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PROCEEDINGS OF THE
AMERICAN MATHEMATICAL SOCIETY
Volume 149, Number 2, February 2021, Pages 619–631
https://doi.org/10.1090/proc/15276
Article electronically published on December 16, 2020
DUALITY AND SYMMETRY OF COMPLEXITY
OVER COMPLETE INTERSECTIONS
VIA EXTERIOR HOMOLOGY
JIAN LIU AND JOSH POLLITZ
(Communicated by Sarah Witherspoon)
Abstract. We study homological properties of a locally complete intersection
ring by importing facts from homological algebra over exterior algebras. One
application is showing that the thick subcategories of the bounded derived cat-
egory of a locally complete intersection ring are self-dual under Grothendieck
duality. This was proved by Stevenson when the ring is a quotient of a regular
ring modulo a regular sequence; we offer two independent proofs in the more
general setting. Second, we use these techniques to supply new proofs that
complete intersections possess symmetry of complexity.
Introduction
Homological algebra over complete intersections is profoundly linked to the ho-
mological algebra over exterior algebras. This was clarified in [4] where Avramov
and Iyengar established a process to obtain homological information over com-
plete intersections from the corresponding results over graded Hopf algebras. Their
techniques provided new, easier proofs of many known results over complete inter-
sections.
For example, the complexity of a module measures the polynomial rate of growth
of its Betti numbers while the injective complexity measures the polynomial rate of
growth of the module’s Bass numbers. Using the process described above, Avramov
and Iyengar easily deduced that over complete intersections the complexity of a
module agrees with its injective complexity, and both of these values are bounded
above by the complexity of the residue field; they also employ their methods to
show the latter is exactly the codimension of the complete intersection.
In this article, we adopt techniques from [4] to acquire further information about
complete intersections. For the rest of the introduction Ris a commutative noe-
therian ring. The first main result is framed in terms of the derived category of R,
denoted D(R).
We let Df(R) denote the full subcategory of D(R) consisting of those complexes
of R-modules whose total homology is finitely generated. It inherits the structure
Received by the editors June 15, 2020, and, in revised form, July 11, 2020.
2020 Mathematics Subject Classification. Primary 13D09; Secondary 13D07, 13H10, 16E45.
Key words and phrases. Complete intersections, thick subcategories, exterior algebra, Koszul
complex, DG algebra, DG module, support variety, duality, complexity.
The first author thanks the China Scholarship Council for financial support to visit Srikanth
Iyengar at the University of Utah.
The second author was supported by the National Science Foundation under Grant No.
1840190.
c
2020 American Mathematical Society
619
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620 JIAN LIU AND JOSH POLLITZ
of a triangulated category from D(R). Recently, there has been much interest
in understanding the structure of thick subcategories of Df(R), see for example
[3, 8, 9, 12, 15, 17, 19]. Our first main result is the following:
Theorem 1. If Ris locally complete intersection, then each thick subcategory of
Df(R)is self-dual under Grothendieck duality. That is, for any thick subcategory
Tof Df(R)and object Min T,RHomR(M,R)is in T,aswell.
Stevenson in [17, 4.11] proved the result under the additional assumption that
Ris a quotient of a regular ring modulo a regular sequence. One can also deduce
Theorem 1 from [17, 4.11] in conjunction with recent results of Letz [12, 3.12 &
4.5]; details are provided in Remark 3.4.
In this article, we present two proofs of Theorem 1 both of which rely on a local-
to-global principle of Benson, Iyengar and Krause (see 1.6) and the structure of
thick subcategories in the derived category of an exterior algebra over a field (cf.
[8]). The first proof uses the theory of cohomological support discussed in Section
2. Namely, we show that the containment of thick subcategories is encoded in the
support varieties of Avramov and Buchweitz defined in [2] (see Theorem 3.1 for a
precise statement).
The second proof makes direct use of the graded Hopf algebra structure of the
exterior algebra to show that thick subcategories over an exterior algebra on gen-
erators of homological degree one are fixed by Grothendieck duality (see Theorem
4.1). It is worth noting both proofs avoid much of the geometric and tensor tri-
angulated category machinery used by Stevenson, making the proofs in this article
simpler even in the case that Ris a quotient of a regular ring modulo a regular
sequence.
As a consequence of Theorem 1 we obtain asmyptotic information over complete
intersections. For example, we recover a result of Avramov and Buchweitz [2, 6.3]
that says the eventual vanishing of Ext is equivalent to the eventual vanishing of
Tor over locally complete intersections (cf. Corollary 3.6). Furthermore, in the
local case we can use Theorem 1 to show complexity is symmetric in Mand N;
recall the complexity of a pair of objects Mand Nof Df(R)isthepolynomialrate
of growth of the minimal number of generators of Extn
R(M,N) (see 5.1 for a precise
definition).
Theorem 2. If Ris complete intersection, then cxR(M,N)=cx
R(N,M)for each
pair of objects Mand Nin Df(R).
This was first proven by Avramov and Buchweitz [2]; an alternative proof was
provided by the second author in [14]. In contrast, we give two new proofs of this
result in this paper; both of which use the homological properties over exterior
algebras. The first proof deduces Theorem 2 from Theorem 1, illustrating how the
containment of thick subcategories and duality can provide asymptotic information.
The second proof directly links the complexity of a pair of objects Mand Nin
Df(R) with the complexity of a pair of objects in the derived category of an exterior
algebra over a field. This proof fills in a missing piece of the work in [4]; cf. [4, 6.9]
and the discussion in Remark 5.4.
1. Background, notation, and terminology
Throughout this article Rwill be a commutative noetherian ring.
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DUALITY AND SYMMETRY OF COMPLEXITY 621
1.1. (Locally) complete intersections. Suppose (R, m,k)islocal. Recallthe
embedding dimension of Ris dimkm/m2, the minimal number of generators for m,
and the codimens ion of R is
dimkm/m2−dim R.
Alocalring(R, m,k)iscomplete intersection if its m-adic completion
Ris iso-
morphic to Q/I where Qis a regular local ring and Iis generated by a Q-regular
sequence. In fact, the presentation can be chosen so that Qand Rhave the same
embedding dimension and Iis generated by celements where cis the codimension
of R.
More generally, Ris locally complete intersection provided that the local ring Rp
is complete intersection for each prime ideal pof R.
1.2. Derived category of a DG algebra. Let Abe a DG R-algebra. We briefly
discuss the derived category of DG A-modules and set notation used throughout
the rest of the article. See [3, Section 3] or [10, Chapter 6] for more details.
Let D(A) denote the derived category of (left) DG A-modules. Recall that D(A)
is a triangulated category with Σbeing the suspension functor; for each Xin
D(A), ΣXis the DG A-module given by ΣXi=Xi−1,a·(Σx)=(−1)|a|ax and
∂ΣX=−∂X.WeletDf(A) denote the full subcategory of D(A) consisting of those
objects Xof D(A) such that H(X) is a finitely generated graded H(A)-module.
Each DG A-module Xadmits a semiprojective resolution. That is, there exists
a surjective quasi-isomorphism P→Msuch that HomA(P, −) preserves surjective
quasi-isomorphisms. For any Yin D(A), we set
RHomA(X, Y ):=Hom
A(P, Y )
where P→Xis a semiprojective resolution and
ExtA(X, Y ):=H(RHomA(X, Y )) ,
which naturally inherits a graded ExtA(Y,Y )-ExtA(X, X )-bimodule structure.
1.3. Koszul complexes. Background on Koszul complexes can be found in [7,
Section 1.6]. We recall the necessary facts here.
For a list of elements x=x1,...,x
nin R, we set KosR(x) to be the Koszul
complex of xon R, which is regarded as a DG R-algebra in the usual way.
When Ris local with maximal ideal m,setKRto be the Koszul complex on
a minimal generating set for m. It is well-defined up to an isomorphism of DG
R-algebras.
Fix a prime ideal pof Rand let Mbe an object of D(R).We set
M(p):=Mp⊗RpKRp
which is a DG KRp-module. Restricting scalars along the morphism of DG algebras
Rp→KRpwe may regard M(p)asanobjectofD(Rp).
1.4. Koszul complexes over complete intersections. Let (R, m,k)becom-
plete intersection of codimension cand Λ be the exterior algebra over kon gen-
erators e1,...,e
cof homological degree 1. Let t:D(R)→D(KR) be the functor
−⊗
RKR.
By [4, 6.4] there is a quasi-isomorphism of DG algebras KRΛ that induces
an equivalence of triangulated categories j:D(KR)→D(Λ); this restricts to an
equivalence Df(KR)≡
−→ Df(Λ) that is compatible with Grothendieck duality (see
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622 JIAN LIU AND JOSH POLLITZ
[3, 3.6] or [4, 2.5]). Hence, when Ris complete intersection we have the following
composition
jt :Df(R)→Df(KR)≡
−→ Df(Λ);
this is the main bridge for importing results over graded exterior algebras to com-
plete intersections. Throughout the rest of the paper, jand twill always denote
the functors introduced here.
1.5. Thick subcategories. Let Abe a DG algebra and Tbe a full subcategory
of D(A). We say Tis thick if it is a triangulated subcategory that is closed under
taking direct summands. For an object Mof D(A), we let thickD(A)(M)denote
the smallest thick subcategory of D(A) containing M. This can be realized as the
intersection of all thick subcategories of D(A) containing M; alternatively, this has
an inductive construction discussed in [3, 2.2.4].
1.6. Local-to-global principle. The main results in the present paper rely on the
following local-to-global principle of Benson, Iyengar and Krause (see by [6, 5.10]).
Namely, for objects Mand Nof Df(R), Mis in thickD(R)(N) if and only if M(p)
is in thickD(Rp)(Np) for each prime ideal pof R. In particular, for Mand Nin
Df(R),
Mis in thickD(R)(N)⇐⇒ M(p)isin thickD(Rp)(N(p))
for each prime ideal pof R.
Indeed, suppose Mis in thickD(R)(N)andpis a prime ideal of R. By applying
−⊗
RKRpwe obtain that
M(p)isin thickD(KRp)(N(p)),
now we need only restrict along Rp→KRpto conclude the forward implication
holds. Conversely, since N(p) is an object of thickD(Rp)(Np) for each prime pwe can
use the local-to-global principle, listed above, to deduce that Mis in thickD(R)(N).
1.7. Homogeneous support. Let Sbe a commutative noetherian graded ring.
We let Spec∗Sdenote the homogeneous spectrum of S. That is, Spec∗Sconsists
of the homogeneous prime ideals of S.ForagradedS-module Xand p∈Spec∗S,
Xpdenotes the homogeneous localization of Xat p. The homogeneous support of
Xis
suppSX={p∈Spec∗S:Xp=0}.
2. Cohomological support varieties
Throughout this section we fix the following notation. Let (R, m,k)becomplete
intersection with codimension cand embedding dimension ν. Let Sdenote the
graded k-algebra k[χ1,...,χ
c]whereeachχihas homological degree −2. We set Λ
to be the exterior algebra over kon generators e1,...,e
cof homological degree 1.
Finally, let jand tbe the functors from 1.4.
2.1. In [16, Theorem 5], Sj¨odin described the graded k-algebra structure of ExtR(k, k).
It contains Sas a polynomial subalgebra in such a way that
ExtR(k, k)∼
=S⊗
kΣ−1kν
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DUALITY AND SYMMETRY OF COMPLEXITY 623
as graded S-modules; see also [1, 10.2.3] for more details. Thus, Sacts on ExtR(k, M )
through the ExtR(k, k)-action for each Min D(R). We define the cohomological
support of Mover Rto be
VR(M):=suppSExtR(k, M ).
2.2. By [4, 5.1] (see also [3, 7.4]), there is an isomorphism of graded k-algebras
ExtΛ(k, k)∼
=S.
For any Xin D(Λ), we define the cohomological support of Xover Λtobe
VΛ(X):=suppSExtΛ(k, X).
These varieties can detect the containment of thick subcategories in Df(Λ).Namely,
in [8, 4.4], Carlson and Iyengar showed
Xis in thickD(Λ)(Y)⇐⇒ VΛ(X)⊆VΛ(Y)
for any pair of objects Xand Yin Df(Λ). This essentially follows from the cele-
brated theorem of Hopkins [11, 11] and Neeman [13, 1.2] (see also [8, 3.2] for the
version needed) and a BGG correspondence (cf. [3, 7.4]).
There is a way to relate the supports defined over Rand Λ.This was first noticed
inthecasethatRis artinain [8, 5.11]; however, the same proof works without any
restriction on the Krull-dimension of Rand so we sketch it for the convenience of
the reader in the following remark and lemma.
2.3. First, there is a canonical injective map η:Ext
Λ(k, k)→ExtR(k, k)ofgraded
k-algebras that can be factored as
ExtΛ(k, k)∼
=
−→ ExtKR(k, k)→ExtR(k, k)
where the isomorphism is induced by the inverse of the equivalence jfrom 1.4.
Moreover, the image of ηis exactly the polynomial subalgebra Sof ExtR(k, k)
mentioned in 2.1. Therefore, the cohomological supports over Rand those over Λ
can naturally be thought of as subsets of the same Spec∗S. Moreover, we have the
following connection.
Lemma 2.4. For each Min D(R),VR(M)=V
R(tM)=V
Λ(jtM).
Proof. First, consider the isomorphisms of graded S-modules
ExtR(k, tM)∼
=ExtKR(tk, tM)
∼
=ExtΛ(jtk, jtM)
∼
=
ν
i=0
Σ−iExtΛ(k, jtM)(ν
i)
where the third isomorphism holds because tk∼
=ν
i=0 Σ−ik(ν
i)and j(k)k,see
[3, 3.9] for the latter. Also, we have the isomorphism of graded S-modules
ExtR(k, tM)∼
=
ν
i=0
ΣiExtR(k, M )(ν
i).
Therefore, the isomorphisms of S-modules show
VR(M)=V
R(tM)=V
Λ(jtM).
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624 JIAN LIU AND JOSH POLLITZ
Proposition 2.5. For Mand Nin Df(R),
tMis in thickD(R)(tN)⇐⇒ VR(M)⊆VR(N).
Proof. The forward direction is trivial from the first equality in Lemma 2.4.
Conversely, assume VR(M)⊆VR(N). Using Lemma 2.4, this reads as
VΛ(jtM)⊆VΛ(jtN).
Thus, 2.2 implies that jtMis an object of thickD(Λ) (jtN).As jis an equivalence we
conclude that tMis an object of thickD(KR)(tN). The result follows by restricting
scalars along the morphism of DG R-algebras R→KR.
We end this section with the following technical lemma which will be put to use
in Section 4. Note that since Λ has trivial differential, for each DG Λ-module X
we can negate differential of Xto obtain a DG Λ-module. Namely, let Xdenote
the DG Λ-module whose underlying graded Λ-module is Xand its differential is
∂X:=−∂X.When Λ is concentrated in even degrees, X∼
=Xas DG Λ-modules
(cf. 4.3). However, as the generators of Λ have degree 1 we do not know whether
these are isomorphic. Instead, we show they have the same cohomological support,
and hence, generate the same thick subcategory.
Lemma 2.6. If Xis in Df(Λ),thenVΛ(X)=V
Λ(X).In particular, we have the
following equality of thick subcategories:
thickD(Λ)(X)=thickD(Λ) (X).
Proof. By [8, 4.2], there exists a semiprojective resolution F
−→ kover Λ such that
Fadmits a DG S-module structure compatible with the S-actiononExt
R(k, Y )for
any Yin D(Λ).As khas trivial differential, the same is true of the semiprojective
DG Λ-resolution equipped with a DG S-module structure F
−→ k.
Define Φ : HomΛ(F, X )→HomΛ(F,X)givenby
α→ (−1)|α|α.
As Sis concentrated in even degrees this is an isomorphism of DG S-modules.
Therefore, H(Φ) establishes the following isomorphism of graded S-modules
ExtΛ(k, X)∼
=ExtΛ(k, X);
so Xand Xhave the same cohomological support. The equality of thick subcate-
gories now follows from 2.2.
3. Duality of thick subcategories via support
In this section we give the first proof of our result on the duality of thick sub-
categories over locally complete intersections (see Theorem 3.3). The main idea
behind it is that the theory of cohomological supports, discussed in Section 2, both
detects containment of thick subcategories and is unaffected by duality. The first
theorem addresses the former point while 3.2 the latter.
Theorem 3.1. Let Rbe locally complete intersection. For M,N in Df(R),
Mis in thickD(R)(N)⇐⇒ VRp(Mp)⊆VRp(Np)
for each prime ideal pof R.
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DUALITY AND SYMMETRY OF COMPLEXITY 625
Proof. First, assume Mis an object of thickD(R)N. Hence, Mpis an object of
thickD(Rp)Npand so it follows easily that VRp(Mp)⊆VRp(Np)forp∈Spec R.
Conversely, suppose VRp(Mp)⊆VRp(Np)foreachp∈Spec R. By Proposition
2.5, M(p)isinthickD(Rp)(N(p)) for each prime ideal pof R. Finally, we apply 1.6
to conclude that Mis in thickD(R)(N).
3.2. It is well-known that cohomological support over complete intersections is
closed under duality. That is,
VR(M)=V
R(RHomR(M,R))
for each Min Df(R), This was shown for closed points of Spec∗Sin [2, 3.3], and
the general setting was shown in [14, 4.1.5]. Alternatively, a new proof is obtained
in the present work by combining Lemma 2.4 and Theorem 4.1; this proof sticks
with the theme of establishing results for complete intersections by passing to an
exterior algebra.
Theorem 3.3. If Ris locally complete intersection, then every thick subcategory
of Df(R)is closed under RHomR(−,R). In particular, for each Min Df(R)
thickD(R)(M)=thickD(R)(RHomR(M,R)).
First proof of Theorem 3.3.As Mis an object of Df(R), there is a natural isomor-
phism
RHomR(M,R)p∼
=RHomRp(Mp,R
p)
for each prime ideal pof R. So 3.2 shows
VRp(Mp)=V
Rp(RHomRp(Mp,R
p)) = VRp(RHomR(M,R)p)
for each prime ideal pof R.Nowweobtain
thickD(R)(M)=thickD(R)(RHomR(M,R))
as an immediate consequence of Theorem 3.1.
Remark 3.4.Theorem 3.3 can also be proved by combining results of Stevenson
and Letz (see [17, 4.11] and [12, 3.12 & 4.5], respectively). Stevenson used his
classification of thick subcategories of the singularity category of a regular ring
modulo a regular sequence in [18, 8.8] to show Theorem 3.3 holds for such rings.
Letz showed for Mand Nin Df(R),
Mis in thickD(R)(N)⇐⇒ M⊗
Rpis in thickD(
Rp)(N⊗R
Rp)
for each prime ideal pof Rwhere
Rpis the pRp-adic completion of Rp.So their
work, indeed, offers a different argument for Theorem 3.3.
In Section 4, we give our second proof of Theorem 3.3. This requires an analysis
of duality over a graded exterior algebra which is discussed there. We end this
section with the following application that recovers a theorem of Avramov and
Buchweitz [2, 6.3]. In Section 5, we strengthen the equivalence of (1) and (2) in
Corollary 3.6 when Ris further assumed to be local. First, we record a well-known
result of exact functors between triangulated categories.
3.5. Fix an exact functor between triangulated categories F:C→C.LetTbe a
thick subcategory of Cand define Tto be the full subcategory of Cconsisting of
objects Xsuch that F(X)isinT. It is clear that Tis a thick subcategory of C.
In particular, if Yis in Tthen thickC(Y)iscontainedinT.
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626 JIAN LIU AND JOSH POLLITZ
Corollary 3.6. Let Rbe locally complete intersection. For Mand Nin Df(R),
the following are equivalent:
(1) Exti
R(M,N)=0for all i0;
(2) Exti
R(N,M)=0for all i0;
(3) TorR
i(M,N)=0for all i0.
Proof. It suffices to show the equivalence of (1) and (3) since Tor is symmetric in
Mand N. Furthermore, the equivalence of (1) and (3) is the same as
RHomR(M,N)isinDf(R)⇐⇒ M⊗L
RNis in Df(R);
so we show the latter equivalence holds. For the rest of the proof let (−)∨be
RHomR(−,R).Observe for any pair of objects Mand Nin D(R),we have the
following adjunction isomorphism:
(†)(M⊗L
RN)∨RHomR(M,N∨).
⇐=:AsM⊗L
RNis in Df(R) and Theorem 3.3, it follows that (M⊗L
RN)∨
is in Df(R).Hence, by (†), RHomR(M,N∨)isinDf(R). By Theorem 3.3, N
is in thickD(R)(N∨) and so applying 3.5 with C=C=D(R), T=Df(R)and
F=RHomR(M,−) we conclude RHomR(M, N)isinDf(R),as well.
=⇒: By Theorem 3.3 and appealing again to 3.5, our assumption is equivalent
to RHomR(M,N∨)isinDf(R).Thus, by (†)wehavethat(M⊗L
RN)isinDf(R)
and hence, Theorem 3.3 implies that (M⊗L
RN)∨∨ is in Df(R), as well. Finally, as
Ris Gorenstein, the natural map
M→M∨∨
is an isomorphism in Df(R). Therefore, M⊗L
RNis in Df(R),as needed.
4. Duality of thick subcategories via exterior algebras
For a finite dimensional Hopf algebra Hover a field kand finite dimensional H-
module M,itiswellknownthatMis a direct summand of M⊗kHomk(M, k)⊗kM
(see, for example, [5, 3.1.10]); the H-module structure on the latter needs both the
antipode and co-multiplication of H. In this section we make use of the DG version
of this fact for a graded exterior algebra.
Throughout kwill be a field and Λ is the exterior algebra on generators e1,...,e
c
of homological degree 1 over k. It is easily checked Λ is a graded Hopf algebra with
co-multiplication and antipode determined by
Δ(ei)=ei⊗1+1⊗eiand σ(a)=(−1)|a|a,
respectively (cf. [4, 5.3]).
The main goal of the section is to prove the next theorem and as an application
we give a second short proof of Theorem 3.3. The proof of Theorem 4.1 requires
some additional setup and can be found after 4.6.
Theorem 4.1. For an o b j ec t Mof Df(Λ),
thickD(Λ)(M)=thickD(Λ)(RHomΛ(M,Λ)).
4.2. Let ρ:Λ→Λ be an anti-endomorphism of graded k-algebras, which is nothing
more than an endomorphism since Λ is graded-commutative. The map ρprescribes
a natural left DG Λ-module structure on the graded k-space M∗=Hom
k(M,k);
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DUALITY AND SYMMETRY OF COMPLEXITY 627
define M∗(ρ) to be the left DG Λ-module whose underlying graded k-space is M∗
and its differential and Λ-action are given by
∂M∗(ρ)(f):=−(−1)|f|f∂M=(−1)|f|+1 f∂M
a·f:=(−1)|a||f|f(ρ(a)−).
We are particularly interested in the relationship between M∗(id) and M∗(σ). Fur-
thermore, as Homk(Λ,k)∼
=Σ−cΛ as DG Λ-modules a direct calculation yields
(1) RHomΛ(M,Λ) ΣcHomk(M, k)=ΣcM∗(id).
4.3. Let Mbe a DG Λ-module. Since Λ has trivial differential we can twist the
differential and Λ-action of Mto obtain an isomorphic DG Λ-module. We define
Mτto be the DG Λ-module whose underlying graded k-space is Mequipped with
differential and Λ-action
∂Mτ:=−∂M
a·m:=(−1)|a|am.
The map M→Mτgiven by
m→ (−1)|m|m
is easily checked to be an isomorphism of DG Λ-modules.
Proposition 4.4. For any Min D(Λ),M∗(σ)and M∗(id)τhave the same under-
lying graded Λ-module while their differentials are negatives of one another.
Proof. This follows directly from the definitions in 4.2 and 4.3.
4.5. Let Mand Nbe left DG Λ-modules, then M⊗kNis a left DG Λ⊗kΛ-module.
We define Homk(M, N) to be a left DG Λ ⊗kΛ-module
∂Homk(M,N)(f):=∂Nf−(−1)|f|f∂M
(a1⊗a2)·f:=(−1)|a2||f|a1f(σ(a2)−).
Hence, both M⊗kNand Homk(M, N) inherit a DG Λ-module structure via Δ.
There is a natural morphism of DG Λ-modules
ϕM,N :N⊗kM∗(σ)→Homk(M,N)
whichisanisomorphisminD(Λ) when H(M) is a finite rank k-space (see [4, 4.8]).
4.6. Let Mbe a DG Λ-module. Consider the morphism of DG Λ-modules
πM:k→Homk(M,M)
mapping 1 to idM.
M∼
=k⊗kMπM⊗M
−−−−−→Homk(M,M)⊗kM.
For an object Min Df(Λ) the composition
M∼
=k⊗kMπM⊗M
−−−−−→Homk(M,M)⊗kMϕ−1
M,M⊗M
−−−−−−→M⊗kM∗(σ)⊗kM
splits in Df(Λ); this is essentially the same argument from the classical case (cf.
[5, 3.1.10]).
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628 JIAN LIU AND JOSH POLLITZ
Proof of Theorem 4.1.As thickD(Λ) (k)=Df(Λ), it follows that
M⊗kM∗(σ)⊗kMis an object of thickD(Λ) (M∗(σ)).
Hence, 4.6 implies that Mis in thickD(Λ)(M∗(σ)), as well. Since (M∗(σ))∗(σ)∼
=M,
then by symmetry we have shown
(2) thickD(Λ)(M)=thickD(Λ) (M∗(σ)) .
Now we only need to observe that
thickD(Λ)(M)=thickD(Λ)(M∗(σ))
=thickD(Λ)(M∗(σ),−∂M∗(σ))
=thickD(Λ)(M∗(id))
=thickD(Λ) (RHomΛ(M,Λ))
where the first equality holds by (2), the second equality is Lemma 2.6, the third
equality is from 4.3 and Proposition 4.4, and the last equality holds by 4.2(1).
Therefore, we have justified the theorem.
As an application of the theory above (in particular, Theorem 4.1) we now
present a second proof of Theorem 3.3.
Second proof of Theorem 3.3.First, if Ris complete intersection we show that for
any Min Df(R),
(3) thickD(R)M⊗RKR=thickD(R)RHomR(M,R)⊗RKR.
Now observe that Theorem 4.1 shows
thickD(Λ) (jtM)=thickD(Λ) (RHomΛ(jtM,Λ))
where jand tare the functors introduced in 1.4. Therefore,
thickD(KR)(tM)=thickD(KR)RHomKR(tM,KR)
and since
RHomKR(tM,KR)tRHom
R(M,R)
it follows that
thickD(KR)(tM)=thickD(KR)(tRHom
R(M,R)) .
So restricting scalars along R→KRfinishes the proof of (3) in the case that Ris
complete intersection.
Now we return to the general setting; namely, assume that Ris locally complete
intersection. Let pbe a prime ideal of R, then by assumption Rpis complete
intersection and
thickD(Rp)(M(p)) = thickD(Rp)RHomRp(Mp,R
p)⊗RpKRp
=thickD(Rp)(RHomR(M,R)(p)) ;
the first equality is from the already established equality (3) and the second equality
is immediate from the isomorphism
RHomRp(Mp,R
p)⊗RpKRpRHomR(M,R)⊗RKRp.
Since these equalities of thick subcategories hold for each prime ideal pof R,an
application of the local-to-global principle in 1.6 establishes the desired result.
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DUALITY AND SYMMETRY OF COMPLEXITY 629
5. Symmetry of complexity
In this section, we offer two proofs of Theorem 5.2; both methods indicate how
the symmetry of complexity over complete intersections follow from studying prop-
erties of exterior algebras. Theorem 5.2 was originally shown in [2, 5.7] using
support varieties and the use of intermediate hypersurfaces. A second proof was
given by the second author in [14, 4.3.1] by studying the cohomological support of
certain DG modules over a graded commutative ring of finite global dimension.
Throughout this section (R, m,k) is a commutative noetherian local ring. Also,
we let (−)∨denote the functor RHomR(−,R).
5.1. Let Mand Nbe in Df(R). The complexity of the pair (M,N), denoted
cxR(M,N), is the least non-negative integer d∈Nsuch that
dimk(Extn
R(M,N)⊗Rk)≤and−1
for all n0andsomea∈R.Thatis,cx
R(M,N) measures the polynomial rate
of growth of the minimal number of generators of Extn
R(M,N).
Theorem 5.2. Let Rbe complete int er section. For each pair of objects Mand N
in Df(R),cxR(M,N)=cx
R(N,M).
Remark 5.3.The first proof of Theorem 5.2 is a straightforward application of
Theorem 3.3 and two standard facts:
(1) when Ris Gorenstein there is a natural isomorphism of graded R-modules
ExtR(M,N)∼
=ExtR(N∨,M∨)
for each Mand Nin Df(R);
(2) when Ris complete intersection
cxR(X, Y )≤cxR(M, N)
whenever Xis in thickD(R)(M)andYis in thickD(R)(N) (see [2, 5.6] or
[14, 4.2.5 & 4.2.9]).
First proof of Theorem 5.2.From Remark 5.3(1) it follows easily that
cxR(M,N)=cx
R(N∨,M∨).
Using Theorem 3.3, we obtain M∨is an object of thickD(R)(M)andN∨is an object
of thickD(R)(N). So Remark 5.3(2) establishes the inequality
cxR(M,N)=cx
R(N∨,M∨)≤cxR(N,M).
By symmetry the result follows.
Remark 5.4.As discussed in the introduction, the theme of [4] is to deduce homo-
logical results over R,whenRis complete intersection, using the bridge
jt :Df(R)→Df(KR)≡
−→ Df(Λ)
from 1.4. However, in [4, 8.9] it is remarked that the authors did not see how to
deduce Theorem 5.2 from studying this bridge. The second proof of Theorem 5.2,
given below, shows that one can in fact arrive at the symmetry of complexity over
the complete intersection Ras a direct consequence of the symmetry of complexity
over Λ.
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630 JIAN LIU AND JOSH POLLITZ
Second proof of Theorem 5.2.Assume Ris complete intersection.
Observe that
ExtR(M,tN)∼
=ExtKR(tM,tN)∼
=ExtΛ(jtM,jtN)
where the first isomorphism is adjunction and the second one uses that jis an
equivalence. Also, by [15, 4.2.7]
cxR(M,N)=cx
R(M,tN),
and so combining this with the isomorphisms above we have
(4) cxR(M,N)=cx
Λ(jtM,jtN).
Similarily, it follows that
(5) cxR(N,M)=cx
Λ(jtN,jtM).
Note [4, 5.3] established
cxΛ(jtM,jtN)=cx
Λ(jtN,jtM);
this equality along with the ones in (4) and (5) establish cxR(M, N)=cx
R(N,M),
as claimed.
Acknowledgments
Both authors are indebted to Srikanth Iyengar for many helpful discussions, as
well as suggesting the two authors collaborate because of their many common inter-
ests. We are also very happy to thank Benjamin Briggs for several useful comments
on an earlier draft of this paper as well as numerous conversations that helped clar-
ify some ideas in Section 4. We also thank Janina Letz and Greg Stevenson for
their comments on a preliminary draft of this work. Finally, we happily thank the
referees for their comments.
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School of Mathematical Sciences, University of Science and Technology of China,
Hefei 230026, Anhui, People’s Republic of China
Email address:liuj231@mail.ustc.edu.cn
Department of Mathematics, University of Utah, Salt Lake City, Utah 84112
Email address:pollitz@math.utah.edu
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