Hom and Ext, Revisited Justin Lyle Lawrence, KS justin.lyle@ku.edu - - PowerPoint PPT Presentation

Hom and Ext, Revisited

Hom and Ext, Revisited

Justin Lyle

Lawrence, KS justin.lyle@ku.edu

April 28, 2018

JL Hom and Ext, Revisited

Hom and Ext, Revisited

• Joint work with Hailong Dao and Mohammad Eghbali

JL Hom and Ext, Revisited

Hom and Ext, Revisited

• Throughout, let (R, m, k) be a commutative Noetherian local

ring and M, N finitely generated R-modules.

JL Hom and Ext, Revisited

Hom and Ext, Revisited

• Throughout, let (R, m, k) be a commutative Noetherian local

ring and M, N finitely generated R-modules.

• There are several results (and open problems!) in the

literature which take the form: if HomR(M, N) has some “nice” properties and Extn

R(M, N) = 0 for some values of n,

then M or N must be free or close to free.

JL Hom and Ext, Revisited

Hom and Ext, Revisited

Some Examples

1 If R is Gorenstein, dim R = 1, M is Maximal Cohen-Macaulay

(MCM) and EndR(M) is free, then M is free. ([Vas68])

JL Hom and Ext, Revisited

Hom and Ext, Revisited

Some Examples

1 If R is Gorenstein, dim R = 1, M is Maximal Cohen-Macaulay

(MCM) and EndR(M) is free, then M is free. ([Vas68])

2 Suppose R is Cohen-Macaulay (CM), I is a CM ideal of

height 1 such that HomR(I, I) is free and Ext1≤i≤d−1

R

(I, I) = Ext1≤i≤d

R

(I, R) = 0. Then M is free. ([GT17])

JL Hom and Ext, Revisited

Hom and Ext, Revisited

Some Examples

1 If R is Gorenstein, dim R = 1, M is Maximal Cohen-Macaulay

(MCM) and EndR(M) is free, then M is free. ([Vas68])

2 Suppose R is Cohen-Macaulay (CM), I is a CM ideal of

height 1 such that HomR(I, I) is free and Ext1≤i≤d−1

R

(I, I) = Ext1≤i≤d

R

(I, R) = 0. Then M is free. ([GT17])

3 Suppose R is Cohen-Macaulay and is a complete intersection

in codimension 1. Furthermore, assume that Q ⊆ R. If M is an R-module that is locally free in codimension one with constant rank, Ext1≤i≤d

R

(M, M) = 0, and Ext1≤i≤2d+1

R

(M, R) = 0, then M is free. [HL04]

JL Hom and Ext, Revisited

Hom and Ext, Revisited

2 and 3 are established cases of a famous open conjecture:

JL Hom and Ext, Revisited

Hom and Ext, Revisited

2 and 3 are established cases of a famous open conjecture: Conjecture (Auslander-Reiten) If Exti>0

R (M, R) = Exti>0 R (M, M) = 0 then M is free.

JL Hom and Ext, Revisited

Hom and Ext, Revisited

To extend these results, we consider two main questions:

JL Hom and Ext, Revisited

Hom and Ext, Revisited

To extend these results, we consider two main questions: Question (1) When does HomR(M, N) have a free summand?

JL Hom and Ext, Revisited

Hom and Ext, Revisited

To extend these results, we consider two main questions: Question (1) When does HomR(M, N) have a free summand? Question (2) When is HomR(M, N) ∼ = Nr?

JL Hom and Ext, Revisited

Hom and Ext, Revisited

To extend these results, we consider two main questions: Question (1) When does HomR(M, N) have a free summand? Question (2) When is HomR(M, N) ∼ = Nr? Our main technique in answering these questions is to consider two categories which behave nicely with respect to localizing and cutting down a regular sequence.

JL Hom and Ext, Revisited

Hom and Ext, Revisited

To extend these results, we consider two main questions: Question (1) When does HomR(M, N) have a free summand? Question (2) When is HomR(M, N) ∼ = Nr? Our main technique in answering these questions is to consider two categories which behave nicely with respect to localizing and cutting down a regular sequence. One nice feature of this approach is that we are often able to remove hypotheses such Cohen-Macaulayness and constant rank.

JL Hom and Ext, Revisited

Hom and Ext, Revisited

Two Key Categories

Henceforth, we set t := depth R. We set Deep(R) := {X | depth M ≥ t}. Our key categories are

JL Hom and Ext, Revisited

Hom and Ext, Revisited

Two Key Categories

Henceforth, we set t := depth R. We set Deep(R) := {X | depth M ≥ t}. Our key categories are Ω Deep(R) := {M | ∃ 0 → M → Rn → X → 0 exact with X ∈ Deep(R)} and

JL Hom and Ext, Revisited

Hom and Ext, Revisited

Two Key Categories

Henceforth, we set t := depth R. We set Deep(R) := {X | depth M ≥ t}. Our key categories are Ω Deep(R) := {M | ∃ 0 → M → Rn → X → 0 exact with X ∈ Deep(R)} and DF(R) := {M | ∃ 0 → R → Mn → X → 0 exact with X ∈ Deep(R)} If M ∈ DF(R) we say M is deeply faithful.

JL Hom and Ext, Revisited

Hom and Ext, Revisited

Observations

• R is in Ω Deep(R) ∩ DF(R).

JL Hom and Ext, Revisited

Hom and Ext, Revisited

Observations

• R is in Ω Deep(R) ∩ DF(R).
• Any t-syzygy module is in Deep(R) and so any (t + 1)-syzygy

is in Ω Deep(R).

JL Hom and Ext, Revisited

Hom and Ext, Revisited

Observations

• R is in Ω Deep(R) ∩ DF(R).
• Any t-syzygy module is in Deep(R) and so any (t + 1)-syzygy

is in Ω Deep(R).

• It is clear that for any X ∈ DF(R), X is faithful, and when

depth(R) = 0 deeply faithful and faithful modules coincide.

JL Hom and Ext, Revisited

Hom and Ext, Revisited

Observations

• R is in Ω Deep(R) ∩ DF(R).
• Any t-syzygy module is in Deep(R) and so any (t + 1)-syzygy

is in Ω Deep(R).

• It is clear that for any X ∈ DF(R), X is faithful, and when

depth(R) = 0 deeply faithful and faithful modules coincide.

• If M is a semi-dualizing module (that is, if the homothety

map R → HomR(M, M) is an isomorphism and Exti>0

R (M, M) = 0), then M ∈ DF(R).

JL Hom and Ext, Revisited

Hom and Ext, Revisited

Observations

• R is in Ω Deep(R) ∩ DF(R).
• Any t-syzygy module is in Deep(R) and so any (t + 1)-syzygy

is in Ω Deep(R).

• It is clear that for any X ∈ DF(R), X is faithful, and when

depth(R) = 0 deeply faithful and faithful modules coincide.

• If M is a semi-dualizing module (that is, if the homothety

map R → HomR(M, M) is an isomorphism and Exti>0

R (M, M) = 0), then M ∈ DF(R).

• Ω Deep(R) and DF(R) behave well with respect to localization

and with cutting down a (general) regular sequence.

JL Hom and Ext, Revisited

Hom and Ext, Revisited

Some Lemmas

In particular,

JL Hom and Ext, Revisited

Hom and Ext, Revisited

Some Lemmas

In particular, Lemma (1) Suppose t > 0. If M ∈ Ω Deep(R) (resp. M ∈ DF(R)) then, for a general R-regular sequence x, we have M ∈ Ω Deep(R) (resp. M ∈ DF(R)).

JL Hom and Ext, Revisited

Hom and Ext, Revisited

Some Lemmas

Lemma (2) Let 0 → X

i1

− → Y

p1

− → Z → 0 be an exact sequence.

JL Hom and Ext, Revisited

Hom and Ext, Revisited

Some Lemmas

Lemma (2) Let 0 → X

i1

− → Y

p1

− → Z → 0 be an exact sequence.

1 If Y ∈ Ω Deep(R) and Z ∈ Deep(R), then X ∈ Ω Deep(R). 2 If X ∈ DF(R) and Z ∈ Deep(R), then Y ∈ DF(R).

JL Hom and Ext, Revisited

Hom and Ext, Revisited

Some Lemmas

Lemma (2) Let 0 → X

i1

− → Y

p1

− → Z → 0 be an exact sequence.

1 If Y ∈ Ω Deep(R) and Z ∈ Deep(R), then X ∈ Ω Deep(R). 2 If X ∈ DF(R) and Z ∈ Deep(R), then Y ∈ DF(R).

Proof. Examine some pushout diagrams.

JL Hom and Ext, Revisited

Hom and Ext, Revisited

Some Lemmas

Lemma (3) Assume that t = 0, 0 = M ∈ Ω Deep(R) = Ω mod(R). The following are equivalent.

1 R | M. 2 M is faithful. 3 Soc(R) Ann(M). 4 M is not a minimal syzygy on R.

JL Hom and Ext, Revisited

Hom and Ext, Revisited

Some Lemmas

Lemma (4) Suppose Ext1≤i≤t

R

(M, R) = 0. Then M∗ free implies M is free.

JL Hom and Ext, Revisited

Hom and Ext, Revisited

Theorem (A) Let M ∈ Deep(R) and N ∈ Ω Deep(R). Assume that HomR(M, N) ∈ DF(R) and Ext1it−1

R

(M, N) = 0. Then R | N.

JL Hom and Ext, Revisited

Hom and Ext, Revisited

Theorem (A) Let M ∈ Deep(R) and N ∈ Ω Deep(R). Assume that HomR(M, N) ∈ DF(R) and Ext1it−1

R

(M, N) = 0. Then R | N. This has a nice corollary, which directly extends (1). Corollary Let M ∈ Deep(R) and N ∈ Ω Deep(R). Furthermore, assume that HomR(M, N) is free and Ext1it−1

R

(M, N) = 0. Then N is free.

JL Hom and Ext, Revisited

Hom and Ext, Revisited

Theorem (A) Let M ∈ Deep(R) and N ∈ Ω Deep(R). Assume that HomR(M, N) ∈ DF(R) and Ext1it−1

R

(M, N) = 0. Then R | N. This has a nice corollary, which directly extends (1). Corollary Let M ∈ Deep(R) and N ∈ Ω Deep(R). Furthermore, assume that HomR(M, N) is free and Ext1it−1

R

(M, N) = 0. Then N is free. Proof. Induct on µ(N), applying the Theorem repeatedly.

JL Hom and Ext, Revisited

Hom and Ext, Revisited

On the other side of the coin the Theorem gives the following:

JL Hom and Ext, Revisited

Hom and Ext, Revisited

On the other side of the coin the Theorem gives the following: Corollary Suppose M ∈ Deep(R), N ∈ Ω Deep(R), Hom(M, N) is free, and Ext1≤i≤t(M, N) = 0. Then M is free. Proof. The previous corollary implies N is free, so Hom(M, N) ∼ = (M∗)n is

• free. Thus M is free by Lemma 4.

JL Hom and Ext, Revisited

Hom and Ext, Revisited

Using these results we are to directly extend (2). In order to do that we need another lemma. Lemma (5) Suppose R is a quotient of a regular local ring that is (S2) and Gorenstein in codimension 1 with Q ⊆ R. Let M ∈ Ω Deep(R) be a reflexive R-module, free in codimension 1, and suppose Ext1≤i≤t−1

R

(M, M) = 0. Then M is free.

JL Hom and Ext, Revisited

Hom and Ext, Revisited

Proof.

• It is harmless to assume R is complete and dim R ≥ 2.

JL Hom and Ext, Revisited

Hom and Ext, Revisited

Proof.

• It is harmless to assume R is complete and dim R ≥ 2.
• We claim that M has constant rank.

JL Hom and Ext, Revisited

Hom and Ext, Revisited

Proof.

• It is harmless to assume R is complete and dim R ≥ 2.
• We claim that M has constant rank. Take p, q ∈ Min R. Since

R satisfies (S2), there is a chain of minimal primes p = p1, p2, . . . , pn = q such that ht(pi + pi+1) ≤ 1 for each i ([HH94]).

JL Hom and Ext, Revisited

Hom and Ext, Revisited

Proof.

• It is harmless to assume R is complete and dim R ≥ 2.
• We claim that M has constant rank. Take p, q ∈ Min R. Since

R satisfies (S2), there is a chain of minimal primes p = p1, p2, . . . , pn = q such that ht(pi + pi+1) ≤ 1 for each i ([HH94]). Ergo, rank Mpi = rank Mpi+1 for each i, since M is free on a minimal prime of pi + pi+1.

JL Hom and Ext, Revisited

Hom and Ext, Revisited

Proof.

• It is harmless to assume R is complete and dim R ≥ 2.
• We claim that M has constant rank. Take p, q ∈ Min R. Since

R satisfies (S2), there is a chain of minimal primes p = p1, p2, . . . , pn = q such that ht(pi + pi+1) ≤ 1 for each i ([HH94]). Ergo, rank Mpi = rank Mpi+1 for each i, since M is free on a minimal prime of pi + pi+1. In particular, rank Mp = rank Mq, and so M has constant rank.

• A trace map argument shows that R is a summand of

EndR(M).

• By Theorem A, M = R ⊕ M′ and m′ satisfies the hypotheses

again.

• Induction on µ(M) gives that M is free.

JL Hom and Ext, Revisited

Hom and Ext, Revisited

With this lemma we can provide a direct improvement of (2) Theorem (B) Suppose R is (S2), Gorenstein in codimension 1, and with Q ⊆ R. Suppose N ∈ mod(R) such that pd Np < ∞ for all p ∈ Spec R with ht p ≤ 1. Set a = min{t, depth N} and suppose Ext1≤i≤t−1

R

(N, N) = Ext1≤i≤2t+1−a

R

(N, R) = 0. Then N is free.

JL Hom and Ext, Revisited

Hom and Ext, Revisited

While our results thus far do not directly extend (3), since I need not be in Ω Deep(R), we can provide an extension of (3) in case R is a normal domain via a Brauer group argument: Theorem (C) Let R be a Cohen-Macaulay local normal domain. Let M be a maximal Cohen-Macaulay module such that HomR(M, M) is free and Ext1≤i≤d−1

R

(M, M) = Ext1≤i≤d

R

(M, R) = 0. Then M is free.

JL Hom and Ext, Revisited

Hom and Ext, Revisited

As far as Question (2) is concerned we have the following: Theorem (4) Assume that depth(M) depth N, Ass(N) = Min(N), and for some s ≥ depth N, Ext1is

R

(M, N) = 0. If Hom(M, N) ∼ = Nr for some r ∈ N, then M/IM ∼ = (R/I)r for I = Ann(N). Furthermore, if one of the following holds:

1 N is faithful. 2 Ass(R) ⊆ Ass(N) and s > 0.

then M ∼ = Rr.

JL Hom and Ext, Revisited

Hom and Ext, Revisited

Shiro Goto and Ryo Takahashi, On the Auslander-Reiten conjecture for Cohen-Macaulay local rings, Proc. Amer. Math.

• Soc. 145 (2017), no. 8, 3289–3296. MR 3652783

Melvin Hochster and Craig Huneke, Indecomposable canonical modules and connectedness, Commutative algebra: syzygies, multiplicities, and birational algebra (South Hadley, MA, 1992), Contemp. Math., vol. 159, Amer. Math. Soc., Providence, RI, 1994, pp. 197–208. MR 1266184 Craig Huneke and Graham J. Leuschke, On a conjecture of Auslander and Reiten, J. Algebra 275 (2004), no. 2, 781–790. MR 2052636 Wolmer V. Vasconcelos, Reflexive modules over Gorenstein rings, Proc. Amer. Math. Soc. 19 (1968), 1349–1355. MR 0237480

JL Hom and Ext, Revisited

Hom and Ext, Revisited

Thank you!

JL Hom and Ext, Revisited