Deligne’s tensor product Questions we answer Existence of Deligne’s tensor Counterexample to the existence Tensor products of finitely cococomplete and abelian categories 1 Ignacio López Franco University of Cambridge Gonville and Caius College Coimbra, 11 July 2012 1 With thanks to P. Deligne.
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Deligne’s tensor productQuestions we answer
Existence of Deligne’s tensorCounterexample to the existence
Tensor products of finitely cococomplete andabelian categories1
Ignacio López Franco
University of CambridgeGonville and Caius College
Coimbra, 11 July 2012
1With thanks to P. Deligne.
Deligne’s tensor productQuestions we answer
Existence of Deligne’s tensorCounterexample to the existence
Plan
Deligne’s tensor product
Questions we answer
Existence of Deligne’s tensor
Counterexample to the existence
Deligne’s tensor productQuestions we answer
Existence of Deligne’s tensorCounterexample to the existence
Deligne’s tensor product of abelian categories
k commutative ring. All the categories and functors will beenriched in k–Mod.
In Catégories tannakiennes (1990) Deligne introduced andused:
Deligne’s tensor productQuestions we answer
Existence of Deligne’s tensorCounterexample to the existence
Deligne’s tensor product of abelian categories
k commutative ring. All the categories and functors will beenriched in k–Mod.
In Catégories tannakiennes (1990) Deligne introduced andused:
Deligne’s tensor productQuestions we answer
Existence of Deligne’s tensorCounterexample to the existence
Deligne’s tensor product of abelian categoriesDefinition (Deligne)Given A,B abelian categories, their tensor product is anabelian category A • B with a bilinear rex in each variable
A × B→ A • B
that induces equivalences for all abelian C
Rex[A • B,C] ' Rex[A,B;C]
A × B //
%%KKKKKKKKKK A • B
�����
C
Deligne’s tensor productQuestions we answer
Existence of Deligne’s tensorCounterexample to the existence
Deligne’s tensor product of abelian categories
Definition (?,Kelly, well-known)Given A,B fin. cocomplete categories, their tensor product isan fin. cocomplete category A � B with a bilinear rex in eachvariable
A × B→ A � B
that induces equivalences for all fin. cocomplete C
Rex[A � B,C] ' Rex[A,B;C]
A × B //
%%KKKKKKKKKK A • B
�����
C
Deligne’s tensor productQuestions we answer
Existence of Deligne’s tensorCounterexample to the existence
ExampleFor k-algebras R, S,
R–Modf × S–Modf⊗k−−→ R ⊗ S–Modf
gives
R ⊗ S–Modf 'R–Modf � S–Modf
'R–Modf • S–Modf if abelian
Deligne’s tensor product has been used inI Representations and classification of Hopf algebras.I Tannaka-type reconstruction results.I Invariants of manifolds.
Deligne’s tensor productQuestions we answer
Existence of Deligne’s tensorCounterexample to the existence
ExampleFor k-algebras R, S,
R–Modf × S–Modf⊗k−−→ R ⊗ S–Modf
gives
R ⊗ S–Modf 'R–Modf � S–Modf
'R–Modf • S–Modf if abelian
Deligne’s tensor product has been used inI Representations and classification of Hopf algebras.I Tannaka-type reconstruction results.I Invariants of manifolds.
Deligne’s tensor productQuestions we answer
Existence of Deligne’s tensorCounterexample to the existence
QuestionsExample (Existence of �)For fin. cocomplete A,B, the tensor A � B exists.
A � B ' Lex[Aop ,Bop; k -Mod]f
Deligne does not show that his tensor product exists in general.We may ask:
1. Does Deligne’s tensor product always exist? No.2. For fin. cocomplete categories A,B, is A � B always
abelian whenever A,B are so? No.3. For abelian A,B, their Deligne tensor product A • B exists
iff A � B is abelian. Yes.2 + 3⇒ 1
Deligne’s tensor productQuestions we answer
Existence of Deligne’s tensorCounterexample to the existence
QuestionsExample (Existence of �)For fin. cocomplete A,B, the tensor A � B exists.
A � B ' Lex[Aop ,Bop; k -Mod]f
Deligne does not show that his tensor product exists in general.We may ask:
1. Does Deligne’s tensor product always exist? No.2. For fin. cocomplete categories A,B, is A � B always
abelian whenever A,B are so? No.3. For abelian A,B, their Deligne tensor product A • B exists
iff A � B is abelian. Yes.2 + 3⇒ 1
Deligne’s tensor productQuestions we answer
Existence of Deligne’s tensorCounterexample to the existence
QuestionsExample (Existence of �)For fin. cocomplete A,B, the tensor A � B exists.
A � B ' Lex[Aop ,Bop; k -Mod]f
Deligne does not show that his tensor product exists in general.We may ask:
1. Does Deligne’s tensor product always exist? No.2. For fin. cocomplete categories A,B, is A � B always
abelian whenever A,B are so? No.3. For abelian A,B, their Deligne tensor product A • B exists
iff A � B is abelian. Yes.2 + 3⇒ 1
Deligne’s tensor productQuestions we answer
Existence of Deligne’s tensorCounterexample to the existence
QuestionsExample (Existence of �)For fin. cocomplete A,B, the tensor A � B exists.
A � B ' Lex[Aop ,Bop; k -Mod]f
Deligne does not show that his tensor product exists in general.We may ask:
1. Does Deligne’s tensor product always exist? No.2. For fin. cocomplete categories A,B, is A � B always
abelian whenever A,B are so? No.3. For abelian A,B, their Deligne tensor product A • B exists
iff A � B is abelian. Yes.2 + 3⇒ 1
Deligne’s tensor productQuestions we answer
Existence of Deligne’s tensorCounterexample to the existence
QuestionsExample (Existence of �)For fin. cocomplete A,B, the tensor A � B exists.
A � B ' Lex[Aop ,Bop; k -Mod]f
Deligne does not show that his tensor product exists in general.We may ask:
1. Does Deligne’s tensor product always exist? No.2. For fin. cocomplete categories A,B, is A � B always
abelian whenever A,B are so? No.3. For abelian A,B, their Deligne tensor product A • B exists
iff A � B is abelian. Yes.2 + 3⇒ 1
Deligne’s tensor productQuestions we answer
Existence of Deligne’s tensorCounterexample to the existence
QuestionsExample (Existence of �)For fin. cocomplete A,B, the tensor A � B exists.
A � B ' Lex[Aop ,Bop; k -Mod]f
Deligne does not show that his tensor product exists in general.We may ask:
1. Does Deligne’s tensor product always exist? No.2. For fin. cocomplete categories A,B, is A � B always
abelian whenever A,B are so? No.3. For abelian A,B, their Deligne tensor product A • B exists
iff A � B is abelian. Yes.2 + 3⇒ 1
Deligne’s tensor productQuestions we answer
Existence of Deligne’s tensorCounterexample to the existence
QuestionsExample (Existence of �)For fin. cocomplete A,B, the tensor A � B exists.
A � B ' Lex[Aop ,Bop; k -Mod]f
Deligne does not show that his tensor product exists in general.We may ask:
1. Does Deligne’s tensor product always exist? No.2. For fin. cocomplete categories A,B, is A � B always
abelian whenever A,B are so? No.3. For abelian A,B, their Deligne tensor product A • B exists
iff A � B is abelian. Yes.2 + 3⇒ 1
Deligne’s tensor productQuestions we answer
Existence of Deligne’s tensorCounterexample to the existence
Existence of Deligne’s product
LemmaFor abelian A,B, if A � B is abelian then A • B exists and is(equivalent to) A � B.
Proof.Need A × B→ A � B to induce
Rex[A � B,C] ' Rex[A,B;C]
for all abelian C. But by definition of � this is true for any fin.cocomplete C. �
Deligne’s tensor productQuestions we answer
Existence of Deligne’s tensorCounterexample to the existence
Existence of Deligne’s product
LemmaFor abelian A,B, if A � B is abelian then A • B exists and is(equivalent to) A � B.
Proof.Need A × B→ A � B to induce
Rex[A � B,C] ' Rex[A,B;C]
for all abelian C. But by definition of � this is true for any fin.cocomplete C. �
Deligne’s tensor productQuestions we answer
Existence of Deligne’s tensorCounterexample to the existence
Existence of Deligne’s product
For a fin. cocomplete A, write A = Lex[Aop , k -Mod]
LemmaIf A • B exists, then A • B is cocomplete abelian and
A × B→ A • B→ A • B
inducesCocts[A • B,C] ' Rex[A,B;C]
for all cocomplete abelian C.
Deligne’s tensor productQuestions we answer
Existence of Deligne’s tensorCounterexample to the existence
Existence of Deligne’s productTheoremFor abelian A,B, TFAE
1. A • B exists.2. A � B is abelian.
Proof.(2⇒1) Lemma.(1⇒ 2) By Lemma, enough to prove A • B ' A � B, i.e.,
A � B ' Lex[Aop ,Bop; k -Mod]
has the universal property of A • B and it is abelian.
Cocts[A � B,C] ' Rex[A � B,C] ' Rex[A,B;C]
�
Deligne’s tensor productQuestions we answer
Existence of Deligne’s tensorCounterexample to the existence
Existence of Deligne’s productTheoremFor abelian A,B, TFAE
1. A • B exists.2. A � B is abelian.
Proof.(2⇒1) Lemma.(1⇒ 2) By Lemma, enough to prove A • B ' A � B, i.e.,
A � B ' Lex[Aop ,Bop; k -Mod]
has the universal property of A • B and it is abelian.
Cocts[A � B,C] ' Rex[A � B,C] ' Rex[A,B;C]
�
Deligne’s tensor productQuestions we answer
Existence of Deligne’s tensorCounterexample to the existence
Existence of Deligne’s productTheoremFor abelian A,B, TFAE
1. A • B exists.2. A � B is abelian.
Proof.(2⇒1) Lemma.(1⇒ 2) By Lemma, enough to prove A • B ' A � B, i.e.,
A � B ' Lex[Aop ,Bop; k -Mod]
has the universal property of A • B and it is abelian.
Cocts[A � B,C] ' Rex[A � B,C] ' Rex[A,B;C]
�
Deligne’s tensor productQuestions we answer
Existence of Deligne’s tensorCounterexample to the existence
Existence of Deligne’s productTheoremFor abelian A,B, TFAE
1. A • B exists.2. A � B is abelian.
Proof.(2⇒1) Lemma.(1⇒ 2) By Lemma, enough to prove A • B ' A � B, i.e.,
A � B ' Lex[Aop ,Bop; k -Mod]
has the universal property of A • B and it is abelian.
Cocts[A � B,C] ' Rex[A � B,C] ' Rex[A,B;C]
�
Deligne’s tensor productQuestions we answer
Existence of Deligne’s tensorCounterexample to the existence
Proof cont.TheoremThe reflection
[(A ⊗ B)op , k-Mod]→ Lex[Aop ,Bop; k-Mod]
is exact if A,B are abelian.
Proof.I Follows from: the reflection[Aop , k -Mod]→ Lex[Aop , k -Mod] is lex.
I Follows from:
Lex[Aop , k -Mod] = Sh(A, J) ⊂ [Aop , k -Mod]
J generated by {e : A ′ → A epi} (because A is abelian).
�
Deligne’s tensor productQuestions we answer
Existence of Deligne’s tensorCounterexample to the existence
Proof cont.TheoremThe reflection
[(A ⊗ B)op , k-Mod]→ Lex[Aop ,Bop; k-Mod]
is exact if A,B are abelian.
Proof.I Follows from: the reflection[Aop , k -Mod]→ Lex[Aop , k -Mod] is lex.
I Follows from:
Lex[Aop , k -Mod] = Sh(A, J) ⊂ [Aop , k -Mod]
J generated by {e : A ′ → A epi} (because A is abelian).
�
Deligne’s tensor productQuestions we answer
Existence of Deligne’s tensorCounterexample to the existence
Proof cont.TheoremThe reflection
[(A ⊗ B)op , k-Mod]→ Lex[Aop ,Bop; k-Mod]
is exact if A,B are abelian.
Proof.I Follows from: the reflection[Aop , k -Mod]→ Lex[Aop , k -Mod] is lex.
I Follows from:
Lex[Aop , k -Mod] = Sh(A, J) ⊂ [Aop , k -Mod]
J generated by {e : A ′ → A epi} (because A is abelian).
�
Deligne’s tensor productQuestions we answer
Existence of Deligne’s tensorCounterexample to the existence
Summary
We showed, for a pair of abelian categories TFAEI Their Deligne tensor product exists.I Their tensor as fin. cocomplete categories is abelian.
Deligne’s tensor productQuestions we answer
Existence of Deligne’s tensorCounterexample to the existence
Counterexample to existence of Deligne’s productEnough to find two abelian A,B with A � B not abelian.
Definition/Theorem (Chase, Bourbaki, 1960s)A k -algebra R is left coherent iff R–Modf is abelian iff every f.g.left ideal is f.p.
Theorem (Soublin, 1968)There exist two commutative coherent Q-algebras R ,S withR ⊗ S not coherent.
Proof.Set R = Q[x], S = (QN)[[u, t ]]. �
So R-Modf � S-Modf is not abelian, and the Deligne’s tensorR-Modf • S-Modf does not exist.
Deligne’s tensor productQuestions we answer
Existence of Deligne’s tensorCounterexample to the existence
Counterexample to existence of Deligne’s productEnough to find two abelian A,B with A � B not abelian.
Definition/Theorem (Chase, Bourbaki, 1960s)A k -algebra R is left coherent iff R–Modf is abelian iff every f.g.left ideal is f.p.
Theorem (Soublin, 1968)There exist two commutative coherent Q-algebras R ,S withR ⊗ S not coherent.
Proof.Set R = Q[x], S = (QN)[[u, t ]]. �
So R-Modf � S-Modf is not abelian, and the Deligne’s tensorR-Modf • S-Modf does not exist.
Deligne’s tensor productQuestions we answer
Existence of Deligne’s tensorCounterexample to the existence
Counterexample to existence of Deligne’s productEnough to find two abelian A,B with A � B not abelian.
Definition/Theorem (Chase, Bourbaki, 1960s)A k -algebra R is left coherent iff R–Modf is abelian iff every f.g.left ideal is f.p.
Theorem (Soublin, 1968)There exist two commutative coherent Q-algebras R ,S withR ⊗ S not coherent.
Proof.Set R = Q[x], S = (QN)[[u, t ]]. �
So R-Modf � S-Modf is not abelian, and the Deligne’s tensorR-Modf • S-Modf does not exist.
Deligne’s tensor productQuestions we answer
Existence of Deligne’s tensorCounterexample to the existence
Counterexample to existence of Deligne’s productEnough to find two abelian A,B with A � B not abelian.
Definition/Theorem (Chase, Bourbaki, 1960s)A k -algebra R is left coherent iff R–Modf is abelian iff every f.g.left ideal is f.p.
Theorem (Soublin, 1968)There exist two commutative coherent Q-algebras R ,S withR ⊗ S not coherent.
Proof.Set R = Q[x], S = (QN)[[u, t ]]. �
So R-Modf � S-Modf is not abelian, and the Deligne’s tensorR-Modf • S-Modf does not exist.
Deligne’s tensor productQuestions we answer
Existence of Deligne’s tensorCounterexample to the existence
Counterexample to existence of Deligne’s productEnough to find two abelian A,B with A � B not abelian.
Definition/Theorem (Chase, Bourbaki, 1960s)A k -algebra R is left coherent iff R–Modf is abelian iff every f.g.left ideal is f.p.
Theorem (Soublin, 1968)There exist two commutative coherent Q-algebras R ,S withR ⊗ S not coherent.
Proof.Set R = Q[x], S = (QN)[[u, t ]]. �
So R-Modf � S-Modf is not abelian, and the Deligne’s tensorR-Modf • S-Modf does not exist.
Deligne’s tensor productQuestions we answer
Existence of Deligne’s tensorCounterexample to the existence
Conclusion
I Deligne’s tensor A • B does not always exist.I When A • B exists it is (equivalent to) A � B.I Better use the product of fin. cocomplete categories �.
Deligne’s tensor productQuestions we answer
Existence of Deligne’s tensorCounterexample to the existence
Bibliography
P. Deligne Catégories tannakiennes The GrothendieckFestschrift, Vol. II. Progr. Math. 87, 111–195. BirkhäuserBoston. 1990
G. M. Kelly Structures defined by finite limits in the enrichedcontext. I Cahiers Topologie Géom. Différentielle, 23. 1982.
T. Kerler and V. Lyubashenko Non-semisimple topologicalquantum field theories for 3-manifolds with corners LNM1765, Springer-Verlag, Berlin, 2001.
V. Lyubashenko Squared Hopf algebras Mem. Amer. Math.Soc. 142. 1999.
J-P. Soublin Anneaux et modules cohérents J. Algebra, 15.1970