Mô đun các đồng cấu và hàm tử mở rộng

HQ va ten h9c vien: Tr§n Thi Thanh Van.. NgucYi huong din khoa h9c: TS.. Tom t�t: Lu�n van "Modun cac d6ng ciu va ham tu ma rmg" da hoan thanh mvc dich va nhi�m vv nghien cuu.. • Trinh b

✸✻ ✤➣ ✤ë♥❣ ✈✐➯♥✱ ❦❤➼❝❤ ❧➺ tæ✐ tr♦♥❣ s✉èt q✉→ tr➻♥❤ ❤å❝ t➟♣ t↕✐ ❧î♣✳

✐✐✐

r

TANG THONG TIN LU�N V N TH.C Si

Tend� tai: Modun cac dBng c�u va ham tr m' rqng

Nganh: D�i s6 va Iy thuy�t s6

HQ va ten h9c vien: Tr§n Thi Thanh Van

NgucYi huong din khoa h9c: TS Nguy�n Ngqc Chau

Casa dao tio: Tru·cmg D�i hqc Srr ph�m - D�i hqc la N�ng

Tom t�t: Lu�n van "Modun cac d6ng ciu va ham tu ma r(mg" da hoan thanh mvc dich va nhi�m vv nghien cuu Cv th�, lu�n van da dit duqc m>t s6 k�t qua sau:

• Tim hi�u ly thuy�t phim tru, ham tu, day n6i cac ham tu, va phep giai xi anhcua m>t modun

• Trinh bay ham tu Hom tren phim tru cac modun tren m>t vanh giao hoan c6dan vi, va khao sat cac tinh chit cua ham tu Hom

• Trinh bay ham tu ma r>ng tren phim tru cac modun, khao sat va chung minhchi ti�t cac tinh chit Cla 10, tu do neu d?C trung tien d� CUa ham tu ma r>ng

Tr kh6a: phim tru, ham tu, phep giai xi anh cua m>t modun, ham tu Hom, ham tu

ma r>ng

Xac nh�n cia giao vien hrr6'ng din N grrcri th fC hi�n d� tai

▼Ð ✣❺❯

✶✳ ▲þ ❞♦ ❝❤å♥ ✤➲ t➔✐

❑❤→✐ ♥✐➺♠ ♣❤↕♠ trò ✈➔ ❤➔♠ tû ❧➛♥ ✤➛✉ t✐➯♥ ✤÷ñ❝ ✤➲ ①✉➜t ❜ð✐ ❤❛✐ ♥❤➔t♦→♥ ❤å❝ ❙✳❊✐❧❡♥❜❡r❣ ✈➔ ❙✳▼❛❝ ▲❛♥❡ ✈➔♦ ♥❤ú♥❣ ♥➠♠ ✶✾✹✵✲✶✾✹✺✱ ✤➣ ❝❤ù♥❣

tä sü ❤ú✉ ➼❝❤ tr♦♥❣ ♥❤✐➲✉ ♥❣➔♥❤ ❦❤♦❛ ❤å❝✱ ✤ç♥❣ t❤í✐ t❤✉ ❤ót sü q✉❛♥ t➙♠

❝õ❛ ♥❤ú♥❣ ♥❤➔ t♦→♥ ❤å❝ ♥ê✐ t✐➳♥❣ tr➯♥ t❤➳ ❣✐î✐✳ ❍✐➺♥ ♥❛②✱ ❧þ t❤✉②➳t ♣❤↕♠trò ✤➣ trð t❤➔♥❤ ♠ët ♥❣➔♥❤ t♦→♥ ❤å❝ ❦❤→ q✉❛♥ trå♥❣ ✈➔ ❤✐➺♥ ❤ú✉ tr♦♥❣

♥❤✐➲✉ ❧➽♥❤ ✈ü❝ ❝õ❛ t♦→♥ ❤å❝✳ ❚✐➳♣ ♥è✐ sü ♣❤→t tr✐➸♥ ❝õ❛ ❧þ t❤✉②➳t ♣❤↕♠ trò

❧➔ ✤↕✐ sè ✤ç♥❣ ✤✐➲✉✱ ♠➔ ❜è♥ trö ❝ët ❝õ❛ ♥â ❧➔ ❝→❝ ❤➔♠ tû ⊗✱ ❍♦♠✱ ❚♦r ✈➔

❊①t✳ ❍➔♠ tû ①♦➢♥ Torn ✤÷ñ❝ ❍✳ ❈❛rt❛♥ ✤➲ ①✉➜t ♥➠♠ ✶✾✹✽✱ ❝á♥ ❤➔♠ tû ♠ðrë♥❣ Extn ✤÷ñ❝ ❣✐î✐ t❤✐➺✉ ❜ð✐ ❙✳ ❊✐❧❡♥❜❡r❣ ✈➔ ❙✳ ▼❛❝ ▲❛♥❡ ✈➔♦ ♥➠♠ ✶✾✹✷✳

❈❍×❒◆● ✶

▼➷✣❯◆

◆❤➡♠ ❧➔♠ ❝ì sð ❝❤♦ ❝→❝ ❝❤÷ì♥❣ s❛✉✱ ❝❤÷ì♥❣ ♥➔② tr➻♥❤ ❜➔② ♠ët sè ❦✐➳♥t❤ù❝ ❝ì ❜↔♥ ✈➲ ♠æ✤✉♥✱ ❝→❝ ❝❤✐ t✐➳t ❧✐➯♥ q✉❛♥ ❝â t❤➸ ①❡♠ tr♦♥❣ ❬✸❪✱ ❬✻❪✳

✶✳✶ ▼æ✤✉♥✱ ✤ç♥❣ ❝➜✉ ♠æ✤✉♥

✣à♥❤ ♥❣❤➽❛ ✶✳✶✳✶✳ ❈❤♦ R ❧➔ ♠ët ✈➔♥❤ ❝â ✤ì♥ ✈à 1 6= 0 ✈➔ M ❧➔ ♠ët ♥❤â♠

❝ë♥❣ ❆❜❡❧✳ ❚❛ ❣å✐ M ❧➔ ♠ët ❘ ✕ ♠æ✤✉♥ tr→✐ ♥➳✉ tç♥ t↕✐ ♠ët →♥❤ ①↕✿

R × M → M(a, x) 7→ ax

◆➳✉ hSi = M ✈➔ S ❤ú✉ ❤↕♥ t❤➻ M ❣å✐ ❧➔ ♠æ✤✉♥ ❤ú✉ ❤↕♥ s✐♥❤✳

✺

✣➦❝ ❜✐➺t✿

◆➳✉ S = ∅ t❤➻ h∅i = {0}✳

◆➳✉ S 6= ∅ t❤➻ hSi =

P

✣à♥❤ ♥❣❤➽❛ ✶✳✷✳✷✳ ❈❤♦ ♠ët ❤å ❝→❝ ♠æ✤✉♥ {Mi}i∈I✳ ❳➨t t➟♣ ❝♦♥ S ❝õ❛Q

✽

0 → Hom(C, X)→ Hom(B, X)g∗ → Hom(A, X) → 0f∗

✈î✐ f∗ = Hom(f, i) ✈➔ g∗ = Hom(g, i) ❝ô♥❣ ❧➔ ♠ët ❞➣② ❦❤î♣ ♥❣➢♥✳

✶✶

▼➺♥❤ ✤➲ ✶✳✹✳✶✺✳ ▼ët ❞➣② ♥û❛ ❦❤î♣ ♥❤ú♥❣ ✤ç♥❣ ❝➜✉ ❝õ❛ ♥❤ú♥❣ ♠æ✤✉♥tr➯♥ R ❧➔ ❦❤î♣ ♥➳✉ ✈➔ ❝❤➾ ♥➳✉ t➜t ❝↔ ❝→❝ ♠æ✤✉♥ ❞➝♥ ①✉➜t ❝õ❛ ♥â ✤➲✉ t➛♠t❤÷í♥❣✳

♥➯♥ t❛ ❝â g∗(β) = α ✈➔ ❞♦ ✤â α ∈ Im(g∗)✳ ❱➻ α ❧➔ ♠ët ♣❤➛♥ tû ❜➜t ❦ý ❝õ❛

Ker(∂)✱ ♥➯♥ t❛ ✤÷ñ❝ Ker(∂) ⊂ Im(g∗) ❱➟② t❛ ✤➣ ❝❤ù♥❣ ♠✐♥❤ ✤÷ñ❝ ✭✶✮✳

✣➸ ❝❤ù♥❣ ♠✐♥❤ ✭✷✮✱ t❛ ✤➸ þ tî✐ ♣❤➛♥ s❛✉

Hn(E) −→ H∂ n−1(C) f∗

−→ Hn−1(D)

✷✵

h : Hom(A, B) → Hom(A′, B′)

✷✶

0 → Hom(C, M )−g∗→ Hom(B, M )−→ Hom(A, M )f∗

✈î✐ f ∗ = Hom(f, idM) ✈➔ g∗ = Hom(g, idM)✱ tr♦♥❣ ✤â idM ❧➔ tü ✤ç♥❣ ❝➜✉

✤ç♥❣ ♥❤➜t ❝õ❛ ♠æ✤✉♥ M✱ ❝ô♥❣ ❧➔ ❦❤î♣✳

✣à♥❤ ❧þ ✶✳✺✳✶✵✳ ◆➳✉ ❞➣② s❛✉ ♥❤ú♥❣ ✤ç♥❣ ❝➜✉ ❝õ❛ ♥❤ú♥❣ ♠æ✤✉♥ tr➯♥ R

0 → A −→ Bf −→ C → 0g

❧➔ ♠ët ❞➣② ❦❤î♣ ♥❣➢♥ ❝❤➫ r❛✱ t❤➻ ❞➣②

0 → Hom(C, M )−g∗→ Hom(B, M )−→ Hom(A, M ) → 0f∗ ✱

tr♦♥❣ ✤â f ∗ = Hom(f, idM) ✈➔ g∗ = Hom(g, idM)✱ ✈î✐ idM ❧➔ tü ✤ç♥❣ ❝➜✉

✶✮ ▼é✐ ✈➟t ❝õ❛ ♣❤❛♠ trò C ❧➔ ♠ët ✈➟t ❝õ❛ ♣❤↕♠ trò P✳

✷✮ ▼é✐ ①↕ ❝õ❛ ♣❤❛♠ trò C ❧➔ ♠ët ①↕ ❝õ❛ ♣❤↕♠ trò P✳

✷✻

✸✮ ❍ñ♣ t❤➔♥❤ ❣❢ ❝õ❛ ❝→❝ ①↕ ❣✱ ❢ tr♦♥❣ ♣❤↕♠ trò C trò♥❣ ✈î✐ ❤ñ♣ t❤➔♥❤ ❝õ❛

❝→❝ ①↕ ✤â tr♦♥❣ ♣❤↕♠ trò P✳

✣à♥❤ ♥❣❤➽❛ ✷✳✶✳✺✳

✶✮ ▼ët ①↕ f : A → B tr♦♥❣ ♠ët ♣❤↕♠ trò P ❣å✐ ❧➔ ❦❤↔ ♥❣❤à❝❤ ❤❛② ✤➥♥❣ ①↕tr♦♥❣ P ♥➳✉ ✈➔ ❝❤➾ ♥➳✉ tç♥ t↕✐ ♠ët ①↕ g : B → A ❝õ❛ P s❛♦ ❝❤♦ t❛ ❝â ✤ç♥❣t❤í✐✿ gf = 1A ✈➔ f g = 1B✳

(A ⊗ −) (1B) = 1A⊗ 1B = 1A⊗B = 1(A⊗−)(B)(A ⊗ −) (gf ) = 1A⊗(gf ) = (1A ⊗ g) (1A ⊗ f ) = [(A ⊗ −) (g)] [(A ⊗ −) (f )]

❍➔♠ tû (A ⊗ −) ❣å✐ ❧➔ ❤➔♠ tû t➼❝❤ t❡♥①ì t❤❡♦ ❜✐➳♥ t❤ù ❤❛✐✳

✷✾

✣➸ ❝❤♦ ❣å♥✱ tø ✤➙② ✈➲ s❛✉ ❦❤✐ ❝❤♦ ♠ët ❤➔♠ tû ♠➔ ❦❤æ♥❣ ♥â✐ ❣➻ t❤➯♠t❤➻ ❤➔♠ tû ✤â ✤÷ñ❝ ❤✐➸✉ ❧➔ ❤➔♠ tû ❤✐➺♣ ❜✐➳♥✳

❚❤➟t ✈➟②✿ δ(1(A,A′ )) = δ(1A, 1A ′) = 1A × 1A ′ = 1(A×A′ ) = 1δ(A,A′ )

✸✷

✐✮ ❉➣② s❛✉ ❧➔ ❦❤î♣

0 → Φ◦(W ) → → Φn(W ) Φ

n (g)

−−−→ Φn(V ) Φ

n (f )

Hom(X, −)(1A) = Hom(1X, 1A) = 1Hom(X,A)

Hom(X, −)(f g) = Hom(1X, f g) = Hom(1X, f )Hom(1X, g)

= [Hom(1X, −)(f )] [Hom(1X, −)(g)]

✸✻

❱➻ Hom(1M, f g)Hom(1M, f ) = Hom(1M, gf ) = Hom(1M, 0) = 0.

◆➯♥ t❛ ❝â Im Hom(1M, f ) ⊆ Ker Hom(1M, g)

◆❣÷ñ❝ ❧↕✐✱ ❣✐↔ sû v ∈ Hom(M, N ) ✈➔ v ∈ KerHom(1M, g)

❑❤✐ ✤â Hom(1M, g)(v) = gv = 0✱ ✈➟② Imv ⊆ Kerg = Imf

✸✼

⇒ f∗(v) = Hom(f, 1X)(v) = vf = 0

❱➟② ∀m′ ∈ M′, vf (m′) = 0 ❤❛② v(Imf ) = 0

❱➻ Imf = Kerg ⇒ v(Kerg) = 0 ❤❛② Kerg ⊆ Kerv✳

❱➻ g ❧➔ t♦➔♥ →♥❤ ♥➯♥ tç♥ t↕✐ ❞✉② ♥❤➜t ♠ët ✤ç♥❣ ❝➜✉ w : M′′ → N s❛♦ ❝❤♦t❛ ❝â✿ v = wg = Hom(g, 1N)(w)

⇒ v ∈ Img∗ ⇒ Kerf∗ ⊆ Img∗

0 → Hom(X, E) → Hom(X, F ) → Hom(X, G) → 0

0 → Hom(G, X) → Hom(F, X) → Hom(E, X) → 0

0 → Hom(Zm,Z) → Hom(Zm,Z) → Hom(Zm,Zm) → 0

tr♦♥❣ ✤â F1 ❧➔ ♠ët ♠æ✤✉♥ tü ❞♦ tr➯♥ R✳ ❇➡♥❣ ♣❤➨♣ q✉② ♥↕♣ t♦→♥ ❤å❝✱ t❛t❤✉ ✤÷ñ❝ ❝→❝ ❞➣② ❦❤î♣ ♥❣➢♥

①➙② ❞ü♥❣ ♥❤÷ tr➯♥ ❧➔ ♥❤ú♥❣ t÷ì♥❣ ✤÷ì♥❣ ❞➙② ❝❤✉②➲♥ ✈➔ ❝→❝ ♣❤➨♣ ❣✐↔✐ ①↕

↔♥❤ C ✈➔ D ❝õ❛ ♠æ✤✉♥ ✤➣ ❝❤♦ X tr➯♥ R ❣å✐ ❧➔ t÷ì♥❣ ✤÷ì♥❣ ✤ç♥❣ ❧✉➙♥ ❤❛②t❤✉ë❝ ❝ò♥❣ ♠ët ❦✐➸✉ ✤ç♥❣ ❧✉➙♥✳

♥➯♥Hom(C, Y )❧➔ ♥û❛ ❦❤î♣ ✈➔ ❞♦ ✤â ♥â ❧➔ ♠ët ❞➣② tr➯♥✳ ❱î✐ ♠å✐ sè ♥❣✉②➯♥

Hom(f, i) = {Hom(fn, i) : Hom(Dn, Y ) −→ Hom(Cn, Y ) | n ∈ Z }

Hom(g, i) = {Hom(gn, i) : Hom(Cn, Y ) −→ Hom(Dn, Y ) | n ∈ Z }

❧➔ ♥❤ú♥❣ ♣❤➨♣ ❜✐➳♥ ✤ê✐ ❞➙② ❝❤✉②➲♥ ❝õ❛ ❝→❝ ❞➣② tr➯♥Hom(C, Y )✈➔Hom(D, Y )✳

❈❤ù♥❣ ❝↔♠ s✐♥❤ r❛ ❝→❝ ✤ç♥❣ ❝➜✉

f∗ : Hn[Hom(D, Y )] −→ Hn[Hom(C, Y )]

✹✾

g∗ : Hn[Hom(C, Y )] −→ Hn[Hom(D, Y )]

✈î✐ ♠å✐ sè ♥❣✉②➯♥ n✳

❱➻ g◦f ✈➔ f◦g ✤ç♥❣ ❧✉➙♥ ✈î✐ ❝→ ♣❤➨♣ ❜✐➳♥ ✤ê✐ ❞➙② ❝❤✉②➲♥ ✤ç♥❣ ♥❤➜t ❝õ❛

❝→❝ ❞➣② ❞÷î✐ C ✈➔ D✳ ◆➯♥ t❛ s✉② r❛ Hom(g, i)◦Hom(f, i) ✈➔

Hom(f, i)◦Hom(g, i) ❝ô♥❣ ❧➔ ✤ç♥❣ ❧✉➙♥ ✈î✐ ❝→❝ ♣❤➨♣ ❜✐➳♥ ✤ê✐ ❞➙② ❝❤✉②➲♥

✤ç♥❣ ♥❤➜t ❝õ❛ ❝→❝ ❞➣② tr➯♥ Hom(D, Y ) ✈➔ Hom(C, Y )✳ ❑❤✐ ✤â t❛ ✤÷ñ❝

▼➺♥❤ ✤➲ ✸✳✷✳✻✳ ❬✸❪ ◆➳✉ ♠æ✤✉♥ X ❝â ♠ët ♣❤➨♣ ❣✐↔✐ ①↕ ↔♥❤ C s❛♦ ❝❤♦

Cn = 0 , ∀n > m t❤➻ t❛ ❝â

Extn(X, Y ) = 0

✈î✐ ♠å✐ n > m ✈➔ ♠å✐ ♠æ✤✉♥ Y tr➯♥ R✳ ❍ì♥ ♥ú❛✱ t❛ ❝â

Extm(X, Y ) ≈ Coker[Hom(∂m, i)]

tr♦♥❣ ✤â Hom(∂m, i) : Hom(Cm−1, Y ) −→ Hom(Cm, Y ) ❧➔ ❍♦♠ ❝õ❛ ✤ç♥❣

❝➜✉ ∂m : Cm −→ Cm−1 tr♦♥❣ C ✈➔ tü ✤ç♥❣ ❝➜✉ ✤ç♥❣ ♥❤➜t i : Y → Y ❝õ❛

♠æ✤✉♥ Y✳

❈❤ù♥❣ ♠✐♥❤✿

❱➻ Hom(Cm, Y ) = 0 ✈î✐ ♠å✐ n > m ♥➯♥ Extn(X, Y ) = 0✳

❍ì♥ ♥ú❛✱ ✈➻ Ker[Hom(∂m+1, i)] = [Hom(Cm, Y )]✱ ♥➯♥ t❛ ❝â

Extm(X, Y ) = Ker[Hom(∂m+1, i)] / Im[Hom(∂m, i)] ≈ Coker[Hom(∂m, i)]

❱➟② t❛ ✤÷ñ❝ ♣❤➨♣ ❣✐↔✐ ①↕ ↔♥❤ C∗ ❜➡♥❣ ❝→❝❤ ❧➜②

✺✸

Extn(X, Y ) = Ker[Hom(∂n∗, i)] / Im[Hom(∂n−1∗ , i)]

= Ker[Hom(∂n−1, i)] / Im[Hom(∂n−2, i)]

= Ker[Hom(∂1, i)] / Im[Hom(∂1∗, i)]

= Im[Hom(∂◦, i)] / Im[Hom(∂1∗, i)]

0 −→ Hom(X, Y ) −−−−−→ Hom(CHom(∂0,i) 0, Y )−−−−−→ Hom(CHom(∂1,i) 1, Y )

❝ô♥❣ ❦❤î♣✱ ❞♦ ✤â Hom(∂◦, i) ❧➔ ✤ì♥ ❝➜✉ ✈➔

Im[Hom(∂◦, i)] = Ker[Hom(∂1, i)]

❱➻ ✈➟② t❛ ✤÷ñ❝

H◦[Hom(C, Y )] = Ker[Hom(∂1, i)] / {0}

= Im[Hom(∂◦, i)] / {0} ≈ Hom(X, Y )

❇ê ✤➲ ✤➣ ✤÷ñ❝ ❝❤ù♥❣ ♠✐♥❤✳

✺✺

Hom(C′, Y′)✳ ❉♦ ✤â Hom(f, k) ❝↔♠ ù♥❣ r❛ ♠ët ✤ç♥❣ ❝➜✉

Hom(f, k)∗n : Extn(X, Y ) −→ Extn(X′, Y′) , ∀n > 0

❚❤❡♦ ▼➺♥❤ ✤➲ ✸✳✶✳✺✱ t❛ t❤➜② r➡♥❣ Hom(f, k)∗n ❦❤æ♥❣ ♣❤ö t❤✉ë❝ ✈➔♦ sü

❧ü❛ ❝❤å♥ ❝õ❛ ♣❤➨♣ ❜✐➳♥ ✤ê✐ ❞➙② ❝❤✉②➲♥ f : C′ → C ✈➔ ✤÷ñ❝ ❤♦➔♥ t♦➔♥ ①→❝

✤à♥❤ ❜ð✐ sè ♥❣✉②➯♥ n ✈➔ ❝→❝ ✤ç♥❣ ❝➜✉ h, k✳ ✣ç♥❣ ❝➜✉ ♥➔② ❣å✐ ❧➔ t➼❝❤ ♠ðrë♥❣ n ❝❤✐➲✉ tr➯♥ R ❝õ❛ ❝→❝ ✤ç♥❣ ❝➜✉ h✱ k ✈➔ ✤÷ñ❝ ❦þ ❤✐➺✉ ❧➔

Extn(h, k) : Extn(X, Y ) −→ Extn(X′, Y′)

✺✻

−−−−−−−−→ Hn[Hom(C, V )] H

n [Hom(i,g)]

−→ Extn(W, Y ) −−→ Extg∗ n(V, Y ) −−→ Extf∗ n(U, Y ) −→ Extδ n+1(W, Y ) −→

tr♦♥❣ ✤â f∗ = Extn(f, i) , g∗ = Extn(g, i) ✈➔ δ ❧➔ ✤ç♥❣ ❝➜✉ ♥è✐✳ ❉➣② ♥➔②

❚❤❡♦ ✣à♥❤ ❧þ ✸✳✷✳✶✸✱ t❛ ❝â ♠ët ❞➣② ❦❤î♣✿

−→ Extn(W, Y ) ξ

n (g)

−−−→ Extn(V, Y ) ξ

n (f )

−−−−→ Extn(U, Y )−−→δ

δ

−−→ Ext(W, Y ) −→

tr♦♥❣ ✤â ξn(f ) = Extn(f, i) , ξn(g) = Extn(g, i)✱ ✈➔ ❞➣② ♥➔② ❜➢t ✤➛✉ ❧➔

0 −→ Hom(W, Y ) −−−−−−→ Hom(V, Y )Hom(g,i) −−−−−−→ Hom(U, Y )Hom(f,i) −−→δ

h ◦ (A)

δ //φ1(Y ) ξ

1 (g) //φ1(F ) = 0

ξ◦(F ) ξ◦(f )//ξ◦(A) δ //ξ1(Y ) ξ

1 (g) //ξ1(F ) = 0

n (g) // 0

0ξn−1(f )// ξn−1(A) δ //ξn(Y ) ξ

n (g) // 0

h n (Z)

φn(Y )

h n (Y )

ξn−1(A′) δ //ξn(Z) ξ

n (α) //ξn(Y )