A GROTHENDIECK TOPOLOGY ON A
SUBCATEGORY OF OPPOSITE CATEGORY OF NONCOMMUTATIVE ALGEBRASYOSHIFUMI TSUCHIMOTO
DEPARTMENT OF MATHEMATICS,
FACULTY OF SCIENCE,
KOCHI UNIVERSITY,
KOCHI 780, JAPAN.
EMAIL:DOCKY@@KDRS1.MATH.KOCHI-U.AC.JP

Contents
1. Introduction
2. Non commutative polynomial algebras
3. Hochschild cohomology
4. Free products
5. faithfully flatness
6. Bi-flat epimorphisms
7. Geometry of the set of left ideals
8. F -étale map
9. The category C
References

1
4
5
6
7
8
13
16
18
22

1. Introduction
The purpose of this paper is to show an existence of a Grothendiecktopology on a subcategory of the opposite category of non commutative
algebras. To define a Grothendieck topology is to define a way to“glue” objects, which will make it possible to define a non-commutative
geometric object which might be called non commutative scheme.
The term “non commutative scheme” already appears in [5] in aslight different context, in which a non commutative scheme is defined
as a ringed space. Our categorical approach provides another look at
non commutative algebraic geometry. It in particular requires (andtherefore offers some motivation for) a study on free products. In
either way, non commutative algebraic geometry seems to play thesame role in non commutative geometry (developed for example by1

2

YOSHIFUMI TSUCHIMOTO

Conne ([2][3])) as algebraic geometry plays in commutative geometry.
It will provide for example singularity theories, homological methods,
and also some concrete examples of non commutative manifolds.
Our point of view in this paper is as follows.
(1) We basically want to regard the opposite of the category (algebra) of (unital associative) algebras as a geometric object.
(2) We define a class of homomorphisms which we call “bi-flat epimorphisms” They play a role which localizations play in thecommutative case.
(3) We notice that a “base extension” of a good homomorphism
may be bad (Counter example 6.1). Therefore we propose wepass to a subcategory C and consider only those “continuousmaps” there. (Note that this does not mean we give up considering “discontinuous maps”. See below.) Then we show that C
has a Grothendieck topology (Main theorem 9.7).
(4) We show that certain class of homomorphisms (which we callbi-faithful “flaky” homomorphisms in this paper) have descent
property. This implies that the set of “affine object valued
maps”, continuous or not, form a sheaf on the site C (Maintheorem 9.8) .
We may make an analogy between this situation and a definition of
measurable maps on a topological manifold. Although “gluing maps”
should be continuous, we may consider any measurable maps on amanifold.
To make the idea more rigorous, let us begin with a review of thedefinition of Grothendieck topology. It is a generalization of the notionof “coverings” in the category of topological spaces.
Definition 1.1. (We essentially repeat the words used in [7]) We saya Grothendieck topology on a category C is given if we are given a set
Cov of families {Ui → U; i ∈ I} of maps in C (called coverings) which
satisfies the following axioms(GT 1) If φ is an isomorphism then {φ} ∈ Cov.
(GT 2) If {Ui → U} ∈ Cov and {Vij → Ui } ∈ Cov for each i then thefamily {Vij } obtained by composition is in Cov.
(GT 3) If {Ui → U} ∈ Cov and V → U is arbitrary morphism of C,
then the fiber product Ui ×U V exists and {Ui ×U V → V } ∈ Cov.
A category equipped with a Grothendieck topology is called a site.
For example, the category (Top) of topological spaces with “opencoverings” in the usual sense forms a site. A basic idea of non commutative geometry is to regard each non commutative algebra A as a

A GROTHENDIECK TOPOLOGY

3“function ring” over a geometric object Geo(A). A category of all suchobjects would be the opposite (algebra)opp of the category of algebras.
For the reason (2) above we pass to a subcategory C of (algebra)opp .
Our first main result then states
Theorem 9.7 A subcategory C of (non commutative algebras)opp
has a Grothendieck topology.
This result makes us possible to “glue” certain functors. It is explained in terms of sheaves on a site.
Definition 1.2. (We follow [7] again.) A sheaf of sets on a site is acontravariant functor C → (Sets) such that for any covering U → X ofan object X in the category C we have an exact sequence of sets
F (X) → F (U)→→ F (U ×X U).
This mechanism may be regarded as a generalization of gluing lemma
for continuous maps on a topological space: Continuous maps definedon a open covering give rise to a continuous map defined on the original
space if they coincide on intersections. We associate each object Geo(A)of C with “Geo(A)-valued maps” functor
F (Geo(B)) = Homalgebra (A; B).
Our second main result states
Theorem 9.8 For any algebra A, “Geo(A)-valued maps” form asheaf on the site C.
Thus we may glue these “affine objects” together as a sheaf to makea “non commutative scheme”.
Technically speaking, there are basically two points we emphasize inthis paper. One is to employ “bi-flat epimorphism” (Definition 6.1) asa non-commutative version of “localization”. (It is worthwhile to mention that “left-flat epimorphisms” play an important role in [5] and isrefered to as “perfect homomorphism” there.) The other is to employ
free product (or its mollified version (Definition 9.2)), rather than tensor product (employed for example in [9]), as an object corresponding
to a product in the geometric sense.
As the result of this paper may be regarded as a step to define noncommutative algebraic geometry, the author attempted to keep thispaper self contained.
Most part of the contents of this paper is obtained when the author took part in the program “operator algebras and its applications”
held in Fields Institute. The author wishes to express his feeling of
thanks to all the participants and the staffs there. They inspired him,
showed him good references, and prepared him a good room for study.

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YOSHIFUMI TSUCHIMOTO

He also thanks Masaki Maruyama, Akira Ishii and Kōta Yoshioka for
discussions on algebraic geometry.
2. Non commutative polynomial algebras
All the rings and homomorphisms which appear in this paper are assumed to be unital.
We fix a ground ring R (not necessarily commutative). It does not seem
there exist a standard notation for non commutative polynomial algebras. We employ in this paper the following notation
Definition 2.1. Let A be a R-algebra. Then the algebra generated
freely by A and indeterminates {Xλ }λ∈Λ , modulo conditions
cXλ = Xλ c (for all c ∈ R, λ ∈ Λ)
is called “the polynomial algebra over A with coefficient ring R andindeterminates {Xλ }” and is denoted as
A[R {Xλ ; λ ∈ Λ}].
As in the commutative case, there is a natural grading on the polynomial algebra.
M
M
A[R {Xλ ; λ ∈ Λ}] =(A[R {Xλ ; λ ∈ Λ}])d ={homogeneous polynomials of degree d}.
dd

We have the following expression for a polynomial algebra with one
indeterminate.
Lemma 2.1. There is an isomorphism
A[ X] ∼
=
R

∞
M
i=0

∞
M
(A ⊗R A ⊗R A ⊗R . . . ⊗R A) =(A⊗R (i+1) )i=0

as graded algebras, where the multiplication of the right hand side isdefined by(a0 ⊗a1 ⊗a2 ⊗. . .⊗ai )×(b0 ⊗b1 ⊗b2 ⊗. . .⊗bj ) = a0 ⊗a1 ⊗a2 ⊗. . .⊗(ai ×b0 )⊗b1 ⊗b2 ⊗. . .⊗bj
Proof. In fact, the isomorphism is obtained by replacing the indeterminate X by tensor sign “⊗”. Namely,
a0 Xa1 X . . . Xan 7→ a0 ⊗ a1 ⊗ . . . ⊗ an .
We use the following lemma, which guarantees us an existence of
certain odd derivations.

A GROTHENDIECK TOPOLOGY

5

Lemma 2.2. Let F be an odd element of P = A[R X]. That means,
M
F ∈
Pn .
n:odd

Then there exists a unique odd derivation δF on P such thatδF (X) = F,δF (a) = 0

for all a ∈ A.

(Recall that a odd derivation on P is an additive map δ such that thecondition
δ(F G) = δ(F )G + (−1)k F δ(G)holds for any F ∈ Pn , G ∈ P . )3. Hochschild cohomology
We recall here briefly a definition of the Hochschild cohomology. Ourdefinition coincides with the one given in [8], and is slightly different
to that given in [1]. The two definitions coincide if the ground ring R
is a commutative field, which is one of the most important cases. Let
I • (A) be a complex defined as
I n (A) = (A[R X])n+1(n ≥ 0)with the odd differential δ obtained as the restriction of the differentialδX (δF with F = X) given in Lemma 2.2. Then we define
Definition 3.1. The Hochschild cohomology of a R-algebra A withcoefficient in a A-bimodule M is the Z(R)(= the center of R)-module
H n (A; M) = H n (Hom•A- bimod (I • (A), M)).
Note that our complex I • here coincides with the complex β(A, A)
given in [8]. We have the following expressions for the zeroth and first
Hochschild cohomology groups.
H 0 (A; M) ∼= M A = {m ∈ M; am = ma for all a ∈ A}
H 1 (A; M) ∼= Der(A; M)/Inn(A; M) ={A → M; R-derivation}{A → M; Inner R-derivation}

We recall another notion which we will use later. The “forgetful” functor
 : (A- bimod) → (R- bimod)
has a left adjoint F defined by
F (M) = A ⊗R M ⊗R A,

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YOSHIFUMI TSUCHIMOTO

So we have a resolvent pair in the sense of [8], denoted R(A- bimod, R- bimod).
The Hochschild cohomology groups may be expressed as relative extension groups as follows.
H i(A; M) = ExtiR(A- bimod,R- bimod) (A; M)
See [8] for details.
4. Free products
The free product, or amalgamation, plays the role of the co-fiberproduct in the category of non commutative algebras. We remind thereader that in this paper we always assume our rings and morphisms
are all unital. Accordingly, “free products” in this paper means unital
ones. We recall the definition for the purpose of later references.
Definition 4.1. Let B, C be A-algebras. Then the free product B ∗A C
of B and C over A is the A-algebra together with two “canonical” Ahomomorphisms
ι1 : B → B ∗A C
ι2 : C → B ∗A C
such that, for any other A-algebra D with two A-homomorphisms
ϕ1 : B → D
ϕ2 : C → D,
there exists a unique A-homomorphism, denoted (ϕ1 , ϕ2 ) from B ∗A C
to D such that we have
(ϕ1 , ϕ2 ) ◦ ιi = ϕi(i = 1, 2).

A good deal of problems occur when dealing with free products since
the free products, unlike tensor products, do not behave bilinearly withrespect to B, C. To improve this, we will introduce a mollified freeproduct for special class of algebras (Definition 9.2). We end this section by stating the following easy lemma relating tensor product andfree product in a case when B = C.
Lemma 4.1. A module homomorphism
B ⊗A B → B ∗A B
mapping b1 ⊗ b2 to ι1 (b1 )ι2 (b2 ) is injective, and the image is a direct
summand of B ∗A B.

A GROTHENDIECK TOPOLOGY

7

Proof. Put(id,id)I∆ = ker(B ∗A B → B),
M∆ = ker(B ⊗A B

multiplication→

B).

Then we have(1) (B ∗A B)/I∆ ∼= B.
= B ⊗A B/M∆ ∼2(2) Both A-bimodules I∆ /I∆ and M∆ serves as the non-commutativeKähler differentials of B over A. (Hence they are isomorphic.)Thus we conclude
2 ∼(B ∗A B)/I∆= B ⊗A B.

5. faithfully flatness
In this section we review basic facts about faithfully flatness. All
results may be proved in analogous ways to the commutative case. We
first recall the definition of flatness.
Definition 5.1. Let A be an R-algebra. An A-algebra C is called leftflat (respectively, right-flat) over A if the functor C⊗A (respectively,⊗A C) is an exact functor. It is called bi-flat if it is left- and right-flat.
Lemma 5.1. Let C be a left-flat A-algebra. Then the followings areequivalent.
(1) For any non-zero left A-module M, we have
C ⊗A M 6= 0.
(2) For any sequence
0→L→M →N →0of left A-module, if a sequence
0 → C ⊗A L → C ⊗A M → C ⊗A N → 0
obtained by tensoring C to the original sequence from the left isexact, then the original sequence is also exact.
(3) For any A-module M, A module homomorphism
M ∋ m 7→ 1 ⊗ m ∈ C ⊗A M
is injective.
(4) For any left ideal I of A we have
B ⊗A (A/I) 6= 0

8YOSHIFUMI TSUCHIMOTO

Definition 5.2. If one (hence all) of the conditions in Lemma 5.1
holds, we call C an algebra left-faithfully flat over A, and the structure
homomorphism A → C is called left-faithfully flat homomorphism.
Right-faithfully flatness is defined in a similar manner. An algebra
or a homomorphism is called bi-faithfully flat if it is left- and rightfaithfully flat.
Lemma 5.2. Left faithfully flat morphisms are injective.
6. Bi-flat epimorphisms
In this section we introduce bi-flat epimorphisms, which will be acentral notion in this paper. This notion is essentially known to specialists. In fact, in [5] one sees that left-flat epimorphisms play animportant role in the theory developed there. Here we examine their
behaviour with respect to free product.
We use the term “epimorphism” in the category theoretical sense.
We first note the following lemma.
Lemma 6.1. For a homomorphism ϕ : A → B, the followings areequivalent(1) We have an isomorphism B ⊗A B ∼= B of B-bimodules via the
product map b1 ⊗ b2 7→ b1 b2 .
(2) We have an isomorphism B ∗A B ∼= B of algebras via the codiagonal map (id, id)(3) ϕ is a (category theoretically) epimorphism. That means, for
any A-algebra C, an A-homomorphism from B to C, if there
exists one, is unique.
Proof. That (2) implies (1) follows from Lemma 4.1. To prove theconverse implication, we note that canonical maps ι1 , ι2 : B → B ∗A B
factors through B ⊗A B → B ∗A B, so that we have in fact ι1 = ι2 .
The equivalence of (2) and (3) follows from the universality of freeproducts.
Definition 6.1. If one (hence all) of the conditions in Lemma 6.1 issatisfied, the homomorphism ϕ is called an epimorphism. It is called
a bi-flat epimorphism (BE) if it is an epimorphism and is bi-flat. If
the homomorphism ϕ referring to is clear, we simply say, by abuse oflanguage, that B is BE over A.
Note that epimorphisms need not be surjective maps in the set theoretic sense. We distinguish in this paper the term “epimorphism” fromthe term “surjective map”. The former is used in the categorical sense
as above while the latter is used in the set-theoretical sense.

A GROTHENDIECK TOPOLOGY

9

Bi-flat epimorphisms play similar role in a theory of non commutative
algebras as localization does in the theory of commutative algebras.
Examples .
(1) Identity maps are bi-flat epimorphisms.
(2) Every localization of a commutative algebras in the usual sense
is a bi-flat epimorphism.
Definition 6.2. Let A be an algebra and S be a subset of A. Thenthe localization AS of A with respect to S is defined as follows.
AS = A[R {Xs ; s ∈ S}]/Xs s = sXs = 1
We also refer to the natural map A → AS as localization.
Remark 6.1. A localization AS by a set S = {s} consisting of a single
element is a “base extension” of a flat homomorphism
R[R X] → R[R X, 1/X]
by a homomorphism
R[R X] → A :That means,

X 7→ s.

AS ∼= A ∗R[R X] R[R X, 1/X].

Lemma 6.2. Any localization A → AS of an algebra is an epimorphism.
Proof. Any element in AS ⊗A AS is a linear combination of elements
which look like−1t ⊗ a0 s−1
1 a1 . . . sn an ,where t ∈ AS , ai ∈ A, si ∈ S. But we have a computation−1t ⊗ a0 s−1
1 a1 . . . sn an−1=ta0 ⊗ s−1
1 a1 . . . sn an−1−1=ta0 s−1
1 s1 ⊗ s1 a1 . . . sn an−1=ta0 s−1
1 ⊗ a1 . . . sn an=...
−1=ta0 s−1
1 a1 . . . sn an ⊗ 1,

which shows that a splitting ⊗1 : AS → AS ⊗A AS of the multiplication
map AS ⊗A AS → AS is surjective.

We note that localizations of non commutative algebras need not be
bi-flat, as the following counter example indicates.

10

YOSHIFUMI TSUCHIMOTOCounter Example 6.1. Let
A = R[R x, t]/tx = (x + 1)t,S = {x}.

Then A → AS = Ax is not bi-flat. In fact, if we let
M
M=
Cen
n∈N

N=

M

Cen

n∈N\{0}be direct sum of one dimensional vector spaces and define action of A
on M obtained byx(en ) = −nen ,t(en ) = en+1 ,then N is stable under this action. But we easily see the followings.
(1) Ax ⊗A M = 0,
(2) Ax ⊗A N = N 6= 0.
This shows that Ax is not left-flat. In particular, in view of Remark
6.1 we see that a “base extension” of a flat homomorphism may not beflat.
However, we have the following result of M. Kashiwara. ([6])
Lemma 6.3. If a subset S of an algebra A satisfies the following properties, then the localization AS is left-flat over A.
(1) The set S is closed under multiplication and contains 1.
(2) For any elements a ∈ A and s ∈ S, there exist elements b ∈ A,t ∈ S such that we haveta = bs.
(3) If furthermore in the above situation a is an element of S, thenwe may choose b, t as above with b ∈ S(4) If a ∈ A, s ∈ S satisfies as = 0, then there exists an elementt ∈ S such that ta = 0.
We have an analogous sufficient condition for right flatness.
Proof. For any A-module M, we have under the conditions of thelemma, an description of localization
AS = S −1 M = S × M/ ∼,where the equivalence relation ∼ is defined as follows(a, m) ∼ (b, n)⇔∃s, t ∈ S; sb = ta, sn = tm.
A GROTHENDIECK TOPOLOGY

11

The author does not know if compositions of localizations are always
localization again. But we have the following lemma.
Lemma 6.4. A composition of two bi-flat epimorphisms A → B, B →C is also a bi-flat epimorphism.
The proof is obvious.

Thus the class of all algebras with bi-flat epimorphisms forms a subcategory of the category of algebras. We will later see that free products
serves as coproducts in this category too.
Lemma 6.5. Let A → B be an epimorphism. Then for any B-leftmodule M1 and right B-module M2 , we have
M2 ⊗A M1 ∼= M2 ⊗B M1
Proof.
M2 ⊗A M1 ∼= M2 ⊗B M1= M2 ⊗B (B⊗A B)⊗B M1 ∼= (M2 ⊗B B)⊗A (B⊗B M1 ) ∼Corollary 6.6. Let ψ : B → C be a homomorphism, and ϕ : A → B
be an epimorphism. Assume that C is left (respectively, right)-flat overA. Then C is also left (respectively, right)-flat over B. In particular,
if both ϕ and ψ ◦ ϕ are bi-flat epimorphisms, then so is ψ.
Proof. The first claim follows easily from the Lemma. Flatness ofψ follows from the first claim. That ψ is an epimorphism is verified asfollows.
C ⊗B C ∼= C.
= C ⊗A C ∼
Lemma 6.7. Let A be an algebra, B, C two A-algebras which are BE
over A. Then the free product B ∗A C is also BE over A. Thus it isalso BE over B and over C.
Proof. Using Lemma 6.1, we obtain an isomorphism(B ∗A C) ∗A (B ∗A C) ∼= B ∗A C.
= (B ∗A B) ∗A (C ∗A C) ∼Thus we see that B ∗A C is also BE over A. To prove the flatness, weuse an explicit description for the free product using tensor products.
Namely, we have the following lemma, which tells us that the freeproduct in our case is actually an inductive limit of bi-flat modules,
and hence is also flat. The last line of the statement is deduced fromthe rest of the lemma by using Corollary 6.6.


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YOSHIFUMI TSUCHIMOTO

Lemma 6.8. Under the assumption of Lemma 6.7, we have an inductive limit expression of the free product(B ⊗A C)⊗A n .
B ∗A C ∼= lim−→
The connecting map for the inductive limit in the right hand side isgiven by
M ⊗A n ∋ m 7→ m ⊗ u ∈ M ⊗A n ⊗A M,
where we put M = (B ⊗A C), and the element u of M is given by
u = 1B ⊗ 1 C .
Proof. We note that for any two elements b1 , b2 in B, we haveb1 ⊗ 1C ⊗ b2 = b1 b2 ⊗ 1C ⊗ 1B ,where both hand sides are considered as elements of B ⊗A C ⊗A B. To
see this, we observe that we have a well-defined map
B ⊗A B → B ⊗A C ⊗A B
defined byb1 ⊗ b2 7→ b1 ⊗ 1C ⊗ b2 ,and recall that B ⊗A B ∼= B. Using this identity (and an analogous
one with B and C interchanged), we observe firstly
x ⊗ u = u ⊗ x in M ⊗A M
for any x in M. Then secondly we see that the canonical identificationmap
M ⊗A m ⊗A M ⊗A n → M ⊗A (m+n)
gives a bilinear map compatible with the inductive system, which gives
rise to a bilinear product(lim M ⊗A n ) ⊗A (lim M ⊗A n ) → lim M ⊗A n
−→
−→
−→
n

n

n

on the inductive limit. Thirdly we may verify that this product isassociative, with the unit element u. We finally note that this algebra
admits an obvious canonical homomorphism from both B and C, andthat it has the universal property of the free product.
Proposition 6.9. The free product B ∗A C serves as the coproduct of
B, C in the category of non commutative R-algebras with bi-flat epimorphisms.

A GROTHENDIECK TOPOLOGY

13

Proof. We have already proved in Lemma 6.7 that all maps appearing in the following commutative diagram are bi-flat epimorphisms.
A −−−→
B




y
y .

C −−−→ B ∗A C
Now let D be an A-algebra which is BE over A with A-homomorphisms
α : B → D and β : C → D. (The homomorphism α, β are automatically bi-flat epimorphisms, in view of Corollary 6.6.) We need to provethat the map (α, β) is a bi-flat epimorphism. But this also follows from
Lemma 6.6.

7. Geometry of the set of left ideals
In this section we observe how the set of left ideals of an algebra
behaves with respect to bi-flat epimorphisms.
We employ the following notations(7.1)

Spl(A) = {left ideal of A},(7.2)

Spr(A) = {right ideal of A}.

Furthermore, for any homomorphism ϕ : A → B between algebras, wedefine “the associate maps”
Spl(ϕ) : Spl(B) → Spl(A),Spr(ϕ) : Spr(B) → Spr(A)defined bySpl(ϕ)(J) = ϕ−1 (J),Spr(ϕ)(J) = ϕ−1 (J).

It is easy to see the following.
Lemma 7.1. A left ideal I of A is an image of Spl(ϕ) if and only if
an identity(7.3)

ϕ−1 (B  ϕ(I)) = I.

holds.
We may use these spaces of ideals to see whether a homomorphism
is faithfully flat. Namely, the following lemma holds.
Lemma 7.2. Let ϕ : A → B be a left-flat homomorphism. Then thefollowings are equivalent.
(1) ϕ is left-faithfully flat.
(2) The associate map Spl(ϕ) : Spl(B) → Spl(A) is surjective.
(3) The associate map Spl(ϕ) : Spl(B) → Spl(A) maps surjectively
on the set of maximal ideals in A.

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YOSHIFUMI TSUCHIMOTO

Proof. This follows easily from Lemma 5.1.

Lemma 7.3. If a homomorphism ϕ : A → B is a bi-flat epimorphism,
then the induced map Spl(ϕ) is injective. To be more precise, we havean equality
J = B  ϕ(ϕ−1 (J))
for any element J of Spl(B).
Proof. We have an exact sequence
0 → A/ϕ−1 (J) → B/J.
Tensoring B over A from the left of this sequence, we obtain the following exact sequence.
0 → B/Bϕ(ϕ−1 (J)) → B/J.
(Note that B ⊗A M ∼= M for any left B-module M)

Our next task is to investigate behaviors of Spl(ϕ). The result will
be summarized in Proposition 7.6.
Lemma 7.4. Let ϕ : A → B, ψ : A → C be bi-flat epimorphisms.
Then we haveSpl(B) ∩ Spl(C) = Spl(B ∗A C),
where both hand sides are considered as subsets of Spl(A).
Proof. Since we already know in Lemma 6.9, that the all maps
appearing in a commutative diagram
A −−−→


y

B


y

.

C −−−→ B ∗A C
are bi-flat epimorphisms, we deduce we have an inclusion
Spl(B ∗A C) ⊂ Spl(B) ∩ Spl(C).
To prove the converse inclusion, we take I ∈ Spl(A) which belongs toSpl(B) ∩ Spl(C). That means, we have equalities
ϕ−1 (Bϕ(I)) = I = ψ −1 (Cψ(I)),
or, equivalently, the following sequences are exact.
(7.4)

0 →A/I → B/Bϕ(I)(7.5)

0 →A/I → C/Cψ(I)A GROTHENDIECK TOPOLOGY

15

We derive from the exact sequence 7.4, using the fact that C is bi-flat
over A, another exact sequence
0 → C ⊗A (A/I) → C ⊗A (B/Bϕ(I)).
But the algebra C ⊗A (A/I) ∼= C/Cψ(I) contains A/I inside (7.5), sothat we have
0 → A/I → C ⊗A (B/Bϕ(I))(∼= C ⊗A B ⊗A (A/I)) :exact.
Tensoring B to the above sequence and continuing the above argument,
we have
0 → A/I → B ⊗A C ⊗A B ⊗A (A/I) :exact.
Continuing the argument again and again we finally conclude that asequence
0 → A/I → lim(C ⊗A B)⊗A n ⊗A A/I−→
is exact. Now the lemma follows from the lemma 6.8.
Lemma 7.5. Let ϕ : A → B be a bi-flat epimorphism. If the associate
map Spl(ϕ) : Spl(B) → Spl(A) is surjective (hence bijective according
to Lemma 7.3), then ϕ is an isomorphism.
Proof. Since {0A } ∈ Spl(A) is in the image of Spl(ϕ), we easily seethat the homomorphism ϕ is injective. On the other hand, we have
B ⊗A (B/ϕ(A)) ∼= B/Bϕ(A) = 0,
which implies B/ϕ(A) = 0, since ϕ is faithfully flat (Lemma 5.1 (4) ,equation (7.3)). Thus the homomorphism ϕ is surjective.
Using Lemma 7.2, we may rephrase the above lemma as “a bi-flat
epimorphism is left-faithful if and only if it is an isomorphism”.
Proposition 7.6. A map{algebra which is BE over A }/ ∼=→{subset of Spl(A)}determined by
B 7→ Spl(B),where the right hand side is considered as a subset of Spl(A) on the
ground of the previous lemma, is injective. Furthermore, this injection
has the following properties.
(1) Free products correspond to intersections via this injection.
(2) “There exists an A-homomorphism B → C” ⇔ Spl(C) ⊂ Spl(B).
(Note that the A-homomorphism B → C in Claim (2) above is unique(Lemma 6.1)

16

YOSHIFUMI TSUCHIMOTO

Proof. Claim (1) is the content of Lemma 7.4. The “ =⇒ ” part ofClaim (2) is clear. To prove the converse of the implication, we apply
claim (1) to the algebra B ∗A C and see that the canonical homomorphism C → B ∗A C satisfies the assumption of the previous lemma
and is therefore an isomorphism. We thus obtain an A-homomorphism
B → C by using the canonical homomorphisms as follows.
B → B ∗A C ∼= C.
This completes the proof
8. F -étale map

Bi-flat epimorphisms satisfy some homological properties. To clarify
this, we generalize in this section the notion of formally étale morphisms
to the non commutative case.
Definition 8.1. Let A, B be two algebras, F a family of B-modules.
A homomorphism ϕ : A → B is called F -étale, if the associated map
for Hochschild cohomology
H i (ϕ) : H i(B; M) → H i (A; M)
gives an isomorphism for any member M of F and for any i.
The choice of appropriate F for defining “genuine” étale maps seems
to require some considerations. In this paper, we will employ the following important classes of modules as F .
(8.1)

Fa = {all B-bimodules}(8.2)

Fc = {M; am = ma for all m ∈ M, and for all a ∈ Z(B)}(Here we denote by Z(•) the set of central elements of the algebra.)
It follows readily from the definition that ϕ is Fa -étale if and only if
it is F -étale for any class F of B-modules. If the homomorphism ϕ isbi-flat, we have the following lemma.
Lemma 8.1. A bi-flat homomorphism ϕ : A → B is F -étale if andonly if the homomorphism
ExtiR(B- bimod,R- bimod) (B, N) → ExtiR(B- bimod,R- bimod) (B ⊗A B, N)
between relative extension groups induced by the multiplication map
B ⊗A B → B
is bijective for any element N ∈ F
The proof is standard and is left to the reader.

A GROTHENDIECK TOPOLOGY

17

Remark 8.1. If the algebras A, B are commutative and the homomorphism ϕ is formally étale in the usual sense, then ϕ is Fc -étale. In fact,
the class Fc in this case is the class of “usual B-modules”, that is Bbimodules for which the left multiplication and the right multiplication
coincide. Then we may use the well-known fact that “the diagonal” B
is a direct summand of the algebra B ⊗A B as an algebra.
Lemma 8.2. Bi-flat epimorphisms are F -étale for any F .
Proof. This is a direct consequence of Lemma 8.1.

Corollary 8.3. Let ϕ be a Fc -étale homomorphism. Then ϕ maps central elements to central elements. In particular, bi-flat epimorphisms
maps central elements to central elements.
Proof. We put i = 0, M = B in the definition of an F -étale map
and obtain an equation
B A = B B (= Z(B) (the center of B)).
On the other hand, ϕ obviously maps central elements of A to elementsin B A . This completes the proof
A motivation for defining F -étale homomorphisms comes from the(infinitesimal) deformation theory of non commutative algebras ([4]).
Let ǫ be a dual number (ǫ2 = 0) which commutes with everything.
Recall that an infinitesimal deformation of an algebra A over a ring R
is by definition given by introducing a R[ǫ]-algebra structure on a R[ǫ]bimodule A[ǫ] which reduces to the original algebra when we let ǫ = 0.
More explicitly speaking, the algebra structure is given by introducing
a multiplication law(8.3)

mǫ (f + ǫg, h + ǫk) = f h + ǫ(f k + gh + ϕ(f, g)),where ϕ : A × A → A is a biadditive homomorphism which satisfies
the following conditions.
ϕ(cf, g) = cϕ(f, g), ϕ(f c, g) = ϕ(f, cg), ϕ(f, gc) = ϕ(f, g)c (f, g ∈ A, c ∈ R)
f ϕ(g, h) − ϕ(f g, h) + ϕ(f, gh) − ϕ(f, g)h = 0 (f, g, h ∈ A) (cocycle condition)
We will refer to the deformed algebra with the multiplication law 8.3 as
Aϕ . We say that two such infinitesimal deformations given by cocycles
ϕ, ψ are equivalent if and only if there exists a R[ǫ]-isomorphism
Aϕ → Aψwhich reduces to the identity when we let ǫ = 0. The reader may easily
verify that the equivalence classes of infinitesimal deformations are the

18

YOSHIFUMI TSUCHIMOTO

same thing as extensions of A by A, and are thus parametrized by thecohomology group
H 2 (A; A).
We may also define deformation of a homomorphism as a homomorphism between deformed algebras which reduces to the homomorphism
between the original algebras.
An étale homomorphism behaves neatly with infinitesimal deformations, as the following lemma shows.
Lemma 8.4. Assume ι : A → B be a {B}-étale homomorphism. Thenfor all infinitesimal deformation ϕ of A, there exists an infinitesimal
deformation ψ of B and an infinitesimal deformation ι̃ of ι such thatthe following diagram commutes.
ι̃

Aϕ −−−→ Bψ




ǫ→0y
yǫ→0
ι

A −−−→ B
Furthermore, ι̃ is unique up to a conjugacy by an infinitesimal deformation id˜B of identity on B.
Proof. Existence is essentially ensured by an existence of a map
H (A; A) → H 2 (B; B) defined as
22

H (A; A)H 2 (A;ι)→2

H 2 (ι;B)

H (A; B)∼=H 2(B; B),

where H 2 (A; ι) and H 2 (ι; B) are additive maps associated to ι in theobvious way. We leave the rest to the reader.

9. The category C
Definition 9.1. A homomorphism ϕ : A → B is called flaky homomorphism if it is a direct sum of bi-flat epimorphisms. Namely, we
have a direct sum decomposition of B
B = ⊕ni=1 Bi
into finite components {Bi } such that each component ϕi of ϕ (withrespect to this decomposition) is a bi-flat epimorphism.
Lemma 9.1. Flaky homomorphisms are Fc -étale. (Hence they map
central elements to central elements.
Proof. An easy homological argument shows that a direct sum of
Fc -étale maps is also Fc -étale. The result now follows from this fact
and Lemma 8.2.


A GROTHENDIECK TOPOLOGY

19

Lemma 9.2. A composition of two flaky homomorphisms are also
flaky.
Proof. Since a direct sum decomposition of an algebra corresponds
to a central idempotent, the result easily follows from the previous
lemma.
The category C is defined as a subcategory of (Non commutative algebras)opp
as follows.
Object(C) = {R-algebras}
Morphism(C) = {opposite of flaky homomorphisms}
To indicate that an algebra A is regarded as an object of C, we employ
the notation Geo(A). A morphism Geo(B) → Geo(A) thus corresponds
to a flaky homomorphism A → B.
This category has fiber products. To see this, we first introduce“mollified free products”.
Definition 9.2. Let A be R-algebra, B,C two A-algebras. The mollified free product of B,C over A is defined as follows.
◦

B ∗A C = (B ∗A C)/h[Z(B), C], [B, Z(C)]i.
We note first the following lemma.
Lemma 9.3. If two homomorphism ϕ : A → B, ψ : A → C are bi-flat◦
epimorphisms, then the mollified free product B ∗A C coincides with thefree product B ∗A C.
Proof. This follows from the fact that B ∗A C is BE over B, C
(Lemma 6.7), and that bi-flat epimorphisms map central elements tocentral elements (Corollary 8.3).
Then we have the following result.
Lemma 9.4. The category C has fiber products. They are given bymollified free products. That is,◦

Geo(B) ×Geo(A) Geo(C) = Geo(B ∗A C).
Furthermore, if we have a decomposition
n
M
B=
Bi ,
C=

i=1
m
M
j=1

Cj

20

YOSHIFUMI TSUCHIMOTO

such that the homomorphisms ϕ, ψ decompose as a sum of bi-flat epimorphisms with respect to these, then we haveM◦
Bi ∗A Cj .
B ∗A C ∼=i,jProof. Let A → B, A → C be two flaky homomorphisms. Given
two flaky A-homomorphisms α : B → D, β : C → D, we have awell-defined homomorphism
B ∗A C → D,since we have a universality for free products. But since α, β are flaky,
images α(Z(B)), β(Z(C)) of the centers are contained in the center
Z(D) of D. We thus have a homomorphism◦

B ∗A C → D.
It remains to prove that homomorphisms◦

B → B ∗A C,◦

C → B ∗A C,◦

B ∗A C → D

are all flaky. We reduce this problem to Lemmas 6.7 and 9.3 by dividing
each algebras with central idempotents.

Corollary 9.5. If a homomorphism A → B is flaky, then a modulehomomorphism◦

B ⊗A B ∋ b1 ⊗ b2 7→ ι1 (b1 )ι2 (b2 ) ∈ B ∗A B
is injective. (Here homomorphisms ι1 , ι2 are natural ones from B to◦
B ∗A B.)
Proof. This is a consequence of the above Lemma and the Lemma
4.1.

Lemma 9.6. Let ϕ : A → B, ψ : A → C be two flaky homomorphisms.
If ϕ is left-faithfully flat, then the “base extension”◦

C → B ∗A C
is also left-faithfully flat.
Proof. Using Lemma 9.4 we may assume that the homomorphism
ψ is a bi-flat epimorphism. Let
B=

n
M
i=1

Bi

A GROTHENDIECK TOPOLOGY21

be a direct sum decomposition with respect to which the homomorphism φ decomposes into a sum of bi-flat epimorphisms. Take a maximal ideal J of C. Since Spl(φ) is surjective, one may find ideals {Ji }ni=1
of Bi such that
(9.1)

φ−1 (

n
M

Ji )(= ∩i (φ−1 (Ji )) = ψ −1 (J)i=1holds. Using the flatness of ψ, we deduce from the above equation thefollowing.
Cψ(∩i (φ−1 (Ji ))) = Cψ(ψ −1 (J)) = J(Lemma 7.3)
Using the maximality of J, we conclude that there exist i0 such that
(9.2)

J = Cψ(φ−1 (Ji0 ))holds. Equations 9.1, 9.2 implies in particular, an equality
ψ −1 (J) = φ−1 (Ji0 ),which shows that J is an element of the set Spl(Bi0 )∩Spl(C)(⊂ Spl(A)).

We now state the main theorem of this paper.
Theorem 9.7. The category C admits a Grothendieck topology. A
covering map is given by a finite set of homomorphisms{Geo(ϕi ) : Geo(Bi ) → Geo(A); i = 1, . . . , n}such that,(1) Each homomorphism ϕi is flaky,(2) The homomorphism A → ⊕Bi obtained from ϕi ’s is bi-faithful.
Proof. (GT1) of Definition 1.1 is obvious. (GT2) results from
Lemma 9.2 and a fact that a composition of faithful homomorphisms
are faithful. (GT3) is a result of Lemma 9.6.
This is an easy consequence of Lemma 9.4.
Theorem 9.8. For any algebra S over the ground ring R, the functor
A 7→ HomR-algebra (A, S)
is a sheaf over the site C. In other words, the sequence
ι1◦

A → B→→ B ∗A B
ι2

is exact. (That is, A is the difference kernel of ι1 , ι2 .)22

YOSHIFUMI TSUCHIMOTO

Proof. The proof is essentially identical with the commutative case.
(We already know in Lemma 5.2 that the homomorphism A → B is
injective.) For the proof of the exactness of the above sequence, it isenough to prove the exactness of(9.3)

A → B→→ B ⊗A B,◦

since B ⊗A B is injectively contained in B ∗A B (Corollary 9.5). To
prove the exactness of 9.3, we may tensorize the sequence with B over
A and consider the exactness of
B → B ⊗A B →→ B ⊗A B ⊗A B.
It is then easy to see that, as in commutative case, this sequence has
actually splitting, and hence is exact (The reader may consult for example Proposition 2.18 of [10] for the precise arguments in commutativecase).

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1956.
[2] A. Conne, Noncommuatative differential geometry, pub.Math. I.H.E.S. 62(1985), 257–360.
[3], Noncommutative geometry, Academic Press, 1994.
[4] M. Gerstenhaber, On the deformation of rings and algebras, Annals of Math.
79, No.1 (1964), 59–103.
[5] J. S. Golan, structure sheaves over a noncommutative ring (lecture notes inpure and applied mathematics 56), Marcel Dekker, inc., 1980.
[6] S. Iitaka, Algebraic geometry, Iwanami (in Japanese), 1970.
[7] D. Kunutson, Algebraic spaces (lecture notes in mathematics 203), Springer
Verlag, 1971.
[8] S. S. MacLane, Homology, Springer Verlag, 1967.
[9] Yuri I. Manin, Topics in non commutative geometry, Princeton University
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[10] J. S. Milne, Étale cohomology, Princeton University Press, 1980.