\magnification=\magstep1
\input starter
\input peterAMSsym.def
\def \Ass{\mathop{\rm Ass}}
\def \Fun{\mathop{\rm Fun}}
\def \Nat{\mathop{\rm Nat}}
\def \Jac{\mathop{\rm Jac}}
\def\inverselimit{{\lim\limits_{\textstyle \longleftarrow}}}

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{\bf Math 8212\hfill Commutative and Homological Algebra 2\hfill  Spring 2022}
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{\bf Homework Assignment 3} Due {\bf Wednesday} 4/13/2022, uploaded to Gradescope.

Each question part is worth 1 point. There are 17 question parts. You are on target for an A if you make a genuine attempt on at least half of them. We define $\Fun(\calC,\calD)$ to be the category whose objects are functors $\calC\to\calD$ and whose morphisms are natural transformations.
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In these questions $p$ is a prime. We will write an element $a_0+a_1p+a_2p^2+\cdots$ of the $p$-adic integers $\ZZ_p^\wedge$, where $0\le a_i\le p-1$, as a string $\cdots a_3a_2a_1a_0.$ with a point to the right of $a_0$.
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1. a. Calculate the 3-adic expansion of $1\over 2$ in $\ZZ_3^\wedge$.
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b. What fraction does the recurring 3-adic integer
$\cdots\overline{0121}01211.$ represent? 
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c. Show that a $p$-adic integer is a negative (rational) integer if and only if it is of the form $\overline{(p-1)} 1a_n\cdots a_3a_2a_1a_0.$
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d. Show that the localization $\ZZ_{(p)}$ of $\ZZ$ at $(p)$ is the subset of $\ZZ_p^\wedge$ consisting of strings
$$\overline{a_m\cdots a_n}\cdots a_3a_2a_1a_0.$$ that eventually recur to the left.
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2. In this question consider the 10-adic topology on $\ZZ$, determined by the powers of the ideal $(10)$, with completion the 10-adic integers $\ZZ_{(10)}^\wedge$, and also the 2-adic topology on $\ZZ$ with completion $\ZZ_{(2)}^\wedge$
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a. Show that a sequence of integers that is a Cauchy sequence in the 10-adic topology is also a Cauchy sequence in the 2-adic topology.
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b. Show that the identity map $1:\ZZ\to\ZZ$ extends to a ring homomorphism $\ZZ_{(10)}^\wedge\to \ZZ_{(2)}^\wedge$.
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c. Determine whether the identity map $1:\ZZ\to\ZZ$ extends to a ring homomorphism $\ZZ_{(2)}^\wedge\to \ZZ_{(10)}^\wedge$.
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d. Using the fact that $\ZZ/10\ZZ \cong \ZZ/2\ZZ \times \ZZ/5\ZZ $ as a product of rings, show that $\ZZ_{(10)}^\wedge \cong A\times B $ for certain rings $A,B$ that are also ideals of $\ZZ_{(10)}^\wedge$, with $A/(A\cap (10))\cong \ZZ/2\ZZ$ and $B/(B\cap (10))\cong \ZZ/5\ZZ$.
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e. Show that $\ZZ_{(10)}^\wedge$ has just two maximal ideals, generated by 2 and 5.
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f. Show that the composite morphism specified as the inclusion of the ideal $A\hookrightarrow \ZZ_{(10)}^\wedge$, followed by the ring homomorphism $\ZZ_{(10)}^\wedge\to \ZZ_{(2)}$ of part b, is surjective. (Consider using Nakayama's lemma.)



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3. Find how many cube roots each of the following numbers has in $\ZZ_{(7)}^\wedge$: 1, 9, -4, 4, 12, 6. Also find how many cube roots each  of the following numbers has in $\ZZ_{(5)}^\wedge$: 1, 2, 3, 4, 5.

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4. Let $I$ be an ideal of $R$. Consider the polynomial $f(x) = 3x^4 + x^2 + 5$ as a function $R\to R$. Show that $f$ is continuous in the $I$-adic topology on $R$. (The $I$-adic topology on $R$ is given by the distance function determined by the powers of $I$.)

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5. For a category $\calC$ and commutative ring $R$ we may take the $R$-linear category $R\calC$ to have the same objects as $\calC$, and with $\Hom_{R\calC}(x,y) = R\Hom_\calC(x,y)$, the set of formal linear combinations of morphism $x\to y$ in $\calC$. Composition is $R$-bilinear. The constant functor $\underline R : R\calC\to R\hbox{-mod}$ is the functor that assigns $R$ to each object of $\calC$, and the identity map $1_R$ to each morphism of $\calC$.
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a. Let $\calC$ be the category $\bullet \gets \bullet \to \bullet$ with three objects, and five morphisms that are the two morphisms shown and the three identity morphisms for the objects. Show that the constant functor on $\calC$ is representable as a linear functor $R\calC\to R\hbox{-mod}$.
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b. Let $\calD$ be the category $\bullet \to \bullet \gets \bullet$ with three objects, and five morphisms, with the two non-identity morphisms pointing in the opposite direction to the last example. Show that the constant functor is not representable.
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c. Show that the inverse limit functor $\inverselimit: \Fun(\calD, R\hbox{-mod}) \to R\hbox{-mod}$ is representable, represented by the constant functor.
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\def\Fun{\mathop{{\rm Fun}}}
6. Let  $\Fun(\calC, \hbox{abgps})$ be the category of functors from $\calC$ to abelian groups, with natural transformations as morphisms. We may take as a definition that a sequence $F_1\to F_2\to F_3$ in $\Fun(\calC, \hbox{abgps})$  is exact if and only if, for all objects $X$ in $\calC$, the sequence of abelian groups $F_1(X)\to F_2(X)\to F_3(X)$ is exact. This is equivalent to other possible definitions of exactness. We may regard the inverse limit construction as a functor $\inverselimit: \Fun(\calC, \hbox{abgps}) \to \hbox{abgps}$. 
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a. Let $\calC$ be the category $\bullet \gets \bullet \to \bullet$ with three objects, and five morphisms that are the two morphisms shown and the three identity morphisms for the objects. Show that the functor $\inverselimit: \Fun(\calC, \hbox{abgps}) \to \hbox{abgps}$ is exact.
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b.  Let $\calD$ be the category $\bullet \to \bullet \gets \bullet$ with three objects, and five morphisms, with the two non-identity morphisms pointing in the opposite direction to the last example. Show (by example, or by giving a reason) that the functor $\inverselimit: \Fun(\calD, \hbox{abgps}) \to \hbox{abgps}$ is not exact in general.

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