieties are nonrealizable. 
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? 
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 nonrealizable tropical ideals. 
Problem 2.55.
For a given tropical ideal $I$, describe the set of tropical ideals which agree with $I$ in low degree. 
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 $1$.
Cite this as: AimPL: Foundations of tropical schemes, available at http://aimpl.org/tropschemes.