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\title{Homework 5}
%\author{Anna Brosowsky}
\date{\vspace{-4em}}

\begin{document}


\maketitle\thispagestyle{fancy}

Remember you are allowed to discuss with classmates (or an AI tool), but that you need to tell me who/what you discussed with + the final submitted writeup should be your own work.


\begin{enumerate}[itemsep=4em]

\item In this question we see some generally useful comm alg facts\footnote{Not included is the useful but harder to prove fact that $A$ is normal if and only if $A$ satisfies $(R_1)$ and $(S_2)$.}. 
In particular, we are NOT assuming char $p$! First, some definitions:
\begin{defn}
Let $A$ be a Noetherian ring and let $k\in \bb N$.
\begin{itemize}
\item $A$ satisfies \emph{Serre's condition $(R_k)$} if for all primes $\mf p$ of height $\le k$, we have $A_{\mf p}$ is a regular local ring.

\item $A$ satisfies \emph{Serre's condition $(S_k)$} if for all primes $\mf p$, we have $\depth(A_{\mf p})\ge \min\{k,\operatorname{ht}(\mf p)\}$.
\end{itemize}
\end{defn}
Now for the problem:
\begin{enumerate}

\item Give an example of a ring that has $(S_1)$ but is not reduced.

\item Give an example of a ring that has $(R_0)$ but is not reduced.



\item Prove that $A$ is reduced if and only if $A$ satisfies $(R_0)$ and $(S_1)$.



\item Prove that $A$ is Cohen-Macaulay if and only if $A$ satisfies $(S_d)$ for $d=\dim A$, if and only if  $A$ satisfies $(S_k)$ for all $k$.\footnote{This has nothing to do with anything else on this HW, but is such a cute fact that I'm including it anyways.}


\end{enumerate}

\item In this question, we are back to assuming $A$ has char $p$.
\begin{enumerate}
\item Let $(A,\mf m)$ be a local char $p>0$ ring of dimension 0. Prove that $A$ is F-injective if and only if it is reduced.


\item Let $(A,\mf m)$ be a local char $p>0$ ring of dimension $>0$. Prove that if $A$ is F-injective then $H^0_{\mf m}(A)=0$.

\item Let $A$ be an $F$-injective ring. Prove that $A$ is reduced.

\end{enumerate}

\item (Exercise 22 of [MP]) Let R be an $\bb N$-graded ring over a field of prime characteristic $p > 0$ with
homogeneous maximal ideal $\mf m$. Show that
\begin{enumerate}
\item If R is $F$-injective, then $[H^i_{\mf m}(R)]_{>0} = 0$ for each $i$. Conclude that $a(R)\le 0$.


\item If R is $F$-rational, then $[H^d_{\mf m}(R)]_{\ge 0} = 0$. Conclude that $a(R)<0$.

\end{enumerate}

\newpage

\item Here are some more useful colon ideal facts:
\begin{enumerate}
\item
Let $I$ unmixed ideal (i.e., no embedded primes/all associated primes of $I$ are minimal over $I$) of a regular local ring $(R,\mf m)$. Prove that
\footnote{Hint: First show that $\Ass(I)=\Ass(I^{[p^e]})$. For the hint and for $\Ass(I^{[p^e]}:I)$, think about the set of non-zero divisors}
\[
\Ass(I^{[p^e]}:I) \subseteq \Ass(I).
\]
Conclude that if $I$ is $\mf p$-primary then $I^{[p^e]}:I$ is $\mf p$-primary.

\item 
Let $(R,\mf m)$ be a regular local ring and $I$ unmixed. Let $I=\bigcap_{i=1}^n Q_i$. Prove that 
\[
I^{[p]}:I = \bigcap_{i} (Q_i^{[p]}:Q_i).
\]


\end{enumerate}

\item As on Thurs 03/26, let $S=k[x,y,z,u,v]_{\mf m}$, let 
\[
I = I_2\begin{pmatrix}
x^n & z & v \\
u   & z & y^n
\end{pmatrix}
= \langle z(x^n-u), z(v-y^n), x^ny^n-uv\rangle
\]
be the ideal of 2x2 minors, and let $R=S/I$. 

\begin{enumerate}
\item Compute the minimal primary decomposition for $I$.\footnote{Hint: Use the $\bb Z^3$-grading coming from the columns, so that $\deg x=e_1$, $\deg u=ne_1$, $\deg z=e_2$, and $\deg y=e_3$, $\deg v=ne_3$.\\
You may use without proof the fact that even in the multigraded setting, associated primes of homogeneous ideals are homogeneous.}


\item 
Prove that if $p\le n$ then $R$ is NOT F-split.\footnote{Hint: Modding out by a monomial ideal can make a complicated equation more understandable.}


\end{enumerate}



\end{enumerate}


\end{document}
