Locally complete intersection maps and the proxy small property
Benjamin Briggs, Srikanth B. Iyengar, Janina C. Letz, Josh Pollitz
aa r X i v : . [ m a t h . A C ] J u l LOCALLY COMPLETE INTERSECTION MAPS ANDTHE PROXY SMALL PROPERTY
BENJAMIN BRIGGS, SRIKANTH B. IYENGAR, JANINA C. LETZ, AND JOSH POLLITZ
Abstract.
It is proved that a map ϕ : R → S of commutative noetherian ringsthat is essentially of finite type and flat is locally complete intersection if andonly S is proxy small as a bimodule. This means that the thick subcategorygenerated by S as a module over the enveloping algebra S ⊗ R S contains aperfect complex supported fully on the diagonal ideal. This is in the spirit ofthe classical result that ϕ is smooth if and only if S is small as a bimodule, thatis to say, it is itself equivalent to a perfect complex. The geometric analogue,dealing with maps between schemes, is also established. Applications includesimpler proofs of factorization theorems for locally complete intersection maps. Introduction
This work concerns the locally complete intersection property for maps betweencommutative noetherian rings. While there are numerous characterizations of thisproperty, none are in terms purely of the structure of the derived category as atriangulated category. Our main results, Theorems A and B, supply just such char-acterizations. To set the stage for the discussion, let ϕ : R → S be a homomorphismof commutative noetherian rings that is flat and essentially of finite type; the lattercondition means that S is a localization of a finitely generated R -algebra.We establish criteria for detecting when ϕ is locally complete intersection, analo-gous to the following criterion for smoothness: ϕ is smooth if and only if S is perfectas a complex over the enveloping algebra S e R := S ⊗ R S via the multiplication map S e R ։ S ; see [12, Proposition (17.7.4)] and [20, Theorem 1]. Perfection of S isequivalent to the condition that S is isomorphic in D ( S e R ), the derived category of S e R , to a bounded complex of finitely generated projective S e R -modules. We prove: Theorem A.
Let ϕ : R → S be a homomorphism of commutative noetherian rings,flat and essentially of finite type. Then ϕ is locally complete intersection if and onlyif the thick subcategory of D ( S e R ) generated by S contains a perfect complex whosesupport equals that of S . By the support of a complex W in D ( S e R ) we mean the set of prime ideals q inSpec( S e R ) such that H( W ) q = 0. A complex (in the derived category of some ring)is proxy small if the thick subcategory it generates contains a perfect complex with Date : 16th July 2020.2010
Mathematics Subject Classification.
Key words and phrases. locally complete intersection map, factorization of locally completeintersection map, Hochschild cohomology, homotopy Lie algebra, proxy small.Partly supported by NSF grants DMS-1700985 (SBI and JCL) and DMS-1840190 (JP). the same support; see Section 1. Thus Theorem A can be rephrased as: ϕ is locallycomplete intersection if and only if S is proxy small in D ( S e R ).There are other reformulations possible. Indeed, it follows from a result of Hop-kins [13] that if the thick category generated by S contains a perfect complex withsupport equal to that of S , then it has to contain every perfect complex whosesupport is contained in that of S ; see 1.12. So Theorem A is equivalent to thestatement ϕ is locally complete intersection if and only if S generates the Koszulcomplex on a finite generating set for the kernel of the multiplication map.Theorem A is a consequence of Theorem 5.2 that applies to maps of finite flat di-mension which is the natural context for the locally complete intersection property.That result is in turn deduced from Theorem 3.2 concerning surjective homomor-phisms; the latter brings out another feature of complete intersections: Theorem B.
Let ϕ : R → S be a surjective homomorphism of finite flat dimension.Then ϕ is complete intersection if and only if any S -complex that is proxy small asan R -complex is also proxy small over S . In other words, ϕ is complete intersection if and only if proxy smallness ascends along ϕ . The forward implication that proxy smallness ascends for complete in-tersection maps is the content of [10, Theorem 9.1], so the result above provides aconverse. For this direction, it suffices to test that the ascent holds for complexeswith finite length homology. When S is local one can even specify a finite collec-tion of S -complexes of finite length homology whose proxy smallness detects thecomplete intersection property for ϕ ; see Theorem 3.11.The notion of proxy small complexes was introduced in [9] as a tool in dualitytheory. It has since become clear that this concept captures also interesting geomet-ric properties of maps, and that this sheds a new light on factorization theorems,for example. Indeed, in Section 4 we use Theorem B, more precisely, Theorem 3.2,to give simple proofs of some fundamental results concerning the factorization of lo-cally complete intersection maps, first established by Avramov [1] as a consequenceof his solution of a conjecture of Quillen concerning cotangent complexes.The statement of Theorem B and its proof are inspired by a result of Pollitz [19,Theorem 5.2] that characterizes local rings that are complete intersection in termsof proxy smallness of objects in its derived category, thereby settling a questionraised in [10, Question 9.10]. Letz [16, Theorem 6.8] used Pollitz’s result directlyto establish Theorem A in the special case where R is a field.Theorems A and B extend to morphism of schemes, but the appropriate notionof proxy smallness involves tensor-generation. This is explained in Section 6. Keep-ing in mind that the multiplication map is the diagonal embedding geometrically,Theorem A readily yields the following result. Theorem C.
Let f : Y → X be a flat, essentially of finite type, separated, morphismof noetherian schemes, and δ : Y → Y × X Y the diagonal embedding. Then f is locallycomplete intersection if and only if the thick ⊗ -ideal of D ( Y × X Y ) generated by δ ∗ O Y contains a perfect complex with support the diagonal. As with Theorem A, but this time using a result of Thomason [25], one canreformulate the theorem above to say that f is locally complete intersection if andonly if δ ∗ O Y tensor-generates a Koszul complex whose support is the diagonal.For any ring S , the perfect S -complexes are precisely the small objects in D ( S )in the sense of category theory: A complex P in D ( S ) is perfect if and only if the ELATIVE COMPLETE INTERSECTIONS 3 functor Hom D ( S ) ( P, − ) commutes with arbitrary direct sums. Thus proxy smallnessof an object also depends only on the triangulated category structure of D ( S ). Itis for this reason that the name “proxy small” is chosen over “proxy perfect”.The derived category of a commutative ring has been a valuable source of inspi-ration for results, if not also their proofs, in other tensor triangulated categories,like the category of spectra, or the stable category of modular representations offinite groups. Theorems A and B open the way to exploring notions of completeintersection rings and maps in these categories; see Remark 5.3.1. Proxy smallness
This section is mainly a collection of definitions and simple observations con-cerning proxy small objects in derived categories of rings and dg (= differentialgraded) algebras. Although the mains results involve only rings, their proofs ex-ploit dg structures extensively. We take [2, 5] as our basic references on this topic.By default the grading will be lower, so differentials decrease degree.Throughout A will be a dg algebra concentrated in non-negative degrees. Such adg algebra A is equipped with an augmentation map A −→ H ( A ). This is a map ofdg algebras where H ( A ), like any ring, is to be viewed as a dg algebra concentratedin degree zero with zero differential. Through this map any dg H ( A )-module (thatis to say, a complex of H ( A )-modules) inherits a structure of a dg A -module.We write D ( A ) for the derived category of left dg A -modules, with its canonicalstructure as a triangulated category, equipped with suspension functor Σ . A thick subcategory of D ( A ) is a triangulated subcategory closed under re-tracts. As the intersection of thick subcategories is again a thick subcategory, foreach object M of D ( A ) there exists a smallest thick subcategory, with respect toinclusion, containing M ; we denote it thick( M ). See [4, §
2] for a constructive de-scription of this category. Following [9, 10], we say that a dg module N is finitelybuilt from M , or that M finitely builds N , if N is in thick( M ). This situation isindicated by writing M | = A N ; we drop the A from the notation if the ambientcategory is unambiguous.A localizing subcategory of D ( A ) is a triangulated subcategory closed underarbitrary coproducts; such a subcategory is thick. Once again mimicking [9, 10],we write M ⊢ A N to indicate that N is in any localizing subcategory generated by M , and say M builds N , or that N is built by M .It is straightforward to verify that the relations ⊢ and | = are transitive; this willbe used without further mention. Evidently if M | = N then M ⊢ N ; the conversedoes not hold for arbitrary pairs of dg A -modules. All objects in D ( A ) are built from A ; in symbols: A ⊢ M for any M in D ( A ).This is a restatement of the fact that every dg module has a semifree resolution;see [11, Chapter 6]. It has long been known—see, for example, [14, Theorem 2.1.3]—that the objects that are finitely built from A are precisely the ‘small objects’ of D ( A ). Recall that a dg A -module M is small (or compact ) provided Hom D ( A ) ( M, − )commutes with arbitrary direct sums. When M and N are small dg A -modules(1.2.1) M ⊢ N implies M | = N .
For a proof, see, for example [18, Lemma 2.3], and also [21, Corollary 3.14].
BRIGGS, IYENGAR, LETZ, AND POLLITZ
Given a morphism of dg algebras ϕ : A → B , then (finite) building in D ( B )implies (finite) building in D ( A ). More precisely, if M ⊢ B N (or M | = B N ) then,viewing M and N as dg A -modules by restricting scalars along ϕ , one has that M ⊢ A N (or M | = A N , respectively). As in [9], a dg A -module M is proxy small if there exists a small dg A -module K such that M | = K and K ⊢ M . We say that K is a small proxy for X . Evidentlysmall objects are proxy small. When A is a commutative noetherian ring, somethese conditions can be expressed can be expressed in terms of support; see 1.13.The following definition is central to this work. Let ϕ : A → B be a morphism of dg algebras. We say that proxy smallnessascends along ϕ if each dg B -module that is proxy small in D ( A ) is proxy small in D ( B ). The phrase proxy smallness descends along ϕ means that each proxy smalldg B -module is also proxy small in D ( A ). Often the focus will be on ascent (ordescent) of proxy smallness for dg modules in a subcategory C of D ( B ), and thenwe speak of “proxy smallness of objects in C ” ascending/descending along ϕ .For example, proxy smallness ascends and descends along ϕ when it is a quasi-isomorphism, for then the base change functor F := B ⊗ L A − : D ( A ) → D ( B ) is anequivalence of categories, with quasi-inverse the restriction from D ( B ) to D ( A ).From the definition it is clear that whether an object in D ( A ) is small or notdepends only the structure of D ( A ) as a triangulated category; in particular, thisproperty is preserved under equivalences of triangulated categories. The same isthus true of proxy small. More precisely: If A, B are dg algebras and F : D ( A ) → D ( B ) is an exact equivalence, then F preserves arbitrary direct sums. Hence a dg A -module M is small if and only if the dg B -module F ( M ) is small. Moreover, X is a small proxy for X , if and only if F ( X ) is a small proxy for F ( M ). Lemma 1.6.
Let A be a dg algebra. The following statements hold. (1) The dg A -module H ( A ) builds any M in D ( A ) with H i ( M ) = 0 for | i | ≫ . (2) If H i ( A ) = 0 for i ≫ , then H ( A ) is proxy small if and only if H ( A ) | = A .Proof. Set B := H ( A ) and let ε : A → B be the augmentation map.(1) We verify this claim by an induction on the number of nonzero homologymodules in M . Set i := inf H( M ); we may assume this is finite, else M ≃
0. In D ( A ) soft truncation yields an exact triangle N −→ M −→ Σ i H i ( M ) −→ where the induced map H n ( N ) → H n ( M ) is bijective for n = i and H i ( N ) = 0.The dg A -module structure on H i ( M ) is the one induced via ε . Now B ⊢ H i ( M ) in D ( B ) and hence also in D ( A ). Since N has one fewer nonzero homology modulesthan M the induction hypothesis yields that B ⊢ A N . The exact triangle abovethen implies that B ⊢ A M .(2) The non-trivial implication is that when B is proxy small then B | = A .Suppose K is a small proxy for B ; in particular, K builds B . By part (1) andthe boundedness hypothesis, B builds A , and it follows that so does K . However K and A are both small objects so K finitely builds A ; see (1.2.1). As B | = K ,transitivity implies once again that B | = A . (cid:3) Example 1.7.
The preceding result does not extend to dg algebras A with H i ( A ) =0 for infinitely many i . For example, if R is any commutative ring and A := R [ x ], ELATIVE COMPLETE INTERSECTIONS 5 viewed as a dg algebra with | x | ≥ R = H ( A ) is smallin D ( A ), and so proxy small. However it does not build A , let alone finitely. Proposition 1.8.
Let R be a commutative noetherian ring, A a dg R -algebra, andlet M, N be dg A -modules. Assume the R -modules H( A ) , H( M ) , and H( N ) arefinitely generated. Then M | = A N if and only if M p | = A p N p for each p in Spec R .Proof. One way to verify this is to mimic the argument for [16, Theorem 4.5] toget the desired result. Another is to invoke the local-to-global principle [6, Theo-rem 5.10], for the triangulated category consisting of dg A -modules with homologyfinitely generated over R , viewed as an R -linear category. (cid:3) In this work the focus is on proxy small objects in D ( R ), the derived category overa commutative noetherian ring R . Next we recollect some results specific to thiscontext. We write D fg ( R ) for the subcategory of D ( R ) consisting of R -complexes M for which the R -module H( M ) is finitely generated. The following remark isobvious, but also obviously useful.While the small objects in D ( A ) form a thick subcategory, the class of proxysmall objects does not seem to have any good structure, except the one below: If M is proxy small, then so is P ⊗ L R M for any perfect complex P .Indeed, this follows from that fact that if M builds, respectively finitely builds, N , then P ⊗ L R M builds, respectively finitely builds, P ⊗ L R N .For any ring R and maximal ideal m of R , the R -module R/ m is proxy small:The Koszul complex on a finite generating set for the ideal m is a small proxy; see[10, Proposition 4.7] for details. Let R be a commutative noetherian ring and Spec R its spectrum, the col-lection of prime ideals in R with the Zariski topology. The closed subsets are givenby V ( I ), where I is an ideal and V ( I ) is the subset of prime ideals containing I .The support of an object M in D ( R ) is the subset of Spec R given bysupp R M := { p ∈ Spec R | k ( p ) ⊗ L R M } where k ( p ) is the residue field, R p / p R p , at p . For M ∈ D fg ( R ) one hassupp R M = { p ∈ Spec R | H( M ) p = 0 } = V (ann R (H( M )))and hence a closed subset of Spec R . For example, if K is the Koszul complex ona finite generating set for an ideal I , then supp R K = V ( I ).For any subset U of Spec R , the set of R -complexes M with supp R M ⊆ U is alocalizing subcategory of D ( R ). In particular if M ⊢ R N , then supp R M ⊇ supp R N .It follows that if M is proxy small, then supp R M is a closed subset of Spec R .The observation above relating supports to building has a perfect converse, es-tablished by Neeman [17, Theorem 2.8]. If M, N are objects in D ( R ) with supp R M ⊇ supp R N , then M ⊢ R N .Using this result and (1.2.1) Neeman deduces the result below concerning finitebuilding, first proved by Hopkins [13] using different techniques. If M, N are small objects in D ( R ) with supp R M ⊇ supp R N , then M | = R N .The preceding results imply the following characterization of proxy small objects. BRIGGS, IYENGAR, LETZ, AND POLLITZ
Corollary 1.13.
Let R be a commutative noetherian ring and M an R -complex.Then M is proxy small if and only if it finitely builds a small object with supportequal to supp R M ; if and only if M finitely builds the Koszul complex on any finitesubset x of R with V ( x ) ⊆ supp R M . (cid:3) Hochschild cohomology
In this section we discuss the (derived) enveloping algebras and Hochschild co-homology of dg R -algebras; [5] is a suitable reference for this material. We aregoing to be interested in two aspects: One is Hochschild cohomology as a sourceof operators on the derived category of A . The other is the smallness, and proxysmallness, of A as a module over A e R . In what follows, dg algebras will be assumedto be graded-commutative: a · b = ( − | a || b | b · a for a, b in A . Given morphisms of dg R -algebras β : B → A and ζ : C → A , a morphism φ : B → C is over A if ζφ = β , that is to say, if the following diagram commutes B CA φβ ζ
We say that B and C are quasi-isomorphic over A to mean that there is a sequenceof quasi-isomorphisms over A linking B to C . In this situation it is easy to see,given the discussion in 1.3, that A is small, respectively, proxy small, in D ( B ) ifand only if it is small, respectively proxy small, in D ( C ). Let A be a dg R -algebra and A e R := A ⊗ L R A , its derived enveloping algebra. When the graded R -module underlying A is flat onecan take A e R = A ⊗ R A .In general, if A ′ ε ′ −→ A is a semiflat dg R -algebra resolution of A , then A e R isrepresented by A ′ ⊗ R A ′ . Different semiflat resolutions yield dg algebras that arequasi-isomorphic over A : Let µ ′ denote the composition of morphisms of dg algebras A ′ ⊗ R A ′ ε ′ ⊗ ε ′ −−−→ A ⊗ R A a ⊗ b ab −−−−−−→ A Then if A ′′ ε ′′ −→ A is another semiflat resolution, then A ′ ⊗ R A ′ and A ′′ ⊗ R A ′′ arequasi-isomorphic over A . For this reason we write µ : A e R → A to denote (somerepresentative) of the map above.From 2.1 it follows that the property that A is small, or proxy small, in D ( A e R )is independent of the choice of a semiflat resolution of A . In fact, this condition isequivalent to A being small, respectively, proxy small, as a dg B -module for anymorphism B → A of dg R -algebras with B and A e R quasi-isomorphic over A . The Hochschild, or Shukla, cohomology of a dg R -algebra A with coefficientsin a dg A -module M is HH( A/R ; M ) := Ext A e R ( A, M ) , where A is viewed as a dg A e R -module via µ . We abbreviate HH( A/R ; R ) toHH( A/R ). This is a graded-commutative R -algebra. In what follows we exploit the ELATIVE COMPLETE INTERSECTIONS 7 fact that it acts on D ( A ), in the sense of [15]. This action comes about as follows:For any class α in HH( A/R ) and M a dg A -module, let χ M ( α ) : M → Σ | α | M be the morphism in D ( A ) defined by the commutative diagram M ⊗ A A M ⊗ A Σ | α | AM Σ | α | M . M ⊗ A α ≃ ≃ χ M ( α ) Thus we get a homomorphism of graded R -algebras χ M : HH( A/R ) −→ Ext A ( M, M ) . We denote this χ AM when the dg algebra A needs to be specified. This map has theproperty that for N in D ( A ) and element ζ ∈ Ext A ( M, N ) one has χ N ( α ) ◦ ζ = ( − | α || ζ | ζ ◦ χ M ( α ) . In particular χ M ( α ) lies in the graded-center of Ext A ( M, M ); see [15] for details.
Fix an α in HH( A/R ) and an M in D ( A ). We write M//α for Cone( χ M ( α )),so there is an exact triangle M χ M ( α ) −−−−−→ Σ | α | M −→ M//α −→ in D ( A ). The result below is one of the main reasons for our interest for the actionof Hochschild cohomology on D ( A ). We do not know if such a statement holdswhen α is an arbitrary element in the center of D ( A ). Lemma 2.5. If M (finitely) builds N , then M//α (finitely) builds
N//α .In particular, if M is proxy small then so is M//α .Proof.
The key point is that the action of α on D ( A ) is induced by a tensor product: M//α = M ⊗ L A A//α .
Hence it commutes with exact triangles, retracts, and (possibly infinite) directsums. This implies the desired result. (cid:3)
Next we record a computation of Hochschild cohomology that will be requiredoften; for example, see Lemma 3.4 and especially its proof.
Let ϕ : R → S be a surjective homomorphism of commutative rings, withkernel I . For each S -module M there is an isomorphism of S -modules δ ϕ ( M ) : Hom S ( I/I , M ) ∼ = −−→ HH ( S/R ; M ) , functorial in M and also in the R -algebra S . The functoriality in the ring argumentmeans the following: Given surjective homomorphisms of rings R e ϕ −→ e S ˙ ϕ −→ S with ˙ ϕ e ϕ = ϕ , for e I = Ker( e ϕ ) the following diagram is commutative(2.6.1) Hom e S ( e I/ e I , M ) Hom S ( I/I , M )HH ( e S/R ; M ) HH ( S/R ; M ) δ e ϕ ( M ) δ ϕ ( M ) BRIGGS, IYENGAR, LETZ, AND POLLITZ Here the S -module M is viewed as an e S -module by restriction of scalars along ˙ ϕ .The map δ ϕ ( M ) is part of a family induced by the universal Atiyah class of ϕ ,and involves the cotangent complex of ϕ . We only need δ which can be definedquite simply: The multiplication map µ : S e R → S embeds in an exact triangle J −→ S e R µ −−→ S −→ in D ( S e R ). For any S -module M viewed as an S e R -module via µ , the exact triangleabove induces isomorphismsExt iS e R ( J, M ) ∼ = Ext i +1 S e R ( S, M ) for i ≥ S e R ( J, M ) ∼ = Hom S ( I/I , M ); with this identification δ ϕ ( M ) isthe isomorphism above for i = 1. The stated functoriality is easily verified.As to the claim, as ϕ is surjective the natural map H ( S e R ) = S ⊗ R S → S is anisomorphism, so from the exact triangle above we obtainH i ( J ) = ( i ≤ Ri ( S, S ) for i ≥ . It is a standard computation that Tor R ( S, S ) ∼ = I/I ; this yields the first isomor-phism below Ext S e R ( J, M ) ∼ = Ext S e R ( I/I , M ) ∼ = Hom S ( I/I , M ) . The second one holds because the action of S e R on I/I and M factors through S .3. Surjective maps
In this section we prove Theorem B from the introduction. Throughout R willbe a commutative noetherian ring. A surjective homomorphism ϕ : R → S is complete intersection if Ker( ϕ ) canbe generated by a regular sequence; it is locally complete intersection if it has thisproperty locally: For each prime q ∈ Spec S , the map ϕ q is complete intersection.There is no distinction between the conditions when R is local. It was proved in[10, Theorem 9.1] that proxy smallness ascends and descends, in the sense of 1.5,along complete intersection maps. The converse is part of the result below: Theorem 3.2.
Let ϕ : R → S be a surjective homomorphism of commutative noe-therian rings. The following conditions on ϕ are equivalent: (1) ϕ is locally complete intersection; (2) S | = S e R S e R ; (3) S is proxy small in D ( S e R ) ; (4) proj dim R S is finite and proxy smallness ascends along ϕ ; (5) proj dim R S is finite and proxy smallness of objects in D fl ( S ) ascends along ϕ . The condition that proj dim R S is finite is equivalent to S being small in D ( R ),so condition (4) involves only the structure of the appropriate derived categories asabstract triangulated categories.The proof of Theorem 3.2 takes some preparation and is given in 3.8. It buildson ideas from [19], and extends that results therein, as is explained in 3.12. ELATIVE COMPLETE INTERSECTIONS 9
Let ϕ : ( R, m , k ) → S be a surjective map of local rings, and let ǫ : S → k bethe canonical surjection. It induces a morphism of dg S -algebras S e R → S ⊗ L R k . Oneof the standard diagonal isomorphisms yields a quasi-isomorphism of dg algebras( S ⊗ L R k ) ⊗ L S e R S ≃ k . This and adjunction yields the isomorphism in the definition of the following ho-momorphism of S -modules ψ S : HH( S/R ; k ) ∼ = Ext S ⊗ L R k ( k, k ) −→ Ext S ( k, k ) . The map heading right is induced by the morphism of dg algebras S → S ⊗ L R k ,and its compatibility with the augmentations to k .It is not hard to verify that the composition of the mapsHH( S/R ; S ) HH(
S/R ; ǫ ) −−−−−−−−→ HH(
S/R ; k ) ψ S −−→ Ext S ( k, k )is nothing but the characteristic map χ k described in 2.3. Lemma 3.4.
Let ( R, m , k ) e ϕ −→ e S ˙ ϕ −→ S be surjective local homomorphisms with ϕ = ˙ ϕ e ϕ . Set e I := Ker e ϕ and I := Ker ϕ . If the induced map e I/ m e I → I/ m I isinjective, then for the induced maps Ext S ( k, k ) Ext ϕ ( k,k ) −−−−−−−→ Ext e S ( k, k ) χ e Sk ←−−− HH ( e S/R ) one has an inclusion Im(Ext ϕ ( k, k )) ⊇ Im( χ e Sk ) .Proof. Set I := Ker( ϕ ). The essence of the proof is that there is a commutativediagram of e S -modulesHom S ( I/I , k ) Hom e S ( e I/ e I , k ) Hom e S ( e I/ e I , e S )HH ( S/R ; k ) HH ( e S/R ; k ) HH ( e S/R )Ext S ( k, k ) Ext e S ( k, k ) ∼ = δ ϕ ( k ) ∼ = δ e ϕ ( k ) ∼ = δ e ϕ ( e S ) ψ S ψ e S χ e Sk Ext ϕ ( k,k ) with surjective maps and isomorphisms as indicated. The δ maps are from 2.6.Given this the desired inclusion can be verified by chasing around the diagram.As to the commutativity of the diagram: The top left square is commutative by(2.6.1), whilst the top right one is commutative by the functoriality of δ e ϕ ( − ). Thevertical maps ψ S and ψ e S are from 3.3, and the commutativity of that square is byfunctoriality of the construction, which is readily verified. The commutativity ofthe triangle has been commented on already in 3.3.It remains to verify the surjectivity of the map in the top left square: Since themap of k -vectorspaces e I/ m e I → I/ m I is injective, it is split-injective. Thereforeapplying Hom k ( − , k ) yields that surjectivity of the map belowHom k ( I/ m I, k ) ։ Hom k ( e I/ m e I, k ) . This is the surjection in the top left square. This completes the proof of the claimsabout the commutative diagram above, and hence that of the result. (cid:3)
In the proof of Theorem 3.2 we also need a criterion for detecting small complexesthrough the action of Hochschild cohomology.
Let ϕ : ( R, m , k ) → S be a local complete intersection map, set I := Ker( ϕ )and N := Hom S ( I/I , S ), the normal module of ϕ . Since the Hochschild cohomol-ogy algebra HH( S/R ) is graded-commutative, the map δ ϕ ( S ) : N → HH ( S/R ),described in 2.6, induces a homomorphism of S -algebrasSym S ( N ) −→ HH(
S/R ) . Since ϕ is complete intersection, the S -module N is free, of rank the codimensionof S in R , and the map above is bijective; see [3, Proposition 2.6]. In particular,the ring HH( S/R ) is noetherian.There is a natural HH(
S/R )-module structure on Ext S ( M, N ), for any
M, N in D ( S ); see 2.3. Given a ideal a of HH( S/R ), an HH(
S/R )-module is a -power torsion if each of its elements is annihilated by a power of a . Since the ring HH( S/R ) isnoetherian, the ideal a can be generated by finitely many elements, say a , . . . , a n ,and a module is a -power torsion if and only if it is ( a i )-power torsion for each i . Lemma 3.6.
Let M be an S -complex and a ⊆ HH(
S/R ) an ideal. If an S -complex N is such that the HH(
S/R ) -module Ext S ( M, N ) is a -power torsion, then so is Ext S ( M, L ) for any L finitely built from N .Proof. The subcategory of D ( S ) with objects W for which the HH( S/R )-moduleExt S ( M, W ) is a -power torsion is thick. This implies the desideratum. (cid:3) Lemma 3.7.
Let φ be complete intersection and M an S -complex. If M is smallin D ( R ) and the HH(
S/R ) -module Ext S ( M, k ) is HH > ( S/R ) -power torsion, then M is small in S .Proof. Since M is small in D ( R ), the HH( S/R )-module Ext S ( M, k ) will be finitelygenerated; see, for example, [3, Corollary 6.2]. It is also HH > ( S/R )-power torsion,hence Ext iS ( M, k ) = 0 for i ≫
0. Thus M is small in S . (cid:3) Proof of Theorem 3.2. (1) ⇒ (3) When R is a local ring, Ker( ϕ ) is generatedby a regular sequence, so the desired result is contained in the proof of [10, Theo-rem 9.1]. The hypothesis is local on Spec S . We claim that so is the conclusion:As ϕ is locally complete intersection its flat dimension is finite so the R -moduleH( S e R ) = Tor R ( S, S ) is finitely generated. Thus Proposition 1.8 applied to the dg S -algebra A := S e R and M := S yields that S is proxy small in D ( S e R ) if and onlyif S p is proxy small in D (( S e R ) p ). It remains to observe that ( S e R ) p ∼ = ( S p ) e R p ∩ R .(2) ⇔ (3) Since R → S is surjective, H ( S e R ) = S . Thus Lemma 1.6 part (2) yieldsthe desired equivalences.(2) ⇒ (4) The assumption that S | = S e R S e R implies proxy smallness ascends along ϕ ; see [9, Theorem VI]. It remains to verify that S is small in D ( R ). Since S | = S e R S e R ,for any S -module M applying ( − ) ⊗ L S M yields M | = R ( S ⊗ L R M ), and hence H( S ⊗ L R M ) is bounded. Since H( S ⊗ L R M ) = Tor R ( S, M ) it follows that flat dim R S < ∞ ,and as S is finite over R we can conclude that S is small over R .(4) ⇒ (5) is a tautology.(5) ⇒ (1) The desired conclusion can be checked locally at the maximal ideals of S , and the hypothesis is easily seen to descend to localization to any such ideal.Thus we may assume ϕ : ( R, m , k ) → ( S, n , k ) is a surjective local homomorphism. ELATIVE COMPLETE INTERSECTIONS 11
Choose a maximal regular sequence x in Ker( ϕ ) \ m Ker( ϕ ) and set e S := R/ ( x ).The map ϕ factors as R e S S , e ϕ ϕ ˙ ϕ where e ϕ is complete intersection and grade e S S = 0; the latter condition implies thateither e S = S or S is not small in D ( e S ); see [8, Corollary 1.4.7]. We shall prove thatunder the hypothesis, S is small in D ( e S ), yielding that ϕ = e ϕ , and hence that ϕ isa complete intersection, as desired.The argument involves a series of reductions. To make it a bit more transparentit helps to write ˙ ϕ ∗ : D ( S ) → D ( e S )for the restriction, and similarly e ϕ ∗ and ϕ ∗ . One has ϕ ∗ ≃ e ϕ ∗ ◦ ˙ ϕ ∗ , as functors.The desired conclusion is that ˙ ϕ ∗ ( S ) is small. Let K be the Koszul complex ona set of generators for the maximal ideal of S . It suffices to verify: Claim.
The e S -complex ˙ ϕ ∗ ( K ) is small.Indeed this is a standard reduction; see [10, Remark 5.6].To verify the previous claim, it suffices to verify: Claim.
The action of the e S -algebra E := HH( e S/R ) on the module Ext e S ( ˙ ϕ ∗ ( K ) , L )is E > -power torsion, for any e S -complex L .Indeed, since ϕ ∗ ( S ) is small in D ( R ), by hypothesis, so is ϕ ∗ ( K ) for it is finitelybuilt out of ϕ ∗ ( S ). Given these properties of K , if the claim holds then Lemma 3.7applied with M := ˙ ϕ ∗ ( K ) yields that ˙ ϕ ∗ ( K ) is small, as desired. Here we use thefact that R → e S is complete intersection, by construction.Observe that we only needed to verify the preceding claim when L = k . However k has a different role to play in the sequel, and the argument may be easier to followfor a general L .For any element s in E , we write k//s for the mapping cone of the image χ e Sk ( s )of s in Ext e S ( k, k ). The preceding claim is a consequence of: Claim.
There exists a set of generators s , . . . , s c for the e S -module E such thateach k//s i finitely builds ˙ ϕ ∗ ( K ).Indeed, it is easy to verify that the E -module Ext e S ( k//s, L ) is annihilated by s for any s ∈ E . Then, given the claim, Lemma 3.6 implies that the E -moduleExt e S ( ˙ ϕ ∗ ( K ) , L ) is ( s i )-power torsion for each i . As noted in 3.5, the ideal E isgenerated by E , so we obtain that Ext e S ( ˙ ϕ ∗ ( K ) , L ) is E > -power torsion, as desired.The next reduction invovles the mapsExt S ( k, k ) Ext ϕ ( k,k ) −−−−−−−→ Ext e S ( k, k ) χ e Sk ←−−− HH ( e S/R )We verify the preceding claim by verifying:
Claim.
There exist elements t , . . . , t c in Ext S ( k, k ), and a set of generators s , . . . , s c for the e S -module E , such that the images of t i and s i under Ext ϕ ( k, k ) and χ e Sk ,respectively, coincide for each i . Indeed, suppose this claim holds. Then the hypothesis means that for each i there is a commutative diagram k Σ kk Σ k ∼ = χ e Sk ( s i ) ∼ =˙ ϕ ∗ ( t i ) in D ( e S ). As ˙ ϕ ∗ is exact it follows that the restriction of the mapping cone of t i ,namely k//t i , is isomorphic to that of s i . That is to say, there is an isomorphism˙ ϕ ∗ ( k//t i ) ∼ = k//s i in D ( e S ).Since k is proxy small in D ( e S ), so is k//s i ; this uses the fact s i comes from E ;see Lemma 2.5. Thus ˙ ϕ ∗ ( k//t i ) is proxy small in D ( e S ). Since R → e S is completeintersection, this implies that the R -complex ϕ ∗ ( k//t i ) ∼ = e ϕ ∗ ( ˙ ϕ ∗ ( k//t i ))is proxy small in D ( R ); this is by the already established implication (1) ⇒ (4).Evidently the S -complex k//t i is in D fl ( S ). Thus, the hypothesis of 3.2 now yieldsthat k//t i is proxy small in D ( S ). Therefore k//t i finitely builds K in D ( S ), for each i ; this is by Corollary 1.13. It follows that ˙ ϕ ∗ ( k//t i ), that is to say, k//s i , finitelybuilds ˙ ϕ ∗ ( K ) for each i , as desired.Finally, the preceding claim itself is an immediate consequence of Lemma 3.4,with s , . . . , s c any set of generators for the the e S -module E . This wraps up theproof that (5) ⇒ (1), and thereby that of Theorem 3.2. (cid:3) In Theorem 3.2, it seems plausible one can relax the hypothesis in (4) to:proj dim R S is finite and any M in D fg ( S ) that is small in D ( R ) is proxy smallin D ( S ). This condition already implies ϕ is quasi-Gorenstein, that is to say:RHom R ( S, R ) ≃ Σ c S in D ( S ), where c = dim R − dim S ; see [10, Theorem 6.7].A shortcoming in the statement of Theorem 3.2(5) is that it involves all of D fl ( S ).If R is local one needs to check the hypothesis only on a finite number of complexesthat can be specified in advance. This is clarified in the discussion below. Let ( R, m , k ) be a local ring and let π ( R ) denote its homotopy Lie algebra;see [2, Chapter 10]. This is a naturally constructed graded subspace of the graded k -vectorspace Ext R ( k, k ), so an element ζ ∈ π n ( R ) represents a morphism k → Σ n k in D ( R ). For what follows we care only about π ( R ), and that can be made explicit.Let ρ : Q → R be a minimal Cohen presentation of R ; that is to say, Q is a regularlocal ring with dim Q equal to the embedding dimension of R . Such a presentationexists when, for example, R is m -adically complete; this is part of Cohen’s structuretheorem [8, Theorem A.21]. With J := Ker( ρ ) the image of the compositionHom R ( J/J , k ) δ ρ ( k ) −−−→ ∼ = HH ( R/Q ; k ) ψ R −−−→ Ext R ( k, k )is precisely π ( R ), and the composite map is a bijection onto its image; see [2,Example 10.2.2] or [22].For any surjective local homomorphism ϕ : R → S one gets a map π ( ϕ ) : π ( S ) −→ π ( R ) , ELATIVE COMPLETE INTERSECTIONS 13 of k -vectorspaces, making π ( − ) into a functor. Moreover the image of the map ψ S : HH ( S/R ; k ) → Ext S ( k, k )is Ker( π ( ϕ )), and this is an isomorphism when R → S is complete intersection.The observation is a key ingredient in the proof of the following result. Theorem 3.11.
Let ϕ : ( R, m , k ) → S be a surjective homomorphism of local ringswith proj dim R S finite. The following conditions are equivalent: (1) ϕ is complete intersection; (2) For each t ∈ Ker( π ( ϕ )) the S -complex k//t is proxy small. (3) For some generating set t , . . . , t c for the k -vectorspace Ker( π ( ϕ )) , the S -complexes k//t i are proxy small.Proof. (1) ⇒ (2) Set I := Ker( ϕ ). Since ϕ is complete intersection, the S -module I/I is free. Thus the natural map HH ( S/R ) → HH ( S/R ; k ) is surjective, for itis identified, via δ ϕ , with the mapHom S ( I/I , S ) −→ Hom S ( I/I , k ) . It follows that the characteristic map χ Sk is surjective onto its image, that is to say,onto Ker( π ( ϕ )). The desired conclusion then follows from Lemma 2.5.(2) ⇒ (3) is a tautology.(3) ⇒ (1) The proof for this implication follows that of the (5) ⇒ (1) in Theo-rem 3.2. We keep the notation from that argument. Recall the factorization R e ϕ −→ e S ˙ ϕ −→ S of ϕ , with e ϕ defined by a maximal regular sequence in Ker( ϕ ) \ m Ker( ϕ ). Recallthat E = HH( e S/R ; k ), and consider the mapsExt S ( k, k ) Ext ϕ ( k,k ) −−−−−−−→ Ext e S ( k, k ) χ e Sk ←−−− HH ( e S/R ) . We pick up the proof at the last claim stated there; rather, we recast it as follows:
Claim.
There exists a set of generators s , . . . , s c for the e S -module E , such thatthe images of t i and s i under Ext ϕ ( k, k ) and χ e Sk , respectively, coincide for each i .Once we verify this claim, using the hypothesis that k//t i is proxy small in D ( S ),and arguing as in the proof of Theorem 3.2, leads to the desired conclusion.As to verifying the claim, the functoriality of π ( − ) gives a commutative diagram π ( S ) π ( e S ) π ( R ) . π ( ˙ ϕ ) π ( ϕ ) π ( e ϕ ) Thus one gets an induced map π ( ˙ ϕ ) : Ker( π ( ϕ ) → Ker( π ( ˙ ϕ )). From these ob-servations, and (2.6.1), one obtains a commutative diagram of k -vectorspacesHH ( S/R ; k ) HH ( e S/R ; k )Ker( π ( ϕ )) Ker( π ( e ϕ )) . ψ S ψ e S ∼ = π ( ˙ ϕ )4 BRIGGS, IYENGAR, LETZ, AND POLLITZ One has an isomorphism on the right because R → e S is complete intersection, byconstruction. It remains to observe that, by the same token, there is an isomorphismHH ( e S/R ; M ) ∼ = HH ( e S/R ) ⊗ S M .
Therefore the natural map E = HH ( e S/R ) −→ HH ( e S/R ; k )is surjective, and any generating set for the k -vectorspace HH ( e S/R ; k ) lifts to agenerating set for the S -module E , by Nakayama’s lemma. The composition of themap above with ψ e S is precisely χ e Sk . Thus choosing s , . . . , s c to be any preimagesof π ( ϕ )( t ) , . . . , π ( ϕ )( t c ) through χ e Sk justifies the claim. (cid:3) A local ring ( S, n ) is complete intersection if for some (equivalently, any)Cohen presentation ϕ : R → b S of the n -adic completion of S , the map ϕ is completeintersection; see [8, Section 2.3]. A commutative noetherian ring S is locally com-plete intersection if it is complete intersection at each prime ideal of S . Theorem 3.2applied locally to Cohen presentations recovers a characterization of complete in-tersections established in [19, Theorem 5.2] and [16, Theorem 6.6]. Corollary 3.13.
Let S be a commutative noetherian ring. The following conditionsare equivalent: (1) S is locally complete intersection; (2) Each object in D fg ( S ) is proxy small; (3) Each object in D fl ( S ) is proxy small. (cid:3) Factorization of locally complete intersection maps
In this section we demonstrate the strength of Theorem 3.2 by deducing somefundamental results on complete intersection rings and morphisms. We restrictourselves to treating surjective local maps for reasons laid out in Remark 5.3.The forward implication in the result below is easy to prove directly from thedefinitions; it is equally simple to deduce it from Theorem 3.2. The converse state-ment is due to Avramov [1, 5.7]. The proof in op. cit. is complicated and involvesnontrivial properties of Andr´e-Quillen homology. The proof presented below is moreelementary and natural from the perspective of ascent of the proxy small property.
Corollary 4.1.
Let R ϕ −→ S ψ −→ T be surjective local homomorphisms.If the maps ϕ and ψ are complete intersection, then so is ψ ◦ ϕ . The converseholds if, in addition, proj dim S T is finite.Proof. We make repeated use of Theorem 3.2 without specific reference.Suppose ϕ and ψ are complete intersection. Since R | = R S and S | = S T , andhence S | = R T , it follows that R | = R T . And given an M ∈ D ( T ), if M is proxysmall in D ( R ), then it is proxy small in D ( S ), since ϕ is complete intersection, andhence in D ( T ), since ψ is complete intersection.Now suppose ψ ◦ ϕ is complete intersection and proj dim S T is finite, that is tosay, S | = S T . The first condition implies R | = R T and then the second one implies R | = R S , by [10, Remark 5.6]. It follows that ψ is complete intersection: Any T -complex that is proxy small in D ( S ) is proxy small in D ( R ), since R | = R S , andhence is proxy small in D ( T ), since ψ ◦ ϕ is complete intersection. ELATIVE COMPLETE INTERSECTIONS 15
Now ψ ◦ ϕ and ψ are both complete intersection; we prove that so is ϕ . Fix M in D ( S ) with nonzero finite length homology such that M is proxy small in D ( R ). It suffices to prove that M finitely builds a small object in D ( S ) withnonzero homology; this is where we need to use the assumption that H( M ) hasfinite length.Since S | = S T one gets that M | = S T ⊗ L S M . Let x be a finite set of elements in R whose images in S form a minimal generating set for the ideal Ker( ψ ), and let K bethe Koszul complex on x , with coefficients in R . Since ψ is complete intersection,the S -complex K ⊗ R S is a resolution of T over S . This justifies the second of thefollowing isomorphisms in D ( R ): K ⊗ R M ≃ ( K ⊗ R S ) ⊗ S M ≃ T ⊗ L S M .
The first one is by associativity of tensor products. Since M proxy small in R itfollows that so is T ⊗ L S M . However as T ⊗ L S M is in D ( T ), the complete intersectionproperty of ψ ◦ ϕ implies that T ⊗ L S M is proxy small in D ( T ) and hence also in D ( S ). Thus, T ⊗ L S M , and hence also M , finitely builds a small S -complex withnonzero homology. (cid:3) Remark . In Corallary 4.1 one cannot weaken the hypothesis in the conversethat proj dim S T is finite to ψ is proxy small. Indeed, let R be a regular local ringand consider surjective local homomorphisms( R, m , k ) ϕ −→ S ψ −→ k . Then ψ ◦ ϕ is complete intersection, and ψ is proxy small, but ϕ is complete inter-section if and only if S is, and that need not be the case.However the proof of Corollary 4.1 does yield also the following: Corollary 4.3.
Let R ϕ −→ S ψ −→ T be surjective local homomorphisms such that S and the map ψ ◦ ϕ are complete intersection. Then the rings R and T , and the map ϕ , are complete intersection.Proof. Since S is complete intersection, T is proxy small in D ( S ), the proof ofCorollary 4.1 shows that proxy smallness for objects in D fl ( T ) ascends along ψ .Thus, since S is complete intersection, so is T . But then ψ ◦ ϕ complete intersectionimplies R is complete intersection, again by the same token. We also know thatthe flat dimension of ϕ is finite, so R and S complete intersection means that ϕ iscomplete intersection. (cid:3) Essentially of finite type maps
A morphism ϕ : R → S of commutative rings is essentially of finite type if itis obtained as the localization of a finitely generated R -algebra; that it to say, ϕ admits a factorization(5.1.1) R e ϕ −−→ e R ˙ ϕ −−→ S where e R = U − R [ x ], where x := x , . . . , x n are indeterminates, U is a multiplica-tively closed subset in R [ x ], and ˙ ϕ is surjective. Such a map is smooth if ϕ is flat,and for each map of rings R → l with l a field, the ring l ⊗ R S is regular; that is tosay, the fibers of ϕ are geometrically regular; see [12]. For example, the map e ϕ inthe factorization above is smooth. The map ϕ is locally complete intersection if the surjection ˙ ϕ : e R → S is locallycomplete intersection in the sense of 3.1. This condition is independent of thefactorization of ϕ ; this fact is also implicit in the proof of Theorem 5.2 below.When ϕ is flat, it is locally complete intersection if and only if all its fibers arelocally complete intersection rings. For details, see [12].The result below is analogous to a classical characterization of smooth maps,recalled in the introduction, namely: ϕ is smooth if and only if it is flat and S is small in D ( S e R ); see [12, Proposition (17.7.4)] and [20, Theorem 1]. A crucialdifference: We do not have to assume ϕ is flat. This comes with a caveat: In thestatement S e R is the derived enveloping algebra. Theorem 5.2.
Let R be a commutative noetherian ring and ϕ : R → S a morphismessentially of finite type. The following conditions are equivalent: (1) ϕ is locally complete intersection; (2) S is proxy small in D ( S e R ) . This result specializes to Theorem A from the Introduction.
Proof.
We reduce to the case where ϕ is surjective. Fix a factorization (5.1.1) of ϕ . As noted before, condition (1) means that ˙ ϕ is locally complete intersection.Consider the morphism of dg algebras θ : S e R −→ S e e R induced by e ϕ . The following claim implies that condition (2) holds for ϕ if andonly if it holds for ˙ ϕ . Claim.
Proxy smallness ascends and descends along θ .The crucial point is that θ can be obtained by base change of a locally completeintersection map, so it is (only a definition away from being) itself locally completeintersection. Here are the details.Consider the natural multiplication map µ : e R e R → e R . The diagonal isomorphismyields a quasi-isomorphism of dg algebras e R ⊗ L e R e R S e R = e R ⊗ L e R e R ( S ⊗ L R S ) ≃ S ⊗ L e R S = S e e R . Thus θ is the base change of µ along the morphism e R e R → S e R , that is to say, thereis a fiber-square e R e R RS e R S e e R . µ ˙ ϕ ⊗ L R ˙ ϕ θ Since e ϕ is smooth it is flat and e R e R may be taken to be the ordinary tensor product, e R ⊗ R e R . Moreover, the map µ is locally complete intersection and, in particular,small. It follows from the fiber-square above that θ is small as well so that proxysmallness descends along ϕ , by [10, Proposition 7.2].As to ascent, let M in D ( S e e R ) be proxy small as a dg module over S e R . ELATIVE COMPLETE INTERSECTIONS 17
Conversely, if S is proxy small over S e R , one gets that e R ⊗ L e R e R S is proxy smallover the dg algebra S e e R . As e ϕ is smooth, e R e R finitely builds e R so S ≃ e R e R ⊗ L e R e R S | = e R ⊗ L e R e R S in D ( S e e R ). In other words, S is proxy small over S e e R .Thus we can replace ϕ by e R → S and assume that ϕ is surjective. At this pointwe can apply Theorem 3.2. (cid:3) Remark . Theorem 5.2 is missing a characterization in terms of the exact functor D ( S ) → D ( R ), akin to the one in Theorem 3.2 (4). It is not reasonable to expectascent and descent of the proxy small property along maps that are not finite.There is a notion of proxy smallness with respect to a map, via the surjectivepart of the smooth-by-surjective factorizations in (5.1.1), but this is not entirelysatisfactory, partly because being essentially of finite type is not a condition thatcan be characterized purely in terms of categorical properties of derived categories.These questions are of interest if one wishes to import the ideas of this paperinto stable homotopy theory, where the requiring that a map of commutative ringspectra to be surjective, or essentially of finite type, is not sensible. There are manyrich examples of complete intersection like behavior in that context [7], and it wouldbe interesting to develop a notion of ascent of proxy smallness that captures them.6. Morphisms of schemes
Let X be a noetherian separated scheme, D ( X ) its derived category of quasi-coherent sheaves, and D b (Coh X ) the bounded derived category of coherent sheaves.We write Perf ( X ) for its full subcategory of perfect complexes. As in the affine case,these are precisely the small objects in D ( X ); see [24, Proposition 1.1]. The derivedtensor product induces an action of Perf ( X ) on D ( X ), as well as on D b (Coh X ). A thick ⊗ -ideal C of D ( X ) is thick subcategory that is closed under the actionof Perf ( X ), that is to say, when F is in C so is P ⊗ L F for any perfect complex P .The notion of a localizing ⊗ -ideal is the obvious one.Given objects F , G in D ( X ) we say that F finitely ⊗ -builds G if the latter is inevery thick ⊗ -ideal containing the former; and F ⊗ -builds G is each localizing ⊗ -ideal containing F also contains G . For a commutative noetherian ring R and theassociated scheme X := Spec( R ), ⊗ -building coincides with the notion of buildingdiscussed earlier, for every thick subcategory of D ( X ) is thick ⊗ -ideal.This leads to the notion of ⊗ -proxy smallness in D ( X ). Since this is the onlyflavor of proxy smallness considered here, we usually drop the qualifier “ ⊗ -”, andspeak of proxy smallness of objects in D ( X ).The following result, implicit in the proof of [23, Lemma 4.1] often reducesquestions of proxy smallness over schemes to the affine case. Lemma 6.2.
Let X = ∪ ni =1 U i be a finite cover. Then any F in D ( X ) is proxy smallif and only if its restriction F | U i is proxy small in D ( U i ) for each i . (cid:3) A quasi-compact scheme X is locally complete intersection if there exists afinite open cover X = ∪ ni =1 U i , such that U i is isomorphic to Spec( R ) where R is alocally complete intersection ring, in the sense of 3.12. Given the definition of locally complete intersection schemes and Lemma 6.2,the following result is an immediate consequence of Corollary 3.13.
Theorem 6.4.
Let X be a noetherian separated scheme. Then the following con-ditions are equivalent (1) X is locally complete intersection; (2) Each object in D b (Coh X ) is proxy small. (cid:3) In the same vein, Theorem 5.2 readily implies the following global statement.
Theorem 6.5.
Let f : Y → X be a flat, essentially of finite type, separated mor-phism of noetherian schemes, and δ : Y → Y × X Y the diagonal embedding. Then f is locally complete intersection if and only if δ ∗ O Y is proxy small in D ( Y × X Y ) . (cid:3) The hypothesis that f is flat, rather than of finite flat dimension as in previoussections, is to avoid invoking the machinery of derived algebraic geometry here. References
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Department of Mathematics, University of Utah, Salt Lake City, UT 84112, U.S.A.
E-mail address : [email protected] Department of Mathematics, University of Utah, Salt Lake City, UT 84112, U.S.A.
E-mail address : [email protected] Fakult¨at f¨ur Mathematik, Universit¨at Bielefeld, 33501 Bielefeld, Germany.
E-mail address : [email protected] Department of Mathematics, University of Utah, Salt Lake City, UT 84112, U.S.A.
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