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If f epsilon Hom

parenthesis v comma

Mu

parenthesis parenthesis

respecting

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g epsilon Hom parenthesis

Xi comma v

parenthesis parenthesis

is a homogeneous morphism whose degree is the matrix

alpha parenthesis

respecting Beta parenthesis

comma f, o, g,

is homogeneous and its degree is the product matrix

alpha beta .

Let alpha equal parenthesis alpha over i j

parenthesis,

1 less than or equal to i less than or equal to m,

1 less than or equal to j less than or equal to n

and Beta equal parenthesis

beta over k L,

1 less than or equal to k less than or equal to n,

1 less than or equal to 1 less than or equal to p

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parenthesis

Xi equals p parenthesis

be the given matrices .

Suppose that f equals

parenthesis

f over 1

until f over m

parenthesis,

g equals parenthesis g

to the power of 1 until g

to the power of n parenthesis,

and let h thus pie until xi

be a morphism,

parenthesis h equals parenthesis

h over 1 to h over p.

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Finally let a equal

parenthesis

a over 1 to a

over p parenthesis

be an element of A to the power of p.

For each index i between 1 and m

parenthesis Mu

equals m parenthesis

we compute the morphism:

X over i equals f over i, o, g,

zero

parenthesis

a over 1 h over 1 to a over p,

h over p parenthesis .

First we get

X over i equals f over i,

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o parenthesis a over 1

to the power of beta over eleven

until

a over p to the power of beta over

1 p g over 1

until

a over 1 to the power of Beta over

i 1

until

a over p to the power of beta over i 1 g

over i .

Then

X over i equals a over 1 to the power of

alpha over i 1

Beta over 1 1

plus

alpha over i 1 Beta i 1

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plus plus

Alpha over i obscure Beta over

alpha 1

until

a over j to the power of alpha over i 1

Beta over i j to A over p

to the power of

alpha over i 1

Beta over 1 p

plus

all over f

over i, o, g, o, h,

Thus f, o, g,

satisfies the homogeneity condition of degree

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alpha beta

parenthesis squared parenthesis

1 point 2 point 2

squared parenthesis

parenthesis.

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