2024 Spr.高代下 midterm(H)

一

Let $k$ be a field and ${\text{Fin Vect}}_k$ the collection of all finite dimensional vector spaces over $k$

Let $S:={\text{Fin Vect}}_k$

Consider a relation $\sim$ over $S$ defined by

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

1-1

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 $a\sim c$

1-2

Calculate SXS／$\backsim$

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

1-3

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／$\backsim$

Moreover, describe the map $\oplus$ explicitly using the representatives you found in 1-2 (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 $(V^{\prime \prime },W^{\prime \prime })$

1-4

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／$\backsim$.

Moreover, describe the map $\otimes$ explicitly using the representatives you found in 1-2

二

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

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

三

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

$$qg=1$$

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 k[i], 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 $g$,

3-1

determine if $f$ and $g$ are coprime in $k[x]$

3-2

calculate $g^{-1}$ if they are coprime in $R$

• i: $g=1$
• ii: $g=x$
• iii: $g=x^2$
• iv: $g=x^3$

四

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

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

4-1

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

4-2

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

4-3

Give a conjectural formula for the minimal polynomial of $A$

4-4

Prove the conjecture

五

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 $f\in \mathrm{Hom}(M,N)$, let $f^{\ast \ast }\in \mathrm{Hom}(M^{\ast \ast },N^{\ast \ast })$ be the double dual of $f$

5-1

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

5-2

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 $\mathrm{lm}(b_i)=b_i^{\ast \ast }$

六

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 $(f\otimes g)(a\otimes b):=f(a)\otimes g(b)$.)

6-1

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

6-2

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

七

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

7-1

For all $x,y,z\in M$

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

7-2

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

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

7-3

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

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

7-4

Prove that if $B$ is either symmetric or alternating

八

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 $1\le i\le j\le n$

8-1

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

8-2

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