PosChar

A few suggested algorithms we might wish to try and implement:

The support of local cohomology modules

* If $V\subset R^n$ is a submodule then we can define $V^[1/p]$ in a similar way to how we did this for ideals. 
* Let $(R, \m)$ be a local ring then there is a natural Frobenius map on $H_{\m}^i(R)$. One cat take a Matlis dual of this, which keeps the Frobenius action. This gives us a map $u:\coker(A)\arrow{}\coker(A^[p])$... 

An algorithm for producing F-pure ideal

Annihilators of Artinian modules compatible with a Frobenius map

Glassbrenner's Criterion for F-regularity

Glassbrenner's Criterion states:

• If there exists an $t$ such that $S_t$ is regular, then $S$ is strongly $F$-regular if and only if $t(I^[p^e] : I)\not\subset m^[p^e]$ for some $e$.

Some immediate things we would need to think aboutL How to pick $t$ and how far in $e$ we need to search.

How to choose t? One way could be thinking about Jacobian matrix as follows; Suppose that $I=(f_1,...,f_r)$ and jacobian matrix is computable by M2, meaning $J=(\partial f_i/ \partial x_j)_{i,j}$. Let $codim I=c$ and set $L=minors(c,J)$. $t$ can be chosen among min. gens. of ideal $L$. Let call $L$ jacobian ideal corresponding to $I$.

Frobenius Powers Given

Given $(R, I, t)$ we can get an ideal $\tau(R, I^t)$ generalizing the test ideal. There is another variant called the Frobenius power, given an ideal $I\subset R$ then $I^[p^e]=\langle f^{p^e} : f\in I\rangle$. In fact we can simply take the $p^e$th power for any generating set of $I$. We can then extend this definition to all real numbers. The steps to define such a thing are roughly:

 1. Define $I^[n]$ for $n\in \mathbb{Z}$ by taking $n$ and writing it in base $p^e$ i.e. $k=k_0+k_1p+\cdots+k_ep^e$ and then setting $I^[n]=I^{k_0}(I^{k_1})^[p]\cdots(I^{k_e})^[p^e]$. (This was implemented in Boise15.)
 2. Define for $I^{\left[\frac{k}{p^e}\right])$.
 3. Define for any real number $\lambda\in \mathbb{R}_{>0}$ by approximating $\lambda$ by rational numbers of the form $k/p^e$ for $e\gg0$. (We will only actually do $\mathbb{Q}_{>0}$.)

There are probably special cases when we can modify existing code -- i.e. monomial ideals -- where we can make things more efficient. Some further ideas and definitions for this: * Speed up computing $I^[1/p^e]$ and $I^[n]$ when $I$ is a monomial ideal. * Setup: Let $R=\mathbb{F}{p}[x_1,\ldots,x_n]$ and $I\subset R$ be an ideal. * Define: $I^{[1/p]}=\min\left{J\subset R ; | ; I\subset J^[p]\right}$ where the minimal is with respect to inclusion. We need that if $I\subset J^[p]{i}$ for all $i\in \N$ then $I\subset \left(\cap J_i\right)^[p]$. (Note we need a regular ring for Frobenius powers and intersections to commute.) * Now $R^p=\mathbb{F}p[x_1^p,\ldots,x_n^p]\subset\mathbb{F}{p}[x_1,\ldots,x_n]=R$ is a free extension with basis $\mathbb{B}={x_1^{a_1}\cdots x_n^{a_n} ; : ; 0\leq a_i\leq p-1}$. (Think of pulling off all of the powers of $x_i^{p}$ via the division algorithm.) * For any $f\in R$ write $f$ in this free basis as $f=\sum_{\mu\in \mathbb{B}} g_{\mu}^p\cdot\mu$. Now if we let $I_f=\langle g_\mu ; : ; \mu\in \mathbb{B}\rangle$, then $f\in I_f^{[p]}$. In fact if $I_f=\langle h_1,\ldots,h_s\rangle$ so that $I_f=\langle h_1^p,\ldots,h_s^p\rangle$ then we can write $f=h_1^pr_1+\cdots+h_s^pr_s$. Letting $\pi_{\mu}:R\arrow{}R^p$ being the projection on $\mu$ we get that $g_{\mu}^p=\pi_{\mu}(f)=h_1^p\pi_{\mu}(r_1)+\cdots+h_s^p\pi_{\mu}(r_s)$ implying that $I_f=\langle f\rangle^{[1/p]}$. * In general if $I=\langle f_1,\ldots,f_m\rangle then $I^[1/p]=\langle f_1\rangle^[1/p]+\cdots+\langle f_m\rangle^[1/p]$.