MATHEMATICAL LECTURES
INST. STUD. THEOR. PHYSICS AND MATHEMATICS
2011 Institute for Research in Fundamental Sciences – IPM
Relative homological algebra in categories
of representations of quivers
A general overview of the Theory of Covers and Envelopes
Sergio Estrada
Universidad de Murcia
[email protected]
Murcia, SPAIN
IPM, Tehran, July 2011
Slide 1
MATHEMATICAL LECTURES
INST. STUD. THEOR. PHYSICS AND MATHEMATICS
• R.Baer, Abelian groups which are direct summands of every
containing group, Bull. Amer. Math. Soc. 46 (1940), 800-806.
Sergio Estrada — Relative Homological Algebra — IPM, Tehran, July 2011
Slide 2
MATHEMATICAL LECTURES
INST. STUD. THEOR. PHYSICS AND MATHEMATICS
• R.Baer, Abelian groups which are direct summands of every
containing group, Bull. Amer. Math. Soc. 46 (1940), 800-806.
• B. Eckmann and A. Schöpf, Über injective modular, Arch. Math. 4
(1953), 75-78.
Sergio Estrada — Relative Homological Algebra — IPM, Tehran, July 2011
Slide 2
MATHEMATICAL LECTURES
INST. STUD. THEOR. PHYSICS AND MATHEMATICS
• R.Baer, Abelian groups which are direct summands of every
containing group, Bull. Amer. Math. Soc. 46 (1940), 800-806.
• B. Eckmann and A. Schöpf, Über injective modular, Arch. Math. 4
(1953), 75-78.
• S. Eilenberg, Homological dimension and syzygies, Ann. Math.
64 (1956), 328-336.
Sergio Estrada — Relative Homological Algebra — IPM, Tehran, July 2011
Slide 2
MATHEMATICAL LECTURES
INST. STUD. THEOR. PHYSICS AND MATHEMATICS
• R.Baer, Abelian groups which are direct summands of every
containing group, Bull. Amer. Math. Soc. 46 (1940), 800-806.
• B. Eckmann and A. Schöpf, Über injective modular, Arch. Math. 4
(1953), 75-78.
• S. Eilenberg, Homological dimension and syzygies, Ann. Math.
64 (1956), 328-336.
• S. Eilenberg and T. Nakayama, On the dimension of modules and
algebras V, Nagoya Math. J. 11 (1957), 9-12.
Sergio Estrada — Relative Homological Algebra — IPM, Tehran, July 2011
Slide 2
MATHEMATICAL LECTURES
INST. STUD. THEOR. PHYSICS AND MATHEMATICS
• R.Baer, Abelian groups which are direct summands of every
containing group, Bull. Amer. Math. Soc. 46 (1940), 800-806.
• B. Eckmann and A. Schöpf, Über injective modular, Arch. Math. 4
(1953), 75-78.
• S. Eilenberg, Homological dimension and syzygies, Ann. Math.
64 (1956), 328-336.
• S. Eilenberg and T. Nakayama, On the dimension of modules and
algebras V, Nagoya Math. J. 11 (1957), 9-12.
• H. Bass, Finitistic dimension and a homological generalization of
semiprimary rings, Trans. Amer. Math. Soc. 95 (1960), 466-488.
Sergio Estrada — Relative Homological Algebra — IPM, Tehran, July 2011
Slide 2
MATHEMATICAL LECTURES
INST. STUD. THEOR. PHYSICS AND MATHEMATICS
From sixties to eighties:
Fuchs and Warfield introduce the pure-injective envelopes.
Enochs defines the torsion-free covers.
Sergio Estrada — Relative Homological Algebra — IPM, Tehran, July 2011
Slide 3
MATHEMATICAL LECTURES
INST. STUD. THEOR. PHYSICS AND MATHEMATICS
From sixties to eighties:
Fuchs and Warfield introduce the pure-injective envelopes.
Enochs defines the torsion-free covers.
General definition of covers and envelopes with respect to a certain
class F :
E. Enochs, Injective and flat covers, envelopes and resolvents, Israel J.
Math. 39 (1981), 190-209.
Sergio Estrada — Relative Homological Algebra — IPM, Tehran, July 2011
Slide 3
MATHEMATICAL LECTURES
INST. STUD. THEOR. PHYSICS AND MATHEMATICS
A abelian category and F a class of objects closed under
isomorphisms.
Definition
An F -precover of M is a morphism
ϕ : F → M such that F ∈ F and every diagram
MATHEMATICAL LECTURES
INST. STUD. THEOR. PHYSICS AND MATHEMATICS
A abelian category and F a class of objects closed under
isomorphisms.
Definition
An F -precover of M is a morphism
ϕ : F → M such that F ∈ F and every diagram
MATHEMATICAL LECTURES
INST. STUD. THEOR. PHYSICS AND MATHEMATICS
A abelian category and F a class of objects closed under
isomorphisms.
Definition
An F -precover of M is a morphism
ϕ : F → M such that F ∈ F and every diagram
MATHEMATICAL LECTURES
INST. STUD. THEOR. PHYSICS AND MATHEMATICS
A abelian category and F a class of objects closed under
isomorphisms.
Definition
An F -precover of M is a morphism
ϕ : F → M such that F ∈ F and every diagram
MATHEMATICAL LECTURES
INST. STUD. THEOR. PHYSICS AND MATHEMATICS
A abelian category and F a class of objects closed under
isomorphisms.
Definition
An F -precover of M is a morphism
ϕ : F → M such that F ∈ F and every diagram
F
Sergio Estrada — Relative Homological Algebra — IPM, Tehran, July 2011
Slide 4
ϕ
/M
MATHEMATICAL LECTURES
INST. STUD. THEOR. PHYSICS AND MATHEMATICS
A abelian category and F a class of objects closed under
isomorphisms.
Definition
An F -precover of M is a morphism
ϕ : F → M such that F ∈ F and every diagram
F
F0
Sergio Estrada — Relative Homological Algebra — IPM, Tehran, July 2011
Slide 4
/M
~>
~
~
~~ 0
~~ ϕ
ϕ
MATHEMATICAL LECTURES
INST. STUD. THEOR. PHYSICS AND MATHEMATICS
A abelian category and F a class of objects closed under
isomorphisms.
Definition
An F -precover of M is a morphism
ϕ : F → M such that F ∈ F and every diagram
FO
f
F0
/M
~>
~
~
~~ 0
~~ ϕ
ϕ
can be completed commutatively, with F 0 ∈ F .
Sergio Estrada — Relative Homological Algebra — IPM, Tehran, July 2011
Slide 4
MATHEMATICAL LECTURES
INST. STUD. THEOR. PHYSICS AND MATHEMATICS
A abelian category and F a class of objects closed under
isomorphisms.
Definition
An F -precover of M is a morphism
ϕ : F → M such that F ∈ F and every diagram
FO
f
F0
/M
~>
~
~
~~ 0
~~ ϕ
ϕ
can be completed commutatively, with F 0 ∈ F .
If ϕ ◦ f = ϕ ⇒ f is an automorphism, ϕ is an F -cover.
Sergio Estrada — Relative Homological Algebra — IPM, Tehran, July 2011
Slide 4
MATHEMATICAL LECTURES
INST. STUD. THEOR. PHYSICS AND MATHEMATICS
A abelian category and F a class of objects closed under
isomorphisms.
Definition
An F -precover of M is a morphism
ϕ : F → M such that F ∈ F and every diagram
FO
f
F0
/M
~>
~
~
~~ 0
~~ ϕ
ϕ
can be completed commutatively, with F 0 ∈ F .
If ϕ ◦ f = ϕ ⇒ f is an automorphism, ϕ is an F -cover.
F -(pre)envelopes are defined in a dual manner.
F : the class of flat modules, F -cover=flat cover.
E : the class of injective modules, E -envelope= injective envelope.
Sergio Estrada — Relative Homological Algebra — IPM, Tehran, July 2011
Slide 4
MATHEMATICAL LECTURES
INST. STUD. THEOR. PHYSICS AND MATHEMATICS
Flat cover conjecture: “Every module over an associative ring has a
flat cover”.
It was known to be true:
For modules over a left perfect ring.
For modules over a Prüfer domain, torsion-free=flat.
E. Enochs, Torsion free covering modules, Proc. Amer. Math. Soc. 14
(1963), 884-889.
Sergio Estrada — Relative Homological Algebra — IPM, Tehran, July 2011
Slide 5
MATHEMATICAL LECTURES
INST. STUD. THEOR. PHYSICS AND MATHEMATICS
Flat cover conjecture: “Every module over an associative ring has a
flat cover”.
It was known to be true:
For modules over a left perfect ring.
For modules over a Prüfer domain, torsion-free=flat.
E. Enochs, Torsion free covering modules, Proc. Amer. Math. Soc. 14
(1963), 884-889.
Before of the later resolution to the conjecture, the most significant
advance was obtained by:
J. Xu, Flat covers of modules, Lecture Notes in Math. 1634,
Springer-Verlag 1996.
Sergio Estrada — Relative Homological Algebra — IPM, Tehran, July 2011
Slide 5
MATHEMATICAL LECTURES
INST. STUD. THEOR. PHYSICS AND MATHEMATICS
Positive solution to the conjecture:
L. Bican, R. El Bashir and E. Enochs, All Modules have flat covers,
Bull. London Math. Soc. 33(4) (2001), 385-390.
Enochs’ proof:
L. Salce, Cotorsion theories for abelian groups, Symposia
Mathematica, Vol. 23 (1979), 11-32.
P.C. Eklof and J. Trlifaj, How to make Ext vanish, Bull. London Math.
Soc. 33(1) (2001), 41-51.
Sergio Estrada — Relative Homological Algebra — IPM, Tehran, July 2011
Slide 6
MATHEMATICAL LECTURES
INST. STUD. THEOR. PHYSICS AND MATHEMATICS
S.T. Aldrich, E. Enochs, J.R. García Rozas and L. Oyonarte, Covers
and envelopes in Grothendieck categories. Flat covers of complexes
with applications, J. Algebra 243(2) (2001), 615-630.
E. Enochs, L. Oyonarte and B. Torrecillas, Flat covers and flat
representations of quivers, Comm. Algebra, 32(4)(2004), 1319-1338.
J.R. García Rozas, J.A. López Ramos and B. Torrecillas, On the
existence of flat covers in R-gr, Comm. Algebra, 29(8) (2001),
3341-3349.
Sergio Estrada — Relative Homological Algebra — IPM, Tehran, July 2011
Slide 7
MATHEMATICAL LECTURES
INST. STUD. THEOR. PHYSICS AND MATHEMATICS
Categories without enough projectives: sheaves of OX -modules over a
topological space, quasi-coherent sheaves on a scheme.
E. Enochs and L. Oyonarte, Flat covers and cotorsion envelopes of
sheaves, Proc. Amer. Math. Soc. 130(5) (2001), 1285-1292.
Sergio Estrada — Relative Homological Algebra — IPM, Tehran, July 2011
Slide 8
MATHEMATICAL LECTURES
INST. STUD. THEOR. PHYSICS AND MATHEMATICS
Categories without enough projectives: sheaves of OX -modules over a
topological space, quasi-coherent sheaves on a scheme.
E. Enochs and L. Oyonarte, Flat covers and cotorsion envelopes of
sheaves, Proc. Amer. Math. Soc. 130(5) (2001), 1285-1292.
Quasi-coherent sheaves?
Existence theorems of covers in categories without enough
projectives.
Qco(X ) is locally κ-presentable.
Sergio Estrada — Relative Homological Algebra — IPM, Tehran, July 2011
Slide 8
INST. STUD. THEOR. PHYSICS AND MATHEMATICS
MATHEMATICAL LECTURES
For a given F in A ,
F ⊥ = {C ∈ Ob(A ) : Ext1 (F , C ) = 0, ∀F ∈ F }.
Analogously ⊥F will denote
⊥
F = {G ∈ Ob(A ) : Ext1 (G, D ) = 0, ∀D ∈ F }.
The pair (F , F ⊥ ) is cogenerated by a set T if:
C∈F⊥
⇔ Ext1 (F , C ) = 0 ∀F ∈ T .
Definition
(F , C ) is a cotorsion theory if
F ⊥ = C and ⊥ C = F .
Sergio Estrada — Relative Homological Algebra — IPM, Tehran, July 2011
Slide 9
INST. STUD. THEOR. PHYSICS AND MATHEMATICS
MATHEMATICAL LECTURES
1
The pair of classes
(P roj , R-Mod) and (R-Mod, I nj )
are cotorsion theories, P roj the class of projective modules and
I nj the class of injective modules. (R-Mod, I nj ) is cogenerated
by the set {R /I : I ≤R R }.
2
The pair (F , C ) composed by flat modules and cotorsion
modules (flat cotorsion theory).
Sergio Estrada — Relative Homological Algebra — IPM, Tehran, July 2011
Slide 10
MATHEMATICAL LECTURES
INST. STUD. THEOR. PHYSICS AND MATHEMATICS
Theorem
Let A be a Grothendieck category and F a class of objects of A
closed under direct sums, extensions and well ordered direct limits.
(F , F ⊥ ) cogenerated by a set ⇒ ∀M ∈ Ob(A ) there exists an
F -cover and an F ⊥ -envelope.
Sergio Estrada — Relative Homological Algebra — IPM, Tehran, July 2011
Slide 11
INST. STUD. THEOR. PHYSICS AND MATHEMATICS
MATHEMATICAL LECTURES
Proof: Every object in a Grothendieck category is small.
Lemma
There exists an ordinal number λ such that ∀λ0 ≥ λ and for every well
ordered inductive system (Eα , fβα ), α < λ0 of injective objects of C ,
lim α<λ0 Eα is also injective ∀λ0 ≥ λ.
→
Proposition
For every object M ∈ Ob(C ) there exists an ordinal number λ such
that ∀λ0 ≥ λ
∼ lim α<λ0 Extn (M , Mα ),
Extn (M , lim α<λ0 Mα ) =
→
→
for every w. o. inductive system (Mα , fβα ), α < λ0 in C .
Sergio Estrada — Relative Homological Algebra — IPM, Tehran, July 2011
Slide 12
MATHEMATICAL LECTURES
INST. STUD. THEOR. PHYSICS AND MATHEMATICS
There is another proof without using homological methods, by means
of “The Small Object Argument”
D.G. Quillen, Homotopical algebra, Lecture
Notes in Mathematics, Vol. 43, Springer-Verlag, 1967.
by using a more general version which appears in
M. Hovey, Model categories, Mathematical Surveys and Monographs,
Vol. 63, American Mathematical Society, Providence, RI, 1998.
So in any case everything reduces to prove that the pair (F , F ⊥ ) is
cogenerated by a set.
Sergio Estrada — Relative Homological Algebra — IPM, Tehran, July 2011
Slide 13
MATHEMATICAL LECTURES
INST. STUD. THEOR. PHYSICS AND MATHEMATICS
Lemma (Eklof)
Let A be an abelian category with direct limits and A, C objects of A .
Let us suppose that
• A = ∪α<λ Aα for an ordinal number λ
• Ext1 (A0 , C ) = 0 and Ext1 (Aα+1 /Aα , C ) = 0 for every α < λ.
Then Ext1 (A, C ) = 0.
Proposition
Let F ∈ F and x ∈ F . Suppose there exists a cardinal ℵ and S ⊆ F
such that.
• x ∈ S, |S | ≤ ℵ
• S, F /S ∈ F .
Then the pair (F , F ⊥ ) is cogenerated by a set.
Sergio Estrada — Relative Homological Algebra — IPM, Tehran, July 2011
Slide 14
MATHEMATICAL LECTURES
INST. STUD. THEOR. PHYSICS AND MATHEMATICS
Representations of quivers
A quiver Q is a directed graph.
A path p of Q is a sequence of arrows. If t (p) = i (q ) we get the path
qp.
P (Q )v , (left) path tree associated to Q with root in v : is a quiver whose
vertices are the paths p of Q begining in v and the arrows the pairs
(p, ap).
A representation by modules of Q is a functor X : Q → R-Mod. A
morphism between X and Y is a natural transformation.
(Q , R-Mod) is the family of representations by modules of a quiver Q,
it is a Grothendieck category with enough projectives.
Sergio Estrada — Relative Homological Algebra — IPM, Tehran, July 2011
Slide 15
MATHEMATICAL LECTURES
INST. STUD. THEOR. PHYSICS AND MATHEMATICS
RQ (path algebra of Q) is the free R-module whose base are the paths
p of Q, and
qp if t (p) = i (q )
q ·p =
0
in other case
If Q has a finite number of vertices, RQ has identity
1 = v1 + · · · + vn ,
if not is a ring with local units.
The categories (Q , R-Mod) and RQ-Mod are equivalent.
Sergio Estrada — Relative Homological Algebra — IPM, Tehran, July 2011
Slide 16
MATHEMATICAL LECTURES
INST. STUD. THEOR. PHYSICS AND MATHEMATICS
Quasi-coherent R-modules
Q = (V , E ) is a quiver.
R is a representation of Q in the category of rings, that is,
for each v ∈ V we have a ring R(v ) and
for an arrow a : v → w ∈ E a homomorphism of rings
R(a) : R(v ) → R(w ).
An R-module M is an R(v )-module M (v ) for v ∈ V
and an R(v )-morphism M (a) : M (v ) → M (w ) for an arrow a : v → w.
Sergio Estrada — Relative Homological Algebra — IPM, Tehran, July 2011
Slide 17
INST. STUD. THEOR. PHYSICS AND MATHEMATICS
MATHEMATICAL LECTURES
M is quasi-coherent if for each edge a the morphism
R(w ) ⊗R(v ) M (v ) → M (w )
is an R(w )-isomorphism.
The category of quasi-coherent R-modules is cocomplete with exact
direct limits and abelian if R(w ) is a flat R(v )-module for v → w.
Sergio Estrada — Relative Homological Algebra — IPM, Tehran, July 2011
Slide 18
MATHEMATICAL LECTURES
INST. STUD. THEOR. PHYSICS AND MATHEMATICS
Consider quasi-coherent sheaves over (X , OX ).
If U are the affine opens U ⊆ X , a quasi-coherent sheaf over (X , OX )
(or a quasi-coherent OX -module) is uniquely determined by
Sergio Estrada — Relative Homological Algebra — IPM, Tehran, July 2011
Slide 19
MATHEMATICAL LECTURES
INST. STUD. THEOR. PHYSICS AND MATHEMATICS
Consider quasi-coherent sheaves over (X , OX ).
If U are the affine opens U ⊆ X , a quasi-coherent sheaf over (X , OX )
(or a quasi-coherent OX -module) is uniquely determined by
An O (U )-module MU for each U
Sergio Estrada — Relative Homological Algebra — IPM, Tehran, July 2011
Slide 19
MATHEMATICAL LECTURES
INST. STUD. THEOR. PHYSICS AND MATHEMATICS
Consider quasi-coherent sheaves over (X , OX ).
If U are the affine opens U ⊆ X , a quasi-coherent sheaf over (X , OX )
(or a quasi-coherent OX -module) is uniquely determined by
An O (U )-module MU for each U
Maps MU → MV if V ⊆ U, V , U ∈ U satisfying
Sergio Estrada — Relative Homological Algebra — IPM, Tehran, July 2011
Slide 19
MATHEMATICAL LECTURES
INST. STUD. THEOR. PHYSICS AND MATHEMATICS
Consider quasi-coherent sheaves over (X , OX ).
If U are the affine opens U ⊆ X , a quasi-coherent sheaf over (X , OX )
(or a quasi-coherent OX -module) is uniquely determined by
An O (U )-module MU for each U
Maps MU → MV if V ⊆ U, V , U ∈ U satisfying
i ) O (V ) ⊗O (U ) MU → MV is an isomorphism ∀V ⊆ U
Sergio Estrada — Relative Homological Algebra — IPM, Tehran, July 2011
Slide 19
MATHEMATICAL LECTURES
INST. STUD. THEOR. PHYSICS AND MATHEMATICS
Consider quasi-coherent sheaves over (X , OX ).
If U are the affine opens U ⊆ X , a quasi-coherent sheaf over (X , OX )
(or a quasi-coherent OX -module) is uniquely determined by
An O (U )-module MU for each U
Maps MU → MV if V ⊆ U, V , U ∈ U satisfying
i ) O (V ) ⊗O (U ) MU → MV is an isomorphism ∀V ⊆ U
ii ) The compatibility condition,
if W ⊆ V ⊆ U, (W , V , U ∈ U ), then
MU
@
@
? @
R
- MW
MV
is commutative.
Sergio Estrada — Relative Homological Algebra — IPM, Tehran, July 2011
Slide 19
MATHEMATICAL LECTURES
INST. STUD. THEOR. PHYSICS AND MATHEMATICS
Proposition
For M a right R-module and N a left R-module, the tensor product
M ⊗R N is the Z-module
(Z(v ) = Z, ∀v ∈ V and Z(a) = idZ for all a ∈ E) such that
M ⊗R N (v ) = M (v ) ⊗R(v ) N (v ),
with M ⊗R N (a) the obvious map.
def .
R F is flat ⇔ − ⊗R F is exact.
F will denote the class of all flat quasi-coherent R-modules.
Sergio Estrada — Relative Homological Algebra — IPM, Tehran, July 2011
Slide 20
MATHEMATICAL LECTURES
INST. STUD. THEOR. PHYSICS AND MATHEMATICS
C. Jensen and H. Lenzing. Model-theoretic algebra with particular
emphasis on fields, rings, modules, Algebra, Logic and Applications,
Vol. 2, Gordon and Breach, 1989.
Proposition
Q = (V , E ) is a quiver, and M a quasi-coherent R-module over Q.
Let κ be an infinite cardinal such that κ ≥ |R(v )| ∀v ∈ V and
κ ≥ |E |, |V |.
Let Xv ⊆ M (v ) be subsets with |Xv | ≤ κ ∀v ∈ V .
There exits a quasi-coherent submodule M 0 ⊆ M with
i) M 0 (v ) ⊆ M (v ) pure, ∀v ∈ V
ii) Xv ⊆ M 0 (v ), ∀v ∈ V and
iii) |M 0 | ≤ κ.
Sergio Estrada — Relative Homological Algebra — IPM, Tehran, July 2011
Slide 21
MATHEMATICAL LECTURES
INST. STUD. THEOR. PHYSICS AND MATHEMATICS
B. Conrad. Grothendieck duality and base change, Lecture Notes in
Mathematics, Vol. 1750, Springer-Verlag, 2000.
Corollary (Gabber)
The category of quasi-coherent R-modules is locally κ-presentable.
Theorem
Every quasi-coherent R-module has a flat cover and a cotorsion
envelope.
Corollary
For a given scheme (X , OX ), every quasi-coherent sheaf on OX
admits a flat cover and a cotorsion envelope.
Sergio Estrada — Relative Homological Algebra — IPM, Tehran, July 2011
Slide 22
MATHEMATICAL LECTURES
INST. STUD. THEOR. PHYSICS AND MATHEMATICS
Gorenstein categories
A Grothendieck category.
Definition
X ∈ Ob(A )
pd X ≤ n ⇔ Exti (X , −) = 0 for i ≥ n + 1.
FPD (A ) = sup{pd X : ∀X , pd X < ∞}
FID (A ) = sup{id X : ∀X , id X < ∞}.
Definition
We will say that A is a Gorenstein category if the following hold:
1) For any object L of A , pd L < ∞ if and only if id L < ∞.
2) FPD (A ) < ∞ and FID (A ) < ∞.
3) A has a generator L such that pd L < ∞.
Sergio Estrada — Relative Homological Algebra — IPM, Tehran, July 2011
Slide 23
MATHEMATICAL LECTURES
INST. STUD. THEOR. PHYSICS AND MATHEMATICS
Definition
E is Gorenstein injective if there exists an exact complex
· · · → E−1 → E0 → E1 → · · ·
such that E = Ker (E0 → E1 ) and it is
Hom(U , −)-exact for all injective U.
Dually we define Gorenstein projective objects.
Definition
Gpd (X ) = n if the first syzygy of X that is Gorenstein projective is the
n-th one and Gpd (X ) = ∞ if there is no such syzygy.
glGpd (A ) = sup{Gpd (X ) : X ∈ Ob(A )}
Then also define Gid (Y ) and glGid (A ).
Sergio Estrada — Relative Homological Algebra — IPM, Tehran, July 2011
Slide 24
MATHEMATICAL LECTURES
INST. STUD. THEOR. PHYSICS AND MATHEMATICS
L = {X ∈ Ob(A ) : id X < ∞}
Theorem
If (A , L ) is a Gorenstein category then
⊥
1 (L , L ) is a cotorsion theory.
⊥ is the class of Gorenstein injective objects of A .
2 L
3 For Each M ∈ Ob (A ) has a special L -precover and a special
L ⊥ -preenvelope (so (L , L ⊥ ) is a complete and hereditary
cotorsion theory).
If FID (A ) = n then Gid (Y ) ≤ n for all objects Y of A .
Sergio Estrada — Relative Homological Algebra — IPM, Tehran, July 2011
Slide 25
MATHEMATICAL LECTURES
INST. STUD. THEOR. PHYSICS AND MATHEMATICS
Theorem
If (A , L ) be a Gorenstein category of dimension at most n having
enough projectives. Then for an object C of A the following are
equivalent:
1) C is an n-th syzygy.
2) C ∈ ⊥ L .
3) C is Gorenstein projective.
As a consequence we get that glGpd (A ) ≤ n and that (⊥ L , L ) is a
complete hereditary cotorsion pair.
Sergio Estrada — Relative Homological Algebra — IPM, Tehran, July 2011
Slide 26
MATHEMATICAL LECTURES
INST. STUD. THEOR. PHYSICS AND MATHEMATICS
A Grothendieck category with enough projectives.
Theorem
Let A be a Grothendieck category with enough projectives. Then the
following are equivalent:
1) A is Gorenstein.
2) glGpd (A ) < ∞ and glGid (A ) < ∞.
Moreover, if (1) (or (2)) holds we have
FID (A ) = FPD (A ) = glGpd (A ) = glGid (A ).
Sergio Estrada — Relative Homological Algebra — IPM, Tehran, July 2011
Slide 27
MATHEMATICAL LECTURES
INST. STUD. THEOR. PHYSICS AND MATHEMATICS
Gexti functors
A Gorenstein category.
• For a given Y there exists a Gorenstein injective resolution, that is
an exact sequence
0 → Y → G0 → G1 → · · ·
such that Hom(−, G) leaves the sequence exact, for all
Gorenstein injective G.
• We define right derived functors Gexti (X , Y ), i ≥ 0 of Hom by
using Gorenstein injective resolutions of Y .
Sergio Estrada — Relative Homological Algebra — IPM, Tehran, July 2011
Slide 28
INST. STUD. THEOR. PHYSICS AND MATHEMATICS
MATHEMATICAL LECTURES
A Gorenstein category with glGid (A ) = n.
i
c (X , Y ), i ∈ Z
Tate cohomology functors Ext
• The n-th cosyzygy of Y , G, is Gorenstein injective, so there exists
E : · · · → E −1 → E 0 → E 1 → · · ·
with G = Ker (E 0 → E 1 ), such that
Hom(U , E)
is exact, U injective.
• E is unique up to homotopy
c i (X , Y ) def
Ext
= i −th cohomology
groups of Hom(X , E)
Sergio Estrada — Relative Homological Algebra — IPM, Tehran, July 2011
Slide 29
MATHEMATICAL LECTURES
INST. STUD. THEOR. PHYSICS AND MATHEMATICS
L. Avramov and A. Martsinkovsky, Absolute, relative and Tate
cohomology of modules of finite Gorenstein dimension, Proc. London
Math. Soc. 85(3) (2002), 393-440.
A. Iacob, Generalized Tate Cohomology, Tsukuba J. Math. 29(2)
(2005), 389-404.
Proposition
If A is a Gorenstein category of dimension at most n then for all
objects X and Y of A there exist natural exact sequences
1
c (X , Y ) → Gext 2 (X , Y ) →
0 → Gext 1 (X , Y ) → Ext1 (X , Y ) → Ext
c n (X , Y ) → 0.
· · · → Gext n (X , Y ) → Extn (X , Y ) → Ext
Sergio Estrada — Relative Homological Algebra — IPM, Tehran, July 2011
Slide 30
MATHEMATICAL LECTURES
INST. STUD. THEOR. PHYSICS AND MATHEMATICS
X ⊆ Pn (A) closed subscheme.
X locally Gorenstein scheme (so R(v ) is Gorenstein ring, i.e.,
commutative noetherian and id R(v ) < ∞, ∀v ).
Theorem
Qco(X ) is a Gorenstein category.
Sergio Estrada — Relative Homological Algebra — IPM, Tehran, July 2011
Slide 31
MATHEMATICAL LECTURES
INST. STUD. THEOR. PHYSICS AND MATHEMATICS
Projective and injective model structure on
Gorenstein categories
Theorem
(Hovey). If (A , L ) is a Gorenstein category then there is a cofibrantly
generated model structure on A with L the full subcategory of trivial
objects and such that the fibrant objects are the Gorenstein injective
objects.
If A has enough projectives then there is a cofibrantly generated
model structure on A with L the trivial objects and such that the
cofibrant objects are the Gorenstein projective objects.
Sergio Estrada — Relative Homological Algebra — IPM, Tehran, July 2011
Slide 32