CHANGE OF RINGS IN DEFORMATION THEORY OF MODULES

ISBN 82–553–1373–7 No. 11 ISSN 0806–2439 April 2003

CHANGE OF RINGS IN

DEFORMATION THEORY OF MODULES

RUNAR ILE

Abstract. Given a B-module M and any presentation B = A/J, the ob- struction theory of M as B-module is determined by the usual obstruction class o_Afor deformingM asA-moduleanda new obstruction class o_J. These two classes give the tool for constructing two obstruction maps which depend on each other and which characterise the hull of the deformation functor.

We obtain relations between the obstruction classes by studying a change of rings spectral sequence and by representing certain classes as elements in the Yoneda complex. Calculation of the deformation functor ofM asB-module, including the (generalised) Massey products, is thus possible within anyA-free 2-presentation ofM.

1. Introduction

In this article we study the following functor of infinitesimal deformations.

Definition 1. LetAbe a (commutative) flatO-algebra whereOis a local complete Noetherian ring with k as residue field. Let A = A⊗Ok and let M be an A- module. DefineArt_O as the category of local commutative ArtinianO-algebrasR with residue field k such that the composition O → R →k equals the canonical map from O to its residue field. Morphisms are maps of local O-algebras. The deformation functor ofM is a covariant functor

Def^A_M:Art_O−→Sets

where Def^A_M(R) is the set of equivalence classes of deformations of M to R. A deformation (or flat lifting) ofM to Ris anAR:=A⊗OR-moduleMR, flat asR- module together with anAR-linear mapπ:MR→M withπ⊗Rk:MR⊗Rk−^'→M.

Two deformations are equivalent if they are isomorphic aboveM. Maps are induced by tensorisation.

Remark 1. One natural choice for O is as the hull of the deformation functor of A as k-algebra, with A the formally versal formal family which in particular is a flatO-algebra with A⊗Ok=A. In the caseO=kwe haveA=Aand the article might be somewhat easier to read with this assumption.

More generally, letF :Art_O→Setsbe a covariant functor withF(k) a one ele- ment set. M. Schlessinger [18] formulated a sufficient and necessary set of criteria for the existence of a complete local ringH, called a (pro-representing)hull, and a for- mally versal formal family{Mn}^∞_n=1, a projective system withMn∈F(H/nmⁿ⁻¹

H ) where n = m²

H+m_O such that the induced map ρ : Hom^cont._O−alg./k(H,−) → F is formally smooth and an isomorphism on the relative Zariski tangent space. F is

2000Mathematics Subject Classification. Primary 13D10, 14B10; Secondary 13D07.

Key words and phrases. Obstruction, Massey product, spectral sequence, mixed characteristic.

Acknowledgement. The author is indebted to Prof. O. A. Laudal for sharing essential insight and for valuable comments on the manuscript.

This article is based on parts of the author’s 2001 Dr. Phil. Thesis at the Dept. of Mathematics, University of Oslo.

1

called pro-representable if ρ is an isomorphism. Most deformation functors have hulls, if the relative Zariski tangent space ofF is finite dimensional.

Example 1. Letρ: Π→Gln(k) be a continuous representation of a profinite group Π satisfying ap-finiteness condition wherepis the characteristic of the finite fieldk.

Define the deformation functor Def_ρ:Art_O→Setsas equivalence classes of liftings e

ρ: Π→Gln(R) of ρ. HereO is a “coefficient ring” with residue field k, typically O = W(k), the Witt ring of k. If Π is the Galois group of a number field, one obtains what B. Mazur has termed deformation theory of Galois representations, cf. [16]. If A=O[[Π]] andM =kⁿ with theA=k[[Π]]-module structure induced from ρ, this deformation functor is canonically isomorphic to the one in Definition 1, (we have to allow for non-commutative algebrasA). By applying the Schlessinger criteria, Mazur proved that Def_ρ in general has a hull, and is pro-representable if ρis absolutely irreducible, see [15].

Example 2. IfO=kis an algebraically closed field of any characteristic andMis a finitely generatedA-module (andAanalgebraic k-algebra, i.e. the Henselisation of a finitely generatedk-algebra at a maximal ideal), locally free on the complement of the closed point, there exist algebraic versal deformations ofM[21]. A. Ishii [11]

has constructed a filtration of the versal base spaces (with reduced structure) of all reflexive modules (including the decomposable ones) over rational surface singu- larities and has determined the local deformation relation of the reflexive modules over the rational double points. These mini-versal base spaces are far from being (locally) “coarse” moduli spaces. In particular the deformation functors of these re- flexive modules restricted to Artinian rings are not pro-representable. Indeed there is only a finite set of isomorphism classes of reflexive modules of fixed rank over a quotient surface singularity, yet the singular versal base has complicated geometry.

Schlessinger did not provide any effective construction of the hull. The only known general method to computeH givenM, is via a naturalobstruction class.

Definition 2. A small lifting situation is a surjective map π : R → S in Art_O where kerπ is contained in the socle ofR, i.e. m_R·kerπ = 0, and a deformation MS ofM toS.

The obstruction class is then an element o_∗= o_∗(π, MS)∈H²⊗kerπwhere H²is the second cohomology group of the objectM. IfF = Def^A_Mthen oA= oA(π, MS) and H²= Ext²_A(M,M). The obstruction class is natural with respect to morphisms of the lifting situation. There exists a lifting ofMS toR(or a prolongation of the deformation MS to the “thicker” Artinian neighbourhood SpecR) if and only if this obstruction class is zero. The obstruction class has been constructed for many deformation functors, e.g. [9, 10, 13], for axiomatic approaches see [1, 5, 7].

If F has a hull, there is a universal elementM1∈F(H1) where H1 =k[H^1∗] = k⊕H^1∗ and H¹ is the relative Zariski tangent space; H¹ ∼=F(k[ε]) (naturally a k-vector space). In the caseF = Def^A_M, H¹= Ext¹_A(M,M) andM1 is given by the universal extension

M1: 0−→M⊗kExt¹_A(M,M)^∗−→M1 π1

−→M−→0.

The construction of H then proceeds through successive “prolongations” ofM1 to thicker ArtinianO-algebras through small lifting situations, at each step calculating the obstruction. If this is done correctly, one obtains power series inT¹, contained in m²

T¹+m_O, one (possibly “0”) for each generator in T²of the relative cotangent space, where Tⁱ is the completion of the free O-algebra which has Hⁱ as relative Zariski tangent space fori= 1,2 . This defines anobstruction map o^∗:T²→T¹, which is naturally compatible with the obstruction class o_∗(see Definition 4), such that H =T¹⊗ˆT²O.

The existence of an obstruction map is provided by O. A. Laudal rather ab- stractly for a deformation functor of a small category of algebras in [13, Thm.

4.2.4] (see also V. P. Palamodov [17, Thm. 5.6] (without proof) for compact ana- lytic manifolds) and for Def^A_M and O=k with explicit Yoneda-representations of the generalised Massey products in [14]. For an axiomatic existence theorem, see [7, Thm. 2.3.10], it shows that the existence of a natural obstruction class together with a natural action of the tangent space on the set of liftings in a small lifting situation, implies the existence of an obstruction map for F. Once we have an obstruction map, the general Krull dimension estimate

dimkH¹≥dimKrullH−dimKrullO ≥dimkH¹−dimkH² follows. (See also [12].)

In practice it is difficult to give non-trivial results about the obstruction map, the usual application is some variation of H² = 0 ⇒ H is smooth. In fact, very few classes of examples of deformation functors have been given for which anything beyond the general Krull dimension estimate is known. By studying modules, one can at least calculate examples as there exists an effective obstruction algorithm.

In the present paper we provide a refinement of the obstruction map for modules which has both theoretical and computational consequences. For an application of these ideas, see [8].

LetB be a flatO-algebra which is a quotient ofAand letJ = ker(A→B) and assumeM is aB=B⊗Ok-module asA-module, i.e. thatJ =J⊗Ok⊆Ann_A(M).

Suppose we want to study the deformation functor of B-modules Def^B_M. TheB- cohomology of M may be complicated while A can be chosen as a simpler ring.

There is a natural injective map Def^B_M → Def^A_M and the ideal J acts on an A- deformation MR of M to R through theAR-action. Let Def^(A,J)_M ⊆Def^A_M be the sub-functor of A-deformations annihilated byJ.

Lemma 1. LetAandB be flat O-algebras andM aB=B⊗Ok-module. LetJ be an ideal in A and assumeB=A/J. Then

Def^B_M∼= Def^(A,J)_M .

The main idea in this paper emerges from Lemma 1: Lift M as an A-module with trivialJ-action and only useA-cohomology to characterise the tangent space and the obstructions. In Theorem 1 we give a new obstruction class o_Jwhich exists (in a small lifting situation) if the obstruction o_Afor liftingM asA-module is zero, such that oJ= 0 if and only if there exists anA-lifting with trivialJ-action. In fact o_Jwill sit in the cokernel of a natural map∂_J: Ext¹_A(M,M)→Hom_A(J,End_A(M)).

Moreover, the kernel of this map is the tangent space of Def^(A,J)_M .

With two natural obstruction classes we can construct two obstruction maps (o^A,o^J), as stated in Theorem 2, which are compatible with the obstructions, Def- inition 4. The obstruction maps are defined if the cohomologyk-vector spaces are of countable dimension, as in [13]. Remark how these maps depend on each other.

In particular, it is not true that o^Ain the pair (o^A,o^J) is induced by o^A for Def^A_Mas we clearly see in Example 5. This example also shows that much of the obstruction space not necessarily is “hit” by obstructions (at least as long as we do not deform over non-commutative Artinian algebras). Theorem 3 compares (o^A,o^J) with the traditional o^B and is based on the relations of the various obstruction classes which are found by investigating maps in a change of rings spectral sequence, which is undertaken in Section 4 and Section 5. In particular, Theorem 4 ties several of our obstruction classes together by a cup product with the obstruction class for lifting M non-flat toA/J². Finally, in Section 6, we give three obstruction calculations.

In Example 6 and 7 we find obstructions in mixed characteristic. In Example 8 the obstruction ideal is given as a regular sequence (at least in an infinite set of cases) with two elements while dimkExt²_B(M,M) = 4.

For actual calculation of the obstruction power series, one can lift a free resolution of the module, see [14]. The universal deformation to the relative Zariski tangent space of the deformation functor is given by perturbing the differentials in the res- olution with Yoneda-representations for a k-basis of Ext¹(M,M). The quadratic obstruction is given in terms of cup products and the higher degree obstruction as generalised Massey products which are represented as composition products in the Yoneda algebra. It is therefore not sufficient for our purposes to work in the derived category, and our results describing maps in the change of rings spectral sequence and the comparison of obstruction classes is done by giving explicit representations in the appropriate Yoneda algebra of a free complex. Our result enables the ob- struction calculus to be performed entirely within a (truncated) Yoneda complex of anA-free resolution of theB-moduleM. A formal proof of this (in the caseO=k) is given in [7, Thm. 3.3.2], see also Example 6–8.

For explicit non-trivial calculations of obstructions (given by cup products) for the Hilbert functor of space curves, see [22, 6]. A. Siqveland gave the local equations for the compactified Jacobian of theE6curve singularity and found the degeneracy diagram of the rank 1 torsion free modules in [19] by calculating the obstruction maps. The Massey product algorithms are given in [20]. Similar ideas have recently been used by I. C. Borge and O. A. Laudal [3] to solve the modular isomorphism problem forp-groups withFp-coefficients. See also [2].

2. The J-obstruction class

In this section we construct 3 obstruction classes for lifting a module in a relative lifting situation.

LetA→Bbe any surjective ring homomorphism and letMandNbeA-modules withA-free resolutionsF andF⁰respectively. The correspondingYoneda complex is the differential graded module Hom^∗_A(F,F⁰) where Homⁿ_A(F,F⁰) = Hom_A(F,F⁰[−n]) with differential ∂ induced from the ones on F and F⁰. Our first objective is to define a lifting of a Tor-action to the Yoneda complex, which will enable us to study the J-action on theA-deformations ofM. Assume thatM andN areB-modules as A-modules and letE be anA-free resolution of the A-moduleB.

Letm:E⊗AF →F andm⁰ :E⊗AF⁰→F⁰ liftB⊗AM ∼=M andB⊗AN ∼=N respectively and see that for e∈ E, m, m⁰ give an m(e) ∈End_A(F) respectively m⁰(e)∈End_A(F⁰). Define

∂A/B:E−→End^∗_A(Hom^∗_A(F,F⁰))

by ∂A/B(e)(φ) =m(e)φ−(−1)^|φ||e|φm⁰(e) whereφ ∈Hom^∗_A(F,F⁰). Clearly∂A/B

depends on the choices made.

Proposition 1. ∂_A/B induces a canonical map of gradedB-modules Tor^A_∗(B,B)−→End^∗_A(Ext^∗_A(M,N))

making Ext^∗A(M,N) aTor^A∗(B,B)-module. In the case M=N, Ext^∗A(M,M) is an algebra-module and Tor^A_p(B,B)acts as degreep-derivations. The map is natural in the sequence (A→B, M, N).

Proof. One calculates

∂A/B(e)(∂φ) =±∂ ∂A/B(e)(φ)

hence we get induced a ∂A/B:E −→End^∗_A(Ext^∗_A(M,N)). It factorises via B⊗AE and one calculates again:

∂A/B(de)(φ) =±∂ ∂A/B(e)(φ) if∂φ= 0. We get a map

H(B⊗AE)−→End^∗_A(Ext^∗_A(M,N))

which is independent of the choices made.

LetJ = ker(A→B) and define

(1) ∂J: Ext¹_A(M,N)−→Hom_A(J,Hom_A(M,N))

to be the adjoint of ∂A/B restricted to J through Tor^A₁(B,B)∼=J/J². For conve- nience we will also use the∂J-notation in adjoint situations, as in the next theorem.

We are now in the position to formulate necessary and sufficient conditions for the existence of deformations of anA-module with trivialJ-action, i.e. aB-module, in a small lifting situation. The standard result here, as given in [14], is to produce a class oB in theB-cohomology of the situation, for modules that would be Ext²_B, which vanish if and only if there is a lifting. We will instead produce two classes (actually three), the oA in Ext²_Awhich is the old obstruction for liftingA-modules, and if oA= 0, a new class oJ, also given byA-cohomology such that oJ= 0 if and only if there exists an A-lifting with trivialJ-action. These two classes will enable us to characterise the hull of Def^B_Mbytwo obstruction maps (see Theorem 2) in the same way as oB gives the tool for constructing the obstruction map in [14].

We formulate the result in the followingrelative lifting situation: SupposeAand B are flatO-algebras whereOis a commutative ring. LetA→B andπ :R →S be surjective maps of O-algebras with kernels the idealsJ and I respectively. Let M be aBS =B⊗OS-module, (similarlyJS=J⊗OS etc.). AssumeI²= 0.

Definition 3. A lifting of M to BR is a BR-module MR and a BR-linear map π :MR→M withπ⊗S :MR⊗RS−^'→M, such that Tor^R₁(MR,S) = 0.

Theorem 1. In the above situation we have:

i) There exists a classo_AS = o_AS(π, M)∈Ext²_AS(M,M⊗SI)such thato_AS= 0 if and only if there exists a lifting of M toAR.

ii) Given a lifting MR ofM toAR, there exists a class o(JS)∈Hom_AS(JS⊗ASM,M⊗SI)

such thato^(JS)= 0 if and only ifMR is aBR-module as an AR-module.

iii) If o_AS= 0 there exists a classo_JS = o_JS(π, M)∈coker∂JS where

∂JS: Ext¹_AS(M,M⊗SI)−→Hom_AS(JS⊗ASM,M⊗SI)

is as given in (1)such that o_JS = 0if and only if there is a lifting of M to BR.

iv) Assume o_AS = 0 = o_JS, then there is a transitive and effective action of ker∂JS on the set of isomorphism classes of liftings of M toBR over M.

Finally, all classes and the action are natural for flat maps of A → B and of R→S and the induced modules. IfM isS-flat,the naturality follows for all maps of R→S.

Remark 2. In the case O = k = A = B (and J = 0) one recovers the standard result and our construction of oSis as in [14].

Remark 3. In fact o_AS is in the image of the natural map Ext²_BS → Ext²_AS as we prove in Lemma 4.

Proof. Since we are mainly interested in the deformation case, we give a proof under the additional assumption of eitherMorI beingS-flat. The complex we use lends itself best to these cases. At the end we comment on the general situation.

i)LetF = (F, d) be anAS-free resolution ofM. By the freeness we can lift the differential to a map ˜d of the gradedAR-free module ˜F which in each degree has the same rank asF, thus ˜d⊗RS=d. If ˜F = ( ˜F ,d) were a complex, it would be a˜ resolution of a lifting ofM toAR. Tensoring ˜F with the short exact sequence 0→ I →R−^π→S→0 ofR-modules gives an exact sequence 0→F⊗SI →F˜−^π→F →0 of graded modules sinceAis flat asO-module. It follows that ( ˜d)²is induced by a map ρ∈Z²Hom_AS(F,F⊗SI) i.e. a 2-cocycle in the Yoneda complex. Define

o_AS = o_AS(π, M) := [ρ]∈Ext²_AS(M,M⊗SI).

This element is independent of the resolutionFand the choice of lifting ( ˜F ,d) and is˜ the obstruction for liftingM alongπ: If o_AS = 0, there is aτ ∈Hom¹_AS(F,F⊗SI)—

this is the place where we use the additional hypothesis—with∂τ =ρ. We perturb d˜byτ and get a differentialdR= ˜d−τ π. HenceFR= ( ˜F , dR) is a complex which is an extension of resolutions (by the additional hypothesis again), thus itself a resolution ofMR:= H0(FR). ClearlyMR⊗RS∼=Mand Tor^R₁(MR,S) = H1(F) = 0, in fact MR is R-flat ifM isS-flat. If there is a lifting MR ofM, o_AS = 0 by the independence of the choices we claimed above.

ii) To find the obstruction forMR to be aBR-module asAR-module, we lift the canonical isomorphismm0: AS⊗ASF →Fto a map of complexesm: ES⊗ASF → F where . . . → E2 → E1 → J gives an A-free resolution of J which, together with J ⊂ A = E0, gives an A-free resolution E of B, and ES := E⊗OS gives an AS-free resolution of BS. The lifting m exists since M is a BS-module as in Proposition 1, i.e. since JS’s action on F is homotopically trivial. Choose an

˜

m : ER⊗ARFR → FR with ˜m⊗RS = m and with ˜m|(ER)0⊗FR the canonical isomorphism AR⊗ARFR ∼= FR. Then we can view ˜m as an attempt to kill the action of JR onFR. We find∂( ˜m) =d(FR)◦m˜−m˜ ◦d(E⊗FR) to be induced by

aρ∈Z⁰Hom_AS(E>1⊗AF,F⊗SI) where E>1= [. . .→E2→E1][+1]. Define

(2) o^(JS):= [ρ]∈H⁰Hom_AS(E>1⊗AF,F⊗SI)∼= Hom_AS(JS⊗ASM,M⊗SI).

This class only depends on the lifting MR and is the obstruction for MR to be a BR-module asAR-module. If o(JS)= 0, there is aτ ∈Hom⁻¹_AS(E>1⊗AF,F⊗SI) with

∂(τ) =ρ. Perturbing ˜mwithτ givesmR= ˜m−τ πwith∂(mR) = 0.

iii) The o^(J) only checks our specific choice of lifting MR given by dR, other choices of MR could be better. To obtain other AR-liftings we perturb dR by ξ ∈Z¹Hom_AS(F,F⊗SI) tod⁰_R=dR+ξπ. This gives a new differential∂⁰ and

∂⁰( ˜m) = (dR+ξπ) ˜m−m(d˜ E⊗F_R⁰)

=dRm˜ −(−1)^|E|m(1⊗d˜ R)−m(d˜ E⊗1) +ξπm˜−(−1)^|E|m(1⊗ξπ)˜

= ρ+∂AS/BS(ξ) π

where∂AS/BS: Hom_AS(F,F⊗SI)−→Hom_AS(E⊗AF,F⊗SI) up to adjointness is the one in Proposition 1. Define the class

o_JS= [o^(JS)]∈coker∂JS = coker Ext¹_AS(M,M⊗SI)→Hom_AS(JS⊗ASM,M⊗SI) , it depends only onMandπand is the obstruction for liftingMtoBRif there exists a lifting ofM toAR. If∂⁰( ˜m) =∂(τ)π, withτ ∈Hom⁰_AS(F,F⊗SI), we can perturb

˜

mto ˜m⁰= ˜m−τ πand∂⁰( ˜m⁰) = ρ+∂AS/BS(ξ)−∂τ

π= 0 so ˜m⁰gives a homotopy to zero for the action of JR onF_R⁰ , i.e.M_R⁰ is a BR-module asAR-module.

iv) It also follows that any ξ⁰ ∈Z¹Hom_AS(F,F⊗SI) with ∂AS/BS(ξ⁰) = 0 gives another lifting to BR by d⁰⁰_R = d⁰_R+ξ⁰π and that the difference d⁰_R−d⁰⁰_R of two

liftings to BR gives an element in ker∂AS/BS. They are isomorphic if and only if this element is zero in Ext¹_AS(M,M⊗SI).

For the general case, the main difference is thatF⊗SI not necessarily is a reso- lution, and a resolutionFRofMRwill give H∗(FR⊗RS)∼= Tor^R_∗(MR,S) and cannot therefore in general be taken as a lifting ˜F ofF. But, in fact only the initial part

(3) F0 F1

d1

oo F2

d2

oo

E1⊗F0 m

;;w

ww ww ww ww

where d1mequals the map induced by the multiplication by (generators of) JS on F, is essential to the existence of liftings toARand toBRas will be exploited later on. The o_AS is induced by ˜d1d˜2 and if o_AS = 0 modify ˜d1 and ˜d2 by τ1 andτ2 as before. MR= H0( ˜F) hasMR⊗RS=M, in a resolution FR forMR we can choose (FR)i = ˜Fi fori= 0,1 and (FR)2= ˜F2⊕K2. Then the Tor-condition follows:

Tor^R₁(MR,S) = H1(FR⊗RS) = ker(F1−→F0)

im F2⊕(K2⊗RS)−→F1 = 0

soMR= H0( ˜F) is certainly a lifting. The o^(JS)is defined as induced by ˜d1m˜−mE1

wheremE1 :E1⊗AF˜0→F˜0 is induced by the multiplication ofJRon ˜F0. The rest

follows as above.

Remark 4. We shall primarily be interested in the deformation situation, Defini- tion 1, and the case of a small lifting situation, Definition 2. If MS is a deforma- tion of M to S in Art_O one has natural isomorphisms like Extⁱ_AS(MS,MS⊗SI) ∼= Extⁱ_A(M,M)⊗kI and Hom_AS(JS⊗ASMS,MS⊗SI)∼= Hom_A(J⊗_AM,M)⊗kI. The ex- istence of suchconstant groups is essential for the existence of an obstruction algo- rithm. With a fixed k-basis the constant cohomology groups will keep track of the different obstruction “polynomials” in (the varying) I. To simplify the notation in the deformation situation, let o_AS = oA, o_JS = oJ and so on.

Example 3. A matrix factorisation (mf) of an element f in a ringA is a pair (φ, ψ) of maps of free modules φ : F → G, ψ : G → F with φψ = f·idG and ψφ=f·idF. LetB =A/(f) thenM = cokerφis aB-module asA-module since f annihilatesM. If f is A-regular then the following 2-periodic complex of free B-modules (necessarily of equal rank ifAis Noetherian and rkG <∞)

(4) G←−^φ F ←^ψ−G←^φ−F ←^ψ−. . .

is a free resolution ofMwhereF =F⊗ABetc. Maximal Cohen-Macaulay modules over a hypersurface singularity are given by mfs of the hypersurface. Mfs were introduced by D. Eisenbud in [4]. A deformation ofMasB-module will be given by a lifting of this resolution, one will therefore have conditions for lifting the equations φ ψ = 0 which create the obstruction oB in Ext²_B(M,M). Instead Theorem 1 offers the possibility of lifting φ corresponding to deformations of M as A-module for which the obstruction oA= 0 since Extⁱ_A(M,M) = 0 fori >1 , such that there is a lifting ofψretaining the relationφψ=f·idG, this gives the non-trivial obstruction o_J (withJ = (f)) in the cokernel of∂J : Ext¹_A(M,M)→End_A(M) where∂J=ψ^∗. Even in this most simple example the advantages are clear: The A-cohomology is easier than the B-cohomology and the relation f = 0 is eliminated from the obstruction calculus. Further simplifications are possible in the rkB(M) = 1-case as we show in [8].

3. The obstruction maps

We define obstruction maps o^A and o^J for the obstructions o_A and o_J in Defini- tion 4 and formulate a structure theorem for the hull of Def^B_M in Theorem 2. A comparison of theA- andJ-obstruction maps with theB-obstruction map is given in Theorem 3.

For greater flexibility we will define obstruction maps as continuous maps be- tween local O-algebras which have countably dimensional Zariski tangent spaces.

LetV be a countably dimensional vector space overkwith a given basis{ei}^∞_i=1. The set of sub-vector spaces ofV which contain almost all the basis elements defines a topology onV such thatV^∗= Hom^cont._k-vec.(V,k) also is countably dimensional, and if we fix the dual basis {e^∗_i}^∞_i=1 forV^∗ thenV^∗∗ ∼=V canonically. LetFreeb O(V^∗) be the free O-algebra in variables{xi}^∞_i=1 completed in the topology given by the basisIijl of open ideals around 0 whereIijl =mⁱ

O+ (x1, x2, . . .)^j+ (xl, xl+1, . . .) . We insist on the continuous identification of the relative cotangent vector space m/(m²+m_O) of Freeb O(V^∗) with V^∗ where xi = e^∗_i hence also a canonical con- tinuous identification of the relative Zariski tangent space of Freeb O(V^∗) with V. Suppose{Hn}^∞_n=1is a projective system of surjections inArt_O. ThenH := lim←−Hn

with the induced topology is a continuous quotient of aFreeb O(V^∗) for someV. Con- versely every continuous quotient of bFreeO(V^∗) can be given as such a projective limit. DefineDefb ^B_M(H) = lim←−Def^B_M(Hn).

Recall the map ∂J with N =M, see (1). Assume for the rest of this section that thek-vector spaces H²_A= im Ext²_B(M,M)→Ext²_A(M,M)

, H²_J= coker∂_Jand H¹= ker∂_J all are of countablek-dimension and for any choice of k-bases letT_A², T_J² andT¹ be the corresponding completeO-algebras with these vector spaces as relative Zariski tangent spaces.

Definition 4. In the situation described before Lemma 1,two obstruction mapsfor the obstructions oA and oJin Theorem 1 (see Remark 4) are continuousO-algebra homomorphisms o^A:T_A²→T¹and o^J :T_J²→T¹satisfying the following conditions.

If H := T¹⊗ˆ_T²

AO⊗ˆ_T²

JO there is a formal deformationMcin Defb ^B_M(H) such that for any small lifting situation, Definition 2, there is a continuous σ :H →S with σ∗Mc=MSand for any suchσwe have that the adjoint o^adjA of o_A(π, MS)∈H²_A⊗kI makes the following diagram commutative

(5) m

T_X²/(m²

T_X² +m_O) ⁼ //H²_X^{∗ o}

adj

X //I_

mT_X²  //

OOOO

T_X^{2 o}

X

//T^{1 θ} //

R

π

H ^σ //S

where θ is continuous and lifts σ and X =A. If o_A(π, MS) = 0 then the adjoint o^adjJ of oJ(π, MS)∈H²_J⊗kI makes the diagram commutative withX=J.

Theorem 2 ([7]). Let A and B be flat O-algebras with B = A/J for an ideal J ⊂ A. LetM be a B = A/J-module where X = X⊗Ok for X =A, B and J.

Then Def^B_M is a functor with two obstructions in H²_A and H²_J such that if H¹, H²_A andH²_J have countablek-dimension there are obstruction maps

o^A:T_A² −→T¹ and o^J:T_J²−→T¹

for the obstructions o_A ando_J. In particular the hull of Def^(A,J)_M ∼= Def^BM is given as

H ∼= T¹⊗ˆ_T²

AO⊗ˆ_T²

JO.

Remark 5. The statement implicitly claims the existence of k-vector bases and hence topologies as described before the Theorem and maps continuous with respect to these topologies.

Example 4. Remark that everyO-algebra in the pro-category ofArt_Ois obtained as the hull of the deformation functor of a module. In fact the following argument is valid for thenon-commutative deformation functor of modules as well as for the commutative one. In the non-commutative case Art_O is the category of local not necessarily commutative Artinian O-algebras (O as in Definition 1) R with k as residue field (i.e. k is the unique simpleR-module). A and B may also be non- commutative O-algebras. A deformation of a left A-module M is defined as in Definition 1 except thatMR is anA−R-bimodule which is a leftA-module and a right R-module, or equivalent, a leftA⊗OR^o-module. Furthermore, bFreeO(V^∗) is the free non-commutative O-algebra, completed in the topology defined by ideals Iijl analogous to the ones in the beginning of this section, e.g. where the “power ideal” (x1, x2, . . .)^j is replaced by the (two sided) ideal generated byj-tensors, and so on.

Fix a maximal idealmin anO-algebraB such thatB/m∼=k. AssumeB/(m²+ m_O) is countably dimensional and let ˆB be the completion ofB in any topology as given in the beginning of this section (or analogous in the non-commutative case), then

Hom^cont._O−alg./k( ˆB,−)−^'→Def^B_k

where φ ∈ Hom^cont._O−alg./k( ˆB,R) is mapped to the B⊗OR^o-module R with mod- ule structure given by (left) multiplication of B⊗OR^o through the composition Bˆ⊗OR^o −−−→^φ⊗id R⊗OR^{o mult.}−−−→ R^o. It gives a deformation of k to R. For the in- verse, any deformation MR of k to R has MR ∼= R as R-modules since MR is R-flat, i.e.R-free of rank 1. HenceRhas a (left)B⊗OR^o-module structure. Define φ: ˆB →R byφ(b) := (b⊗1)•1R=r∈R forb∈B. Thenφ(b⁰b) =b⁰b⊗1•1R = (b⁰⊗1)(b⊗1)•1R= (b⁰⊗1)•r= (b⁰⊗1)(1⊗r)•1R= (1⊗r)(b⁰⊗1)•1R= 1⊗r•r⁰ = r⁰r= (b⁰⊗1•1R)(b⊗1•1R) =φ(b⁰)φ(b),φ(1) = 1R andφis additive. Iff :O →B and g : O → R define the O-algebra structures, φ(f(λ)a) = f(λ)a⊗1•1R = a⊗g(λ)•1R= (1⊗g(λ))•(a⊗1•1R) =g(λ)(a⊗1•1R) =g(λ)φ(a), henceφgives a well definedO-algebra homomorphism ˆB→Rabovekand ˆBpro-represents Def^B_k. In particular

Bˆ∼= T¹⊗ˆT_A²O⊗ˆT_J²O

for obstruction maps o^A and o^J. For instance, if B=O thenB =k and Def^B_k(R) is a one element set for allR and Def^B_k is pro-represented byO.

Remark that if B is non-commutative, we can still deform k over commutative Artinian O-algebras, but then the completion ˆB will be in the idealsIijl+ [B, B]

and hence in that case be a commutativeO-algebra, indeed ˆB= (B/[B, B])^ˆ. In [7] we define axiomatically a functor with n obstructions and corresponding obstruction maps and prove the existence of such maps in the countably dimensional case. Theorem 2 is an instance of this. A single obstruction map defined for the obstruction oA has been constructed by O. A. Laudal in [14].

We next state a theorem describing how closely related the obstruction map o^B, defined byB-cohomology, is to o^A and o^J, defined byA-cohomology. LetT_B² be a local complete O-algebra with relative Zariski tangent space Ext²_B(M,M) for any

topology as in Theorem 2. In the next sections (Proposition 2 and Lemma 4) we show that there is a canonical isomorphism ker∂_J ∼= Ext¹_B(M,M) and a natural exact sequence of A-modules 0→coker∂_J →Ext²_B(M,M) →Ext²_A(M,M). Hence there is a “short exact sequence” of continuous maps T_A² ,→T_B² T_J². Our main comparison result reads:

Theorem 3. With assumptions as in Theorem 2,given a pair of obstruction maps o^A :T_A² →T¹ ando^J: T_J²→T¹ for the obstructions oA and oJ, defining the hull of Def^(A,J)_M ,there exists an obstruction map o^B:T_B² −→T¹ for the obstructionoB, defining the hull of Def^B_M,such that

(6) o^B|^TA² = o^A, and o^B⊗ˆ

T_A²O= o^J⊗ˆ

T_A²O as maps T_J²−→T¹⊗ˆ

T_A²O. Conversely, given an obstruction map o^B, there exists a pair of obstruction maps o^A ando^J such that the following diagram of continuous maps is commutative:

T_A²

o^A

A

AA AA AA A

//T²_B

o^B

//T²_J

o^J

~~}}}}}}}}

T¹

Remark 6. The o^B-map is in general not the “union” of o^A and o^J, but there is always a pair of obstruction maps (o^A,o^J) such that o^Bcan be taken as the “union”

of o^A and o^J.

We do not necessarily get a trivial o^J1+J2 even if o^Ji is trivial for i= 1,2 . The reason for this is simply that the natural map coker∂J1+J2 → ⊕coker∂Ji does not have to be injective, an explicit example is given in [7, Ex. 4.1.4].

Remark 7. If a choice of o^A for Def^A_M continued to T¹ is trivial, one can choose (o^A,o^J) for Def^(A,J)_M such that o^A is trivial. But even if o^A in (o^A,o^J) is trivial, o^A for Def^A_M continued to T¹ may be far from trivial as Example 5 shows. There is no way one can find o^A “first” and then find o^J as this has no meaning. It is not clear to the author whether o^A in the pair (o^A,o^J) and the locus it defines has any interesting interpretation.

Proof. Suppose (o^A,o^J) is given, assume dimkker∂J<∞and letT_n¹=T¹/mⁿ⁻¹n wheren=m²+m_O. LetG^An =Tn¹/(f_iⁿ⁻¹)m+ (g_jⁿ⁻¹)mwhich maps surjectively to G^J_n =T_n¹/(f_iⁿ) + (g_jⁿ⁻¹)maboveHn−1=T_n−1¹ /(f_iⁿ⁻¹) + (g_jⁿ⁻¹) wheref_iⁿ= o^A(yi) and gⁿ_j = o^J(zj) inT_n¹. Observe thatG^A_n is “maximal” (with fixed relative Zariski tangent space) such thatπ^A_n :G^A_n Hn−1andMn−1, a versal lifting ofMtoHn−1, together give a small lifting situation. G^A_n is the test algebra for the oA-obstruction.

Similarly G^J_n is maximal such that π^J_n : G^J_n Hn−1 and Mn−1 together give a small lifting situation with o_A(π_n^J, Mn−1) = 0. G^J_n is the test algebra for the o_J- obstruction. By Proposition 2 and Lemma 4 we have a “short exact sequence”

T_A² ,→T_B² T_J², the last map has a section and we let{yi} ∪ {zj}also denote the

“generators” inT_B². We want to define o^B. While we let o^B(yi) := o^A(yi) which is OK by Lemma 4, we find o^B(zj) by induction. Let I_n^A= kerπ^A_n and I_n^J = kerπ_n^J. Then oB(π_n^A, Mn−1)∈H²_B⊗I_n^A maps to oJ(π_n^A, Mn−1)∈H²_J⊗I_n^J alongI_n^AI_n^J by Theorem 4. We have chosen representatives gj (and fi) inT¹, likewise there is a choice for o^B(zj) inT¹. Mapped toI_n^A, o^B(zj) andgⁿ_j may only differ by an element in Kn:= ker(I_n^AI_n^J). But Kn = (f_iⁿ), hence we can define o^B to “leveln” by

o^B(zj) :=gⁿ_j +X

i

a⁽ⁿ⁾_ji f_iⁿ∈T_n¹

witha⁽ⁿ⁾_ji ∈ O/mⁿ

O and where oB(π_n^A, Mn−1) =X

i

y_i^∗⊗f_iⁿ+X

j

z_j^∗⊗(gⁿ_j +X

i

a⁽¹⁾_ji f_iⁿ)∈H²_B⊗I_n^A.

By naturality of o_B this extends o^B(zj) defined on level n−1 if we choosea⁽ⁿ⁾_ji ∈ O/mⁿ as a lifting of a⁽ⁿ⁻¹⁾_ji ∈ O/mⁿ⁻¹. In the limit we get o^B(zj) =gj+P

ajifi

whereaji= lim←−{a⁽ⁿ⁾_ji }inOinT¹. This is stronger than our claim. For the general case choose a sequence of finite dimensional k-vector spacesV1 ⊂V2 ⊂. . . ⊂H¹ with S

Vs = H¹. This gives a topology on H¹ and hence on H^1∗ = lim←−V_s^∗ and on T¹ = lim←−T_s¹ where T_s¹ has Vs as relative Zariski tangent space. For each s the argument above may be applied to the composition of o^A and of o^J with the continuous mapT¹T_s¹. By induction ons(andn) one proves that o^B_s+may be chosen compatible with o^B_s. Set o^B= lim←−{o^B_s}.

Given o^B, let o^A(yi) := o^B(yi) and let o^J(zj) := o^B(zj) then (o^A,o^J) are obstruc- tion maps for the two obstructions oA and oJ. This follows from Theorem 4 and

Lemma 4.

4. The change of rings spectral sequence

The spectral sequence connects theA- and theB-cohomology and also provides a framework for describing relations between the various obstruction classes. In the following we give detailed descriptions of the maps α,d2 andγ by representations in the Yoneda complex.

Lemma 2. LetA→B be a ring homomorphism andN, M anA- and aB-module respectively. Then there is a first quadrant cohomological spectral sequence

E^pq₂ = Ext^p_B(M,Ext^q_A(B,N))⇒Ext^∗_A(M,N).

In particular there is a canonical 5-term exact sequence which,in the caseB =A/J andN is aB-module asA-module,becomes

0−→Ext¹_B(M,N)−→Ext¹_A(M,N)−→^α Hom_A(J,Hom_A(M,N))

d2

−→Ext²_B(M,N)−→^γ Ext²_A(M,N) (7)

Proof. Let G = G· M be a B-projective resolution of M and N ,→ I^· = I an A-injective resolution of N. Then the II-filtration of Hom_B(G,Hom_A(B,I)) gives a spectral sequence which collapses at stage 2 to the total cohomology. The spectral sequence obtained from the I-filtration gives the E2-terms. The 5-term exact sequence is the standard one with E⁰¹₂ ∼= HomA(J,Hom_A(M,N)).

Let ε: (F, d) M be anA-free resolution of M, E →B an A-free resolution of B; . . . E2 → E1 → A B. Recall the definition of m : E⊗AF → F before Proposition 1. We change the notation by s:=m>1:E>1⊗F →F where E>1 = [. . . → E2 → E1][+1] and let mE1 : E1⊗F →F be the multiplication with J = (f1, . . . , fr) on F-map pulled back along E1 J. Then ∂(s) equals mE1 when restricted toE1⊗F and is zero elsewhere. Hence the map∂J is described simply as induced by the pullback alongs:E1⊗F0→F1, (them in (3)).

Proposition 2. If M andN are B-modules asA-modules then

∂J: Ext¹_A(M,N)−→Hom_A(J,Hom_A(M,N))

given in (1)is the edge map αin the change of rings spectral sequence in Lemma 2. In particular there are canonical isomorphisms

ker∂J∼= Ext¹B(M,N) and coker∂J∼= imd2⊆Ext²_B(M,N)

where d2 is the spectral sequence differential in Lemma 2.

Proof. Letξ ∈Hom_A(F1,N) be a cocycle representing the class [ξ]∈Ext¹_A(M,N).

Then there is a ρ ∈ Hom_A(F0,I⁰) extending ιξ where ι : M ,→ I⁰ is the coaug- mentation map. There is also a τ ∈Hom_A(M,I¹) extendingd⁰ρ, clearly [τ] = [ξ].

From ρd1=ιξ we get ρmE1 =ιξs. The mapιξs represents∂J([ξ]). If ε=ε⊗AB one is left to prove that the connecting Hom_B(M,Ext¹_A(B,M))−^'→End_B(M) is rep- resented by taking τ ε to ρmE1. Applying Hom_A(F0,Hom_A(−,I^·)) to the short exact sequence 0 → J → A → B → 0 gives an exact sequence of complexes.

Observe Hom_A(F0,Hom_A(B,I^·)) ∼= HomA(F0,I^·) and Hom_A(F0,Hom_A(J,I^·)) ∼= Hom_A(J⊗F0,I^·)Hom_A(E1⊗F0,I^·) hence:

Hom_A(E1⊗F0,I¹) Hom_A(F0,I¹)3τ ε

m^∗_E

oo 1 oo Hom_A(F0,I¹)3τ ε←0

Hom_A(E1⊗F0,I⁰)3ρmE1

OO

Hom_A(F0,I⁰)3ρ

m^∗_E

oo 1 OO

Hom_A(F0,I⁰)←−0

oo OO

Hom_A(E1⊗F? 0,N)3ξs

ι_∗

OO

Remark 8. IfAis anO-algebra for any commutative ringO, there is a restriction of derivations map

D∈DerO(A,Hom_O(M,N))−→Hom_A(J,Hom_A(M,N))3D|J

which one checks is well defined if M and N are annihilated by J. The inner derivations maps to zero and ifAisO-flat we have

DerO(A,W)/(inner derivations)∼= HH¹(A; W)∼= Ext¹A(M,N)

where HH^∗(A; W) is the Hochschild cohomology with values in theA-bimoduleW = Hom_O(M,N). Via this identification our ∂J equals the restriction of derivations map. This is proved by constructing a specific lifting of the multiplication of J- map on a non-reduced bar complex. Indeed lets:E1⊗AA⊗OM→A⊗OA⊗OMbe defined bys(ei⊗1⊗m) = 1⊗fi⊗mwhereJ = (f1, . . . , fr) andE1=⊕^i=r_i=1Aei. Then

∂J=s^∗. We are indebted to Prof. O. A. Laudal for suggesting this interpretation of

∂Jat an early stage, which again led to the above change of rings spectral sequence.

Example 5. LetP be a regular local complete Noetheriank-algebra, and letJA

and JB be ideals in P with 0 6= JA ⊆ mJB and JB ⊆ m². Let A = P/JA, B =P/JB,O=kandM=kwhich is assumed to be the residue field ofB. Then Def^X_M is pro-represented byX forX =P, Aand B, see Example 4. In particular the idealJAis given by the image of the maximal ideal under the obstruction map o^A. But if J := JB·A ⊆ A then o^A in the pair of obstruction maps (o^A,o^J) is trivial. The reason for this can be seen from the obstruction calculus. Since o^J potentially can generateJB as obstruction ideal and o^A only generatesJA⊆mJB, the o_A-obstruction will be one or more “steps” behind o_J, the latter thus takes care of all the obstruction. This phenomenon can also be deduced from the 5-term exact sequence. For transparency assume JA andJB are generated by regular sequences of length aand b. A- andB-free minimal resolutions of kmay be produced from theP-free Koszul resolution ofktogether with “Eisenbud systems”, i.e. systems of homotopies for killing the action of the ideals on the Koszul complex, see [4, Thm.

7.2], [7, Chap. 7.4] and Lemma 3 below. Then the 5-term exact sequence is:

0−→m_B/m²

B

∗ '−→m_A/m²

A

∗ 0

−→(J⊗k)^∗−→^d2 k^b⊕k(ⁿ2)−→k^a⊕k(ⁿ2)