A Noether-Deuring theorem for derived categories

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by  Alexander Zimmermann

A NOETHER-DEURING THEOREM FOR DERIVED CATEGORIES
ALEXANDERZIMMERMANN
Abstract. We prove a Noether-Deuring theorem for the derived category of bounded
complexes of modules over a Noetherian algebra.
2
1 Introduction
0
2 The classical Noether-Deuring theorem states that given an algebra A over a field K and

n a finite extension field L of K, two A-modules M and N are isomorphic as A-modules, if
a L ⊗ M is isomorphic to L ⊗ N as an L ⊗ A-module. In 1972 Roggenkamp gave a
J K K K
nice extension of this result to extensions S of local commutative Noetherian rings R and
3
modules over Noetherian R-algebras.
1
ForthederivedcategoryofA-modulesnosuchgeneralisationwasdocumentedbefore. The

] purpose of this note is to give a version of the Noether-Deuring theorem, in the generalised
T
version given by Roggenkamp, for right bounded derived categories of A-modules. If there
R
is a morphism α ∈ Hom (X,Y), then it is fairly easy to show that for a faithfully
. D(Λ)
h
flat ring extension S over R the fact that id ⊗ α is an isomorphism implies that α is
t S
a an isomorphism. This is done in Proposition 1. More delicate is the question if only an
m
isomorphism in Hom (S ⊗ X,S ⊗ Y) is given. Then we need further finiteness
[ D(S⊗RΛ) R R
conditions on Λ and on R and proceed by completion of R and then a classical going-down

2 argument. This is done in Theorem 4 and Corollary 8.
v For thenotation concerningderived categories werefer toVerdier[5]. Inparticular, D(A)
8 (resp D−(A), resp Db(A)) denotes the derived category of complexes (resp. right bounded
5
4 complexes, resp. bounded complexes) of finitely generated A-modules, K−(A−proj) (resp.
3 Kb(A−proj), resp K−,b(A−proj)) is the homotopy category of right bounded complexes
.
2 (resp. bounded complexes, resp. right bounded complexes with bounded homology) of
1 finitely generated projective A-modules. For acomplex Z wedenotebyH (Z)thehomology
i
1
of Z in degree i, and by H(Z) the graded module given by the homology of Z.
1
:
v 1. The result
i
X
We start with an easy observation.
r
a Proposition 1. Let R be a commutative ring and let Λ be an R-algebra. Let S be a
commutative faithfully flat R-algebra. Denote by D(Λ) the derived category of complexes
L
of finitely generated Λ-modules. Then if there is α ∈ Hom (X,Y) so that id ⊗ α ∈
D(Λ) S R
L L
Hom (S⊗ X,S⊗ Y) is an isomorphism in D(S⊗ Λ), then α is an isomorphism
D(S⊗RΛ) R R R
in D(Λ).
Proof. Let Z be a complex in D(Λ). Since S is flat over R the functor S ⊗ − :
R
L
R−Mod −→ S −Mod is exact, and hence the left derived functor S ⊗ − coincides with
R
the ordinary tensor product functor S ⊗ −. We can therefore work with the usual tensor
R
product and a complex Z of Λ-modules.
L
We claim that since S is flat, S⊗ − induces an isomorphism S⊗ H(Z) ≃ H(S⊗ Z).
R R R
If ∂ is the differential of Z, then
Z
0 −→ ker(∂ ) −→ Z −∂→Z im(∂ ) −→ 0
Z Z
Date: November11, 2011; revised December 28, 2011.
2010 Mathematics Subject Classification. Primary 16E35; Secondary 11S36, 13J10, 18E30, 16G30 .
1
2 ALEXANDERZIMMERMANN
is exact in the category of Λ-modules.
Since S is flat,
0 −→ S ⊗ ker(∂ ) −→ S ⊗ Z idS−⊗→R∂Z S ⊗ im(∂ ) −→ 0
R Z R R Z
is exact. Hence
ker(id ⊗ ∂ ) = S ⊗ ker(∂ ) and im(id ⊗ ∂ ) = S ⊗ im(∂ ).
S R Z R Z S R Z R Z
This shows the claim.
Sinceid ⊗ αisanisomorphism,itsconeC(id ⊗ α)isacyclic. Moreover, C(id ⊗ α) =
S R S R S R
S ⊗ C(α) by the very construction of the mapping cone. But now,
R
0 = H(C(id ⊗ α)) = H(S ⊗ C(α)) = S ⊗ H(C(α)).
S R R R
Since S is faithfully flat, this implies H(C(α)) = 0 and therefore C(α) is acyclic. We
conclude that α is an isomorphism in D(Λ) which shows the statement.
α
Remark 2. Observe that we assumed that X −→ Y is assumed to be a morphism in D(Λ).
αˆ
The question if the existence of an isomorphism S⊗ X −→ S⊗ Y in D(S⊗ Λ) implies
R R R
L
the existence of a morphism α : X −→ Y in D(Λ) so that id ⊗ α is an isomorphism is
S R
left open. Under stronger hypotheses this is the purpose of Theorem 4 below. The proof
follows [4] which deals with the module case.
Lemma 3. If S is a faithfully flat R-module and Λ is a Noetherian R-algebra, then for all
objects X and Y of Db(Λ) we get
Hom (S ⊗ X,S ⊗ Y)≃ S ⊗ Hom (X,Y).
Db(S⊗RΛ) R R R Db(Λ)
Proof. Since S is flat over R, the functor S ⊗ − preserves quasi-isomorphisms and
R
therefore we get a morphism
S ⊗ Hom (U,V)−→ Hom (S ⊗ U,S ⊗ V)
R Db(Λ) Db(S⊗RΛ) R R
in the following way. Given a morphism ρ in Hom (U,V) represented by the triple
Db(Λ)
α β
U ←− W −→ V , for a quasi-isomorphism α and a morphism of complexes β, and s ∈ S
(cid:16) (cid:17)
then map s⊗ρ to S ⊗ U id←S⊗−αS ⊗ W −s⊗→β S ⊗ V . This is natural in U and V.
R R R
(cid:16) (cid:17)
We use the equivalence of categories K−,b(Λ−proj) ≃ Db(Λ) and suppose therefore that
X and Y are right bounded complexes of finitely generated projective Λ-modules. But
S ⊗RHomΛ(Λn,U) = S ⊗RUn = (S ⊗RU)n = HomS⊗RΛ((S ⊗RΛ)n,S ⊗RU)
which proves the statement in case X or Y is in Kb(A−proj) since then a homomorphism
is given by a direct sum of finitely many homogeneous mappings in those degrees where the
complexes do both have non zero components. Now, tensor product commutes with direct
sums.
We come to the general case. Recall the so-called stupid truncation τ of a complex.
N
Let Z be a complex in K−,b(Λ − proj), denote by ∂ its differential and let N ∈ N so
that H (Z) = 0 for all n ≥ N. We denote the homogeneous components of ∂ so that
n
∂n : Zn −→ Zn−1 for all n. Let τNZ be the complex given by (τNZ)n = Zn if n ≤ N and
(τ Z) = 0 else. The differential δ on τ Z is defined to be δ = ∂ if n ≤ N and δ = 0
N n N n n n
else. Now, ker(∂ ) =: C (Z) is a finitely generated Λ-module. Therefore we get an exact
N N
triangle, called in the sequel the truncation triangle for Z,
τ Z −→ Z −→ C (Z)[N +1] −→ (τ Z)[1]
N N N
for all objects Z in K−,b(A−proj). Obviously τ (S⊗ Z)= S⊗ τ Z and since S is flat
N R R N
over R also C (S ⊗ Z)= S ⊗ C (Z).
N R R N
We choose N so that H (X) = H (Y) = 0 for all n ≥ N. To simplify the nota-
n n
tion denote for the moment the bifunctor HomK−,b(Λ−proj)(−,−) by (−,−), the bifunctor
HomK−,b(S⊗RΛ−proj)(−,−) by (−,−)S and the bifunctor S ⊗R HomK−,b(Λ−proj)(−,−) by
A NOETHER-DEURING THEOREM FOR DERIVED CATEGORIES 3
S(−,−). Further, put S⊗ X =:X and S⊗ Y =:Y . From the long exact sequence ob-
R S R S
tained byapplying(X ,−) tothetruncationtriangle ofY , wegetacommutative diagram
S S S
with exact lines (†)
(XS,CN(YS)[N])S → (XS,τNYS)S → (XS,YS)S → (XS,CN(YS)[N+1])S → (XS,τNYS[1])S
↑ ↑ ↑ ↑ ↑
S(X,CN(Y)[N]) → S(X,τNY) → S(X,Y) → S(X,CN(Y)[N +1]) → S(X,τNY[1])
Since τ (Y ) is a bounded complex of projectives,
N S
(X ,τ Y ) = S ⊗ (X,τ Y) and (X ,τ Y [1]) = S ⊗ (X,τ Y[1]).
S N S S R N S N S S R N
Weapply(−,C (Y )[k]) ,forafixedinteger k, tothetruncationtriangleforX andobtain
N S S S
an exact sequence
(τ X [1],C (Y )[k]) → (C (X )[N +1],C (Y )[k]) → (X ,C (Y )[k]) →
N S N S S N S N S S S N S S
→ (τ X ,C Y [k]) → (C (X )[N],C (Y )[k])
N S N S S N S N S S
and a commutative diagram analogous to the diagram (†).
Now, for morphisms between finitely presented Λ-modules M and N we do have that the
natural map
S ⊗RHomΛ(M,N) −→ HomS⊗RΛ(S ⊗RM,S ⊗RN)
is an isomorphism. Indeed, let
P −→ P −→ M −→ 0
1 0
be the first terms of a projective resolution of M as a Λ-module. Then
S ⊗ P −→ S ⊗ P −→ S ⊗ M −→ 0
R 1 R 0 R
are the first terms of a projective resolution of S⊗ M as an S⊗ Λ-module, and, denoting
R R
by SM := S ⊗ M, SN := S ⊗ N, SP := S ⊗ P for i ∈ {0,1}, and SΛ := S ⊗ Λ, we
R R i R i R
get
Hom (SM,SN) ֒→ Hom (SP ,SN) → Hom (SP ,SN)
SΛ SΛ 0 SΛ 1
↑ ↑ ↑
S ⊗ Hom (M,N) ֒→ S ⊗ Hom (P ,N) → S ⊗ Hom (P ,N)
R Λ R Λ 0 R Λ 1
is a commutative diagram with exact lines. The second and the third vertical morphisms
are isomorphisms. Indeed,
S ⊗ Hom (Λn,N) = S ⊗ Nn
R Λ R
= (S ⊗ N)n
R
= HomS⊗RΛ(S ⊗RΛn,S ⊗RN)
and since P and P are direct factors of Λn, for certain n gives the result. Therefore the
0 1
left most vertical homomorphism is an isomorphism as well.
Given a projective resolution P• −→ M of M, denote by ∂n : ΩnM ֒→ Pn−1 the embed-
dingof then-thsyzygy of M intothedegreen−1homogeneous componentoftheprojective
resolution. Then
ExtnΛ(M,N) = HomΛ(ΩnM,N)/(HomΛ(Pn−1,N)◦∂n)
and therefore
Hom (ΩnM,N)
S ⊗ Extn(M,N) = S ⊗ Λ
R Λ R(cid:18)HomΛ(Pn−1,N)◦∂n(cid:19)
(S ⊗ Hom (ΩnM,N))
R Λ
=
(S ⊗R(HomΛ(Pn−1,N)◦∂n))
= HomS⊗RΛ(S ⊗RΩnM,S ⊗RN)
HomS⊗RΛ(S ⊗RPn−1,S ⊗RN)◦(1S ⊗∂n)
= Extn (S ⊗ M,S ⊗ N)
S⊗RΛ R R
for all n ∈N, natural in M and N.
4 ALEXANDERZIMMERMANN
The case k = N +1 shows then
(C (X )[N +1],C (Y )[N +1]) = S ⊗ (C (X)[N +1],C (Y)[N +1])
N S N S S R N N
and
(C (X )[N],C (Y )[N +1]) = S ⊗ (C (X)[N],C (Y)[N +1]).
N S N S S R N N
By the case for bounded complex of projectives we get that the natural morphism is an
isomorphism for
(τ X [1],C (Y )[N +1]) ≃ S ⊗ (τ X[1],C (Y)[N +1])
N S N S S R N N
and
(τ X ,C (Y )[N +1]) ≃ S ⊗ (τ X,C (Y)[N +1]).
N S N S S R N N
Therefore also
(X ,C (Y )[N +1]) ≃ S ⊗ (X,C (Y)[N +1])
S N S S R N
and by the very same arguments, using k = N, we get
(X ,C (Y )[N]) ≃ S ⊗ (X,C (Y)[N]).
S N S S R N
This shows that we get isomorphisms in the two left and the two right vertical morphisms
of (†) and hence also the central vertical morphism is an isomorphism. Hence
(X ,Y ) ≃ S ⊗ (X,Y)
S S S R
and the lemma is proved.
Theorem 4. Let R be a commutative Noetherian ring, let S be a commutative Noetherian
R-algebra and suppose that S is a faithfully flat R-module. Suppose S⊗ rad(R)= rad(S).
R
Let Λ be a Noetherian R-algebra, let X and Y be two objects of of Db(Λ) and suppose that
End (X) is a finitely generated R-module. Then
Db(Λ)
L L
S ⊗ X ≃ S ⊗ Y ⇔ X ≃ Y.
R R
Remark 5. We observe that if R is local and S = Rˆ is the rad(R)-adic completion, then S
is faithfully flat as R-module and S ⊗ rad(R)= rad(S).
R
Proof of Theorem 4. According to the hypotheses we now supposethat End (X) and
Db(Λ)
EndD−(Λ)(Y) are finitely generated R-module and that S ⊗Rrad(R) = rad(S). Since S is
flat over R, tensor product of S over R is exact and we may replace the left derived tensor
product by the ordinary tensor product. We only need to show ”⇒” and assume therefore
that X and Y are in K−,b(Λ−proj), and that S ⊗ X and S ⊗ Y are isomorphic.
R R
Let X := S⊗ X and S⊗ Y =: Y in Db(S⊗ Λ) to shorten the notation and denote
S R R S R
by ϕ the isomorphism X −→ Y . Since then X is a direct factor of Y by means of ϕ ,
S S S S S S
the mapping
n
ϕ = s ⊗ϕ : X −→ Y
S i i S S
Xi=1
for s ∈S and ϕ ∈ Hom (X,Y) has a left inverse ψ : Y −→ X so that
i i Db(Λ) S S
ψ◦ϕ = id .
S XS
Then,
0 −→ rad(R)−→ R −→ R/rad(R)−→ 0
is exact and since S is flat over R we get that
0 −→ S ⊗ rad(R)−→ S −→ S ⊗ (R/rad(R)) −→ 0
R R
is exact. This shows
S ⊗ (R/rad(R)) ≃ S/(S ⊗ rad(R)).
R R
By hypothesis we have S ⊗ rad(R) = rad(S), identifying canonically S ⊗ R ≃ S. Then
R R
there are r ∈ R so that 1 ⊗r −s ∈ rad(S) for all i∈ {1,...,n}.
i S i i
A NOETHER-DEURING THEOREM FOR DERIVED CATEGORIES 5
Put
n
ϕ := r ϕ ∈ Hom (X,Y).
i i Db(Λ)
Xi=1
Then
n n
ψ◦(1 ⊗(r ϕ ))−1 ⊗id = (ψ◦(1 ⊗r ϕ )−ψ◦(s ⊗ϕ ))
S i i S X S i i i i
Xi=1 Xi=1
n
= (1 ⊗r −s )·(ψ◦(id ⊗ϕ ))
S i i S i
Xi=1
∈ rad(S)⊗ End (X)
R Db(Λ)
(cid:16) (cid:17)
and since End (X) is a Noetherian R-module, using Nakayama’s lemma we obtain that
Db(Λ)
n
ψ ◦( 1 ⊗r ϕ ) is invertible in S ⊗ End (X). Hence id ⊗ ϕ is left split and
i=1 S i i R Db(Λ) S R
therePfore
X id−S⊗→Rϕ Y −→ C(id ⊗ ϕ) −0→ X [1]
S S S R S
is a distinguished triangle, with C(id ⊗ ϕ) being the cone of id ⊗ ϕ. However,
S R S R
C(id ⊗ ϕ) = S ⊗ C(ϕ)
S R R
and hence
X id−S⊗→Rϕ Y −→ S ⊗ C(ϕ)−0→ X [1]
S S R S
is a distinguished triangle.
Since ϕ is an isomorphism, ϕ has a right inverse χ : Y −→ X as well. Now, since
S S S S
X ≃ Y , since S is faithfully flat over R, and since End (X) is finitely generated as
S S Db(Λ)
R-module, using Lemma 3 we obtain that End (Y) is finitely generated as R-module
Db(Λ)
as well. The same argument as for the left inverse ψ shows that (id ⊗ϕ)◦χ is invertible
S
in S ⊗ End (Y). Hence
R Db(Λ)
X id−S⊗→Rϕ Y −0→ S ⊗ C(ϕ)−0→ X [1]
S S R S
is a distinguished triangle. This shows that S ⊗ C(ϕ) is acyclic, and hence
R
0= H(S ⊗ C(ϕ)) = S ⊗ H(C(ϕ)).
R R
Since S is faithfully flat over R also H(C(ϕ)) = 0, which implies that C(ϕ) is acyclic and
therefore ϕ is an isomorphism.
This proves the theorem.
Let A be an algebra over a complete discrete valuation ring R which is finitely generated
as a module over R. We shall need a Krull-Schmidt theorem for the derived category of
bounded complexes over A. This fact seems to be well-known, but for the convenience of
the reader we give a proof.
Proposition 6. Let R be a complete discrete valuation ring and let A be an R-algebra,
finitely generated as R-module. Then the Krull-Schmidt theorem holds for K−,b(A−proj).
Proof. We first show a Fitting lemma for K−,b(A−proj).
Let X be a complex in K−,b(A−proj) and let u be an endomorphism of the complex
X. Then X = X′ ⊕X′′ as graded modules, by Fitting’s lemma in the version for algebras
over complete discrete valuation rings [1, Lemma 1.9.2]. The restriction of u on X′ is an
automorphism in each degree and the restriction of u on X′′ is nilpotent modulo rad(R)m
ι 0
for each m. Therefore u is a diagonal matrix in each degree where ι :X′ −→ X′
(cid:18) 0 ν (cid:19)
is invertible, and ν : X′′ −→ X′′ is nilpotent modulo rad(R)m for each m in each degree.
∂ ∂
The differential ∂ on X is given by 1 2 and the fact that u commutes with ∂ shows
(cid:18) ∂3 ∂4 (cid:19)
6 ALEXANDERZIMMERMANN
that ∂ ι = ν∂ and ∂ ν = ι∂ . Therefore, ∂ ιs = νs∂ and ∂ νs = ιs∂ for all s. Since ν is
3 3 2 2 3 3 2 2
nilpotentmodulorad(R)m foreach mineach degree, andιisinvertible, ∂ = ∂ =0. Hence
2 3
the differential of X restricts to a differential on X′ and a differential on X′′. Moreover, X′
and X′′ are both projective modules, since X is projective.
Now, X, and therefore also X′′ is exact in degrees higher than N, say. We fix m ∈ N
and obtain therefore that u is nilpotent modulo rad(R)m in each degree lower than N. Let
M be the nilpotency degree. Then, since X′′ is exact in degrees higher than N, modulo
m
rad(R)m the restriction of the endomorphism uMm to X′′ is homotopy equivalent to 0 in
degrees higher than N. We get therefore that the restriction of u to X′′ is actually nilpotent
modulo rad(R)m for each m.
Hence,theendomorphismringofanindecomposableobjectislocalandtheKrull-Schmidt
theorem is an easy consequence by the classical proof as in [3] or in [1].
This shows the proposition.
Remark 7. If R is a field and A is a finite dimensional R-algebra, then we would be able
to argue more directly. Indeed, X′ = im(uN) and X′′ = ker(uN) for large enough N. Then
it is obvious that X′ and X′′ are both subcomplexes of X. Observe that R may be a field
in Proposition 6.
For the next Corollary we follow closely [4].
Corollary 8. Let R be a commutative semilocal Noetherian ring, let S be a commutative
R-algebra so that Sˆ := Rˆ ⊗ S is a faithful projective Rˆ-module of finite type. Let Λ be
R
a Noetherian R-algebra, finitely generated as R-module, and let X and Y be two objects
of Db(Λ) and suppose that End (X) and End (Y) are finitely generated R-module.
Db(Λ) Db(Λ)
Then
L L
S ⊗ X ≃ S ⊗ Y ⇔ X ≃ Y.
R R
Proof. IfS⊗LX ≃ S⊗LY in Db(S⊗ Λ), weget Sˆ⊗LX ≃ Sˆ⊗LY in Db(Sˆ⊗ Λ). Since
R R R R R R
R is semilocal with maximal ideals m ,...,m we get Rˆ = s Rˆ for the completion
1 s i=1 mi
Rˆ of R at m . Now, Sˆ is projective faithful of finite type, anQd so there are n ,...,n with
mi i 1 s
s
Sˆ ≃ (Rˆ )ni
mi
Yi=1
and therefore Sˆ⊗L X ≃ Sˆ⊗L Y implies
R R
s s
(Rˆ )ni ⊗L X ≃ (Rˆ )ni ⊗L Y.
mi R mi R
Yi=1 Yi=1
Hence
(Rˆ ⊗L X)ni ≃ (Rˆ ⊗L Y)ni
mi R mi R
for each i, and therefore by Proposition 6
Rˆ ⊗L X ≃ Rˆ ⊗L Y
mi R mi R
for each i. By Theorem 4 we obtain X ≃ Y.
We get cancellation of factors from this statement.
Corollary 9. Under the hypothesis of Theorem 4 or of Corollary 8 we get X⊕U ≃ Y ⊕U
in Db(Λ) implies X ≃ Y.
Proof. This is clear by Corollary 8 in combination with Proposition 6.
Remark 10. In [2] we developed a theory to roughly speaking parameterise geometrically
objects in Db(A) by orbits of a group action on a variety. For this purpose we need to
assume that A is a finite dimensional algebra over an algebraically closed field K, so that it
is possible to use arguments and constructions from algebraic geometry. Using Theorem 4
we can extend the theory to non algebraically closed fields K as well.
A NOETHER-DEURING THEOREM FOR DERIVED CATEGORIES 7
Acknowledgement: I wish to thank the referee for useful suggestions which lead to sub-
stantial improvements.
References
[1] David Benson, Representations and Cohomology I, Cambridge UniversityPress 1997.
[2] BerntToreJensen,XiupingSuandAlexanderZimmermann,Degenerations forderivedcategories,Jour-
nal of Pureand Applied Algebra 198 (2005) 281-295
[3] Serge Lang, Algebra,Addison-Wesley 1993.
[4] Klaus W. Roggenkamp, An extension of the Noether-Deuring theorem, Proceedings of the American
Mathematical Society 31 (1972) 423–426.
[5] Jean-Louis Verdier,Des cat´egories d´eriv´ees des cat´egories ab´eliennes,Ast´erisque 239 (1996).
Universit´e de Picardie,
D´epartement de Math´ematiques et LAMFA (UMR 7352 du CNRS),
33 rue St Leu,
F-80039 Amiens Cedex 1,
France
E-mail address: [email protected]