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\DeclareMathOperator{\sgn}{sgn} \DeclareMathOperator{\Ext}{Ext} \DeclareMathOperator{\TropB}{TropB} \DeclareMathOperator{\weight}{wt} \DeclareMathOperator{\Span}{span} \DeclareMathOperator{\Coh}{Coh} \DeclareMathOperator{\Pic}{Pic} \DeclareMathOperator{\Fuk}{Fuk} \DeclareMathOperator{\str}{star} \DeclareMathOperator{\Ob}{Ob} \DeclareMathOperator{\Coh}{Coh} \DeclareMathOperator{\CritVal}{CritV} \DeclareMathOperator{\Sing}{Sing} \DeclareMathOperator{\FS}{FS} \DeclareMathOperator{\Vect}{Vect} \DeclareMathOperator{\grad}{grad} \DeclareMathOperator{\Supp}{Supp} \DeclareMathOperator{\Bl}{Bl} \DeclareMathOperator{\Spec}{Spec} \DeclareMathOperator{\Tw}{Tw} \DeclareMathOperator{\Int}{Int} \DeclareMathOperator{\Arg}{\mathbf{M}}\begin{filecontents}{references.bib} @article{ballard2012hochschild, title={Hochschild dimensions of tilting objects}, author={Ballard, Matthew and Favero, David}, journal={International Mathematics Research Notices}, volume={2012}, number={11}, pages={2607--2645}, 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To each tropical Cartier divisor we can construct an associated Weil divisor: take the graph of $\phi$, which is a polyhedral complex; extend this to a balanced graph by adding in weighted edges going to the tropical boundary. \begin{figure} \label{fig:TropicalWeilDivisors} \centering \begin{tikzpicture} \draw (-3,-1) -- (2.5,-1); \draw (-3,0.5) -- (-1.5,0.5) -- (0,2) -- (2.5,2); \draw (-1.5,0.5) -- (-1.5,-1) (0,2) -- (0,-1); \node at (3,2) {graph($\phi$)}; \node at (-1.5,-1.5) {$1$}; \node at (0,-1.5) {$-1$}; \end{tikzpicture} \caption{A tropical Cartier divisor gives a Weil divisor by balencing the graph of the tropical function. } \end{figure} From now on, everything happens in $\RR^n$. A map $f: (Q\subset \RR^n)\to (P\subset \RR^m)$ is a morphism of cycles if it is a map of polyhedral complexes which preserves the affine structure. From this we obtain the following: \begin{proposition} \label{prp:TropicalPullback} Given $\phi: P\to \RR$ a tropical rational function, the pullback $\phi^*f: Q\to \RR$ is a rational function on $Q$. Furthermore, the pullback function induces a map $\phi^*: \mathrm{Div}(P)\to \mathrm{Div}(Q)$. \end{proposition} \begin{proposition} \label{prp:TropicalPushforward} Given $Z\subset X$ a cycle in $X$, we define $f_*Z\subset Y$ the pushforward of $Z$ which satisfies the properites: \begin{itemize} \item $|f_*Z|=f(|Z|)$ \item By picking a sufficiently fine representative of $Z$, we have \[f_*Z = \{f(\sigma) \st \sigma\in Z \text{so that $\sigma$ is contained in a maximal cell on which $f$ is injective}\}\] where the weights \[w(\sigma')=\sum_{\substack{\sigma\in Z^{top}\\ f(\sigma)=\sigma}} w_Z\sigma \cdot \left|\frac{\Lambda_\sigma'}{f(\Lambda_\Sigma')}\right|.\] \end{itemize} The pushforward is balanced. \end{proposition} \begin{example} \label{exm:TropicalPullbackAndPushforward} Consider the pair of pants in $\RR^2$, and the map $f(q_1, q_2)=q_1+q_2$. \begin{figure} \label{fig:TropicalPairOfPantsPushforward} \centering \begin{tikzpicture} \draw (-0.5,0) -- (-2,0) (-0.5,0) -- (-0.5,-1.5) (-0.5,0) -- (0.5,1); \draw (3.5,0) -- (6.5,0); \draw (-2,1) rectangle (0.5,-1.5); \node (v1) at (-0.5,-2.5) {$\mathbb R^2$}; \node (v2) at (5.5,-2.5) {$\mathbb R$}; \draw[->] (v1) edge node[fill=white]{$\begin{pmatrix}1& 1 \end{pmatrix}$} (v2); \node at (4.5,1) {$(f_1)_*X=\{\{x\leq 0\}, \{0\}, \{x\geq 0\}\}$}; \node[fill, circle, scale=.2] at (5,0) {}; \draw[->] (5,0) -- (6,0); \draw[->>] (5,0) -- (4.5,0); \end{tikzpicture} \caption{Pushforward of a function to the tropical pair of pants} \end{figure} \end{example} \begin{proposition} \label{prp:ProjectionFormula} Let $\phi\in \mathrm{Div}(Y)$ and $Z\in Z_k(X)$, then $f_*(f^*(\phi)\cdot Z)= \phi\cdot f_*(Z)$ \end{proposition} \begin{definition}{stable intersection} Let $X, Y$ be tropical intersection. The stable intersection is defined by $X\cdot Y=\lim_{\eps\to 0} X\cap (Y+\eps v)$ over all intersections of $X, Y+\eps v$ which are transverse. \end{definition} Outside of the setting of $\RR^n$, it is not possible to define this intersection using this formula. We can define (generally) the intersection $\mathrm{Div}(\phi)\cdot Z$, but that is it. \section{rational equivalence in tropical geometry} \label{art:RationalEquivalenceInTropicalGeometry} \begin{definition} \label{def:TropicalRationalEquivalence} Let $Z\subset Q$ be a subcycle (bounded) rationally equivalent to $0$ if there exists a cycle $Y, \dim(Y)=\dim(Z)$ and a morphism $Y\to Q$ and a (bounded) rational function $\phi\in \mathcal K^*_Y$ such that \[f_*(\deg(\phi)\cdot Y)=Z.\] \end{definition} If you're familiar with rational equivalence in algebraic geometry, the main difference is that we ask the cycle $Y$ to be a subvariety of $X$. Originally, the definition used that $Y$ was a subset of $X$, but unfortunately this definition is not compatible with pushforward. The belief is that the definition of tropical rational function is too rigid (so our definition for tropical rational equivalence) is a bit more flexible. \begin{definition} \label{def:TropicalChowGroup} Let $X$ be a tropical variety. We define the tropical chow groups $A^{(b)}_\bullet(X)=Z^{(b)}_\bullet(X)/\sim^{(b)}$ \end{definition} Let $Q\subset Q\times \RR$. Let $\phi_p$ be the pullback of $\max(x, p')$. Then $F_p=\mathrm{Div}(\phi\cdot F)\subset Q\times \{Q\}\cong Q$ Given $P_1, 2\in Q$ we say that the are $\sim^\RR$ if there exists a cycle $F$ and points $q_1, q_2\in \RR$ so that $P_i=F_{q_i}$. \begin{proposition} \label{prp:CharacterizationOfBoundedRationalEquiavlence} A cycle $P_1\sim^b 0$ if and only if $Z_1\sim^\RR 0$ \end{proposition} \begin{proof} \label{prf:CharacterizationOfBoundedRationalEquiavlence} Suppose that $P_1\sim^\RR P_2$. Then there exists $F\subset Q\times \RR$ with $F_{q_i}=P_i$. Take $R=F$, and define $\phi:=\pi^*\psi$. We see that $(\pi_Q)_* (\pi^*(\psi^*\psi)\cdot F)=P_1-P_2$. NOw suppose instead that $P_1\sim^b P_2$. Then there exists $R$ and $\phi\in \mathcal K_R$ such that $f_*(\mathrm{Div}(\phi))=P_1-P_2$. Tke $F:=(f\times \id)_* (\text{graph}(\phi))$. Cruicially, the function $\phi$ is bounded so that the intersection with a sufficiently large slice is empty \begin{figure} \label{fig:ALargeSliceIsEmpty} \centering \begin{tikzpicture} \draw (-1.5,0) -- (1.5,0); \draw (-1.5,0.5) -- (-0.5,0.5) -- (0.5,2) -- (1.5,2); \node at (2.5,0) {$R$}; \draw (-0.5,0.5) -- (-0.5,-0.5) (0.5,2) -- (0.5,-0.5); \draw[dashed] (-1.5,-0.5) -- (1.5,-0.5); \draw[dashed] (-1.5,3) -- (1.5,3); \end{tikzpicture} \caption{sufficiently large slices have empty intersection} \end{figure} \end{proof} \begin{remark} Observe that if $R\subset P$ has $R\sim^b 0$ and $g: P\to Q$ is a map of tropical varities that $g_*Z\sim^b0$. However, consider the morphism this gives an example of $Z\sim 0$ but $f_*Z\not\sim 0$. \begin{figure} \label{fig:APullbackOfAZeroCycleMayNotBeZero} \centering \begin{tikzpicture} \draw (-2.5,0.5) ellipse (1.5 and 0.5); \draw (2.5,0.5) .. controls (2,0) and (0.5,0) .. (0.5,0.5) .. controls (0.5,1) and (2,1) .. (2.5,0.5) .. controls (3,0) and (4.5,0) .. (4.5,0.5) .. controls (4.5,1) and (3,1) .. (2.5,0.5); \node[circle, scale=.2, fill] at (-3,1) {}; \node[circle, scale=.2, fill] at (-3,0) {}; \draw (-4,-2) -- (-1.5,-2); \draw (-4,-1.5) -- (-3.5,-1.5) -- (-3,-1) -- (-2.5,-1) -- (-2,-1.5) -- (-1.5,-1.5); \node[circle, scale=.2, fill] at (-3.5,-2) {}; \node[circle, scale=.2, fill] at (-3,-2) {}; \node[circle, scale=.2, fill] at (-2,-2) {}; \node[circle, scale=.2, fill] at (-2.5,-2) {}; \node[below] at (-3.5,-2) {$+$}; \node[below] at (-3,-2) {$-$}; \node[below] at (-2,-2) {$+$}; \node[below] at (-2.5,-2) {$-$}; \begin{scope}[shift={(0,-2)}] \draw (2.5,0.5) .. controls (2,0) and (0.5,0) .. (0.5,0.5) .. controls (0.5,1) and (2,1) .. (2.5,0.5) .. controls (3,0) and (4.5,0) .. (4.5,0.5) .. controls (4.5,1) and (3,1) .. (2.5,0.5); \end{scope} \node at (0.5,-1.5) {$+$}; \node at (4.5,-1.5) {$+$}; \node at (2.5,-1.5) {$-2$}; \end{tikzpicture} \caption{} \end{figure}\end{remark} \section{bounded rational equiavlence in \(\RR^n\)} \label{art:BoundedRationalEquiavlenceInRrN} \begin{definition} \label{def:TropicallyNumericalEquivalenct} Let $Q\subset \RR^n$ be a cycle of codimension $k$. We obtain a map \begin{align*} d_Q: Z_k(\RR^n)\to \ZZ\\ P\mapsto \deg(P\cdot Y) \end{align*} \end{definition} We say that $Q_1$ is numerically equivalent to $Q_2$ if $d_{Q_1}=d_{Q_2}$. \end{definition} Observe that if $Q_1\sim^b Q_2$, this is the same as saying that $Q_1-Q_2\sim^b 0$, which implies that $d_{Q_1}-d_{Q_2}=0$. It is natural to ask if the converse hold. We first understand the question locally, i.e. the intersection of fans. \begin{proposition} \label{prp:DDeterminesAFan} If $Q_1, Q_2\subset \RR^n$ are fans in $\RR^n$, then $d_{Q_1}=d_{Q_2}$ tells us that $Q_1 = Q_2$. \end{proposition} \begin{corollary} If $Q_1\sim^b Q_2$ are bounded equivalent fans, they are the same fan. \end{corollary} The plan: \begin{enumerate} \item Associate to a cycle $Q$ a fan $Q'$, \item Show that $Q\sim^b Q'$, \item use the previous results. \end{enumerate} For the first part: Let $\sigma\subset \RR^n$ be the polyhedron. Define the recession cone \[\text{Rec}(\sigma):= \{v\in \RR^ \st q+\RR_{\geq 0} v \subset \sigma \forall q\in \sigma\}\] For sufficiently fine polyhedral structure on $Q$, one can show that the $\text{Rec}(X):=\{\text{Rec}(\sigma), \sigma\in X\}$ comes with the structure of a balanced fan. This essentially is just ``zooming out'' really far away from your variety. \begin{theorem} \label{thm:CyclesAndRecessionCones} Let $Q\subset \RR^n$ be a cycle. It is bounded rationally equivallent to its recession cone, that it: $Q\sim^b \text{Rec}(Q)$ \end{theorem} The main proof of the theorem is the following. \begin{proposition} \label{prp:TropicalCyclesAsSums} Any cycle $Q$ can be written as a sum, \[Q=\sum(Q_i+\vec v_i)\] where each $Q_i$ is a fan. \end{proposition} \begin{proof} \label{prf:TropicalCyclesAsSums} Every $q\in Q$ has a neighborhood $\text{Star}_Q(p)$ which looks like a fan. For any fan $P$, we define its splitting dimension $\dim_s(P)$ to be the maximum $k$ so that \[P=\sum P_i\] where the $P_i$ have at least $k$-dimensional translation invariance. We then define $\dim_s(q):= \dim_s(\text{Star}_Q(p)$. The proof then proceeds by removing first the star fans of points with star fan dimension 0, and then proceeding through the tropical subvariety. \end{proof} We obtain the proof of the theorem: \begin{align*} \text{Rec}(Q)=& \text{Rec}(\sum Q_i + \vec v_i) \\ = &\sum \text{Rec}(Q_i + \vec v_i) = \sum Q_i \sim^b \sum Q_i+\vec v_i = Q \end{align*} In summary, the following are equivalent: \begin{align*} Q_1\sim^b Q_2 && d_{Q_1} = d_{Q_2} && \text{Rec}(Q_1)= \text{Rec}(Q_2) \end{align*} \printbibliography \end{document}