2024 Spr.高代下 midterm(H)

Problem 1.Let $k$ be a field and Fin Vect $k$ the collection of all finite dimensional vector spaces over $k$ .Let $S:=$ Fin Vect $k$ .Consider a relation $\sim$ over $S$ defined by

$$(V,W)\sim (V^{\prime },W^{\prime })\text{ if and only if }V\oplus W^{\prime }\cong V^{\prime }\oplus W$$

(i)Prove that $\backsim$ is an equivalent relation.In other words, (1)$a\sim a$ ；(2)$a\sim b$ implies $b\sim a$ ；(3)$a\sim b$ and $b\sim c$ implies $a\sim c$

(ii)Calculate SXS／～.In other words, find all equivalence classes, and find a representative(will be refered to as distinguished represantative in other subproblems) for each equivalence class

(iii)For $(V,W),(V^{\prime },W^{\prime })\in SXS$, let

$$(V,W)\oplus (V^{\prime },W^{\prime }):=(V\oplus V^{\prime },W\oplus W^{\prime })$$

Prove that it induces a well-defined map on the quotient SXS／～.Moreover, describe the map $\oplus$ explicitly using the representatives you found in(ii)(i.e.for each distinguished representative( $V,W$ )and each distinguished representative $(V^{\prime },W^{\prime })$, find a distinguished representative $(V^{\prime \prime },W^{\prime \prime })$ such that $(V,W)\oplus (V^{\prime },W^{\prime })$ is equivalent to $(V^{\prime \prime },W^{\prime \prime }))$

(iv)For $(V,W),(V^{\prime },W^{\prime })\in SXS$, let

$(V,W)\otimes (V^{\prime },W^{\prime }):=((V\otimes V^{\prime })\oplus (W\otimes W^{\prime }),(V\otimes W^{\prime })\oplus (V^{\prime }\otimes W))$

Prove that it induces a well-defined map on the quotient SXS／～.Moreover, describe the map $\otimes$ explicitly using the representatives you found in(ii)

Problem 2.Let $A,B$ be submodules of some $k$-module $X$ .Prove that

$$\frac{A+B}{B}\cong \frac{A}{A\cap B}$$

Problem 3.Let $k$ be a field and consider $k[x]$ .Let $f\in k[x]$ be a fixed monic polynomial.Recall that, for $g\in k[x]$ such that $f$ and $g$ are coprime in $k[x]$, if we run Euclidean algorithm on $f$ and $g$, we can find $p,q\in k[x]$ such that

$$pf+qg=1$$

Now, consider $R:=k[x]/(f)$ .Then, the formula above gives

$$9g=1$$

in $R$, where $q,g,1$ are treated as their equivalence classes in $R=k[x]/(f)$ .In other words, for every $g$ that is coprime with $f$ in ki］, there exists $g^{-1}$ in $R$ . Moreover, this $g^{-1}$ can be calculated using Euclidean algorithm.Let $f=x^4+x^3+x^2+x+1$ . For each of the following $g$, (1)determine if $f$ and $g$ are coprime in $k[x],(2)$ calculate $g^{-1}$ if they are coprime in $R$

(i) $g=1$

(ii) $g=x$

(iii) $g=x^2$

(iv) $g=x^3$

Problem 4. Let $k$ be a field, and let $a_0,\cdots ,a_{n-1}\in k$. Consider the $n\times n$ matrix $A$ given by

$$A_{ij}=\{\begin{array}{ll}{a}_{j-i},& i\le j,\\ 0,& i>j\end{array}$$

(i) Calculate the minimal polynomial of $A$ when $a_1=1$ (assume that $n⩾2$ )

(ii) Calculate the minimal polynomial of $A$ when $a_1=0$ and $a_2=1$ (assume that $n⩾3$ )

(iii) Give a conjectural formula for the minimal polynomial of $A$

(iv) Prove the conjecture

Problem 5. For a k-module $M$, let $l_M:M\to M^{\ast \ast }$ given by $m\mapsto e{v}_m$, where ${\mathrm{ev}}_m$ : $M^{\ast }\to k$ given by $f\mapsto f(m)$. For $f\in \mathrm{Hom}(M,N)$, lot $f^{\ast \ast }\in \mathrm{Hom}(M^{\ast \ast },N^{\ast \ast })$ be the double dual of $f$

(i) Prove that $l_N$ $o$ $f$ = $f^{\ast \ast }$ $o$ $l_m$ as elements in Hom(M, $N^{\ast \ast }$)

(ii) Suppose that $M$ has a basis $b_1,\cdots ,b_d$. Let $b_1^{\ast },\cdots ,b_d^{\ast }\in M^{\ast }$ be the dual basis of $b_1,\cdots ,b_d$. Let $b_1^{\ast \ast },\cdots ,b_d^{\ast \ast }\in M^{\ast \ast }$ be the dual basis of $b_1^{\ast },\cdots ,b_{\alpha }^{\ast }$. Prove that $\mathrm{lm}\left(b_i\right)=b_i^{\ast \ast }$

Problem 6. Let $M$ be a finite free $k$-module equipped with a homomorphism $\mathrm{\Delta }=M\to M\otimes M$. Let $\mathrm{idm}\in \mathrm{Hom}(M,M)$ be the identity map on $M$. The pair $(M,\mathrm{\Delta })$ is called coormmutative if the following diagram commutes

graph TD;
    M[Left-Up] --> |$\Delta$| $M \otimes N$ [Right-Up]
    M[Left-Up] --> |$\Delta$| $M \otimes M$ [Left-Down]
    $M \otimes N$ --> |$Delta \otimes id_{m}$| $M \otimes M \otimes M$ [Right-Down]
    $M \otimes M$ --> |$id_{m} \otimes \Delta$| $M \otimes M \otimes M$ [Right-Down]

(Recall that for $f\in \mathrm{Hom}(A,A^{\prime })$ and $g\in \mathrm{Hom}(B,B^{\prime })$, we define $f\otimes g\in \mathrm{Hom}(A\otimes B,B^{\prime }\otimes B^{\prime })$ given by $(f\otimes g)(a\otimes b):=f(a)\otimes g(b)$.)

(i) Suppose that $M$ has basis $a,b,c,d$, and define the homomorphism $\mathrm{\Delta }$ as follows

$$\begin{array}{rl}& \mathrm{\Delta }=M\to M\otimes M\\ & a\mapsto a\otimes a+b\otimes c\\ & b\mapsto a\otimes b+b\otimes d\\ & c\mapsto c\otimes a+d\otimes c\\ & d\mapsto c\otimes b+d\otimes d\end{array}$$

Prove that it is cocommutative

(ii) Consider the map $\varphi \in \mathrm{Hom}(M^{\ast }\otimes M^{\ast },M^{\ast })$ given by

$$\varphi (f\otimes g)=(m\mapsto (f\otimes g)(\mathrm{\Delta }(m))$$

We call the pair $(M^{\ast },\varphi )$ commutative if $\varphi (f\otimes g)=\varphi (g\otimes f)$ for all $f,g\in M^{\ast }$. Prove that, $(M^{\ast },\varphi )$ is commutative if and only if $(M,\mathrm{\Delta })$ is cocommutative

Problem 7. Let $k$ be an integral domain (ie. for $a,b\in k,ab=0$ implies either $a=0$ or $b=0$ ). Let $M$ be a $k$-module, and $B\in (M,M^{\ast })$. We call $B$

• reflexive if $B(x,y)=0$ implies that $B(y,x)=0$ for all $x,y\in M$

• symmetric if $B(x,y)=B(y,x)$ for all $x,y\in M$

• alternating if $B(x,y)=0$ for all $x\in M$

The goal of this problem is to prove that:

$B$ is reflexive if and only if $B$ is either symmetric or alternating

The sufficiency is easy, and you do not need to provide a proof. You are asked to prove the necessity by proving the following statements. Assume now that $B$ is reflexive

(i) For all $x,y,z\in M$

$$B(x,y)B(z,x)=B(x,z)B(y,x)$$

(ii) For all $x,y\in M$, if $B(x,x)\ne 0$, then

$$B(x,y)=B(y,x)$$

(iii) Suppose that for some $x,y\in M,B(x,y)\ne B(y,x)$. If $B(z,z)\ne 0$ for some $z\in M$, then

$$B(x+z,x+z)=0$$

(iv) Prove that if $B$ is either symmetric or alternating

Problem 8. Let $M$ be a bilinear space, where $M=k^n$ as a $k$-module ( $k$ is a field where

2 is invertible). Suppose that

$$⟨e_i,e_j⟩=j-i$$

for $1⩽i⩽j⩽n$

(i) Assume that $n=5$ and $M$ is symmetric. Find an explicit basis for $M$ under which the Gram matrix is diagonal

(ii) Assume that $n=5$ and $M$ is alternating. Find an explicit basis for $M$ under which the Gram matrix is block diagonal matrix, where each block is either the $2\times 2$ alternating matrix with diagnoal entries 0 and anti-diagonal entries $\pm 1$, or $|x|$ matrix with a single entry 0