Detecting Koszulness and related homological properties from the algebra structure of Koszul homology

DETECTING KOSZULNESS AND RELATED HOMOLOGICAL PROPERTIES FROM THE ALGEBRA STRUCTURE OF KOSZUL HOMOLOGY

AMANDA CROLL, ROGER DELLACA, ANJAN GUPTA, JUSTIN HOFFMEIER, VIVEK MUKUNDAN, DENISE RANGEL TRACY, LIANA M. S¸EGA, GABRIEL SOSA, AND PEDER THOMPSON

ABSTRACT. Letkbe a field andRa standard gradedk-algebra. We denote byH^Rthe homology algebra of the Koszul complex on a minimal set of generators of the irrelevant ideal ofR. We discuss the relationship between the multiplicative structure ofH^Rand the property thatRis a Koszul algebra. More generally, we work in the setting of local rings and we show that certain conditions on the multiplicative structure of Koszul homology imply strong homological properties, such as existence of certain Golod homomorphisms, leading to explicit computations of Poincar´e series. As an application, we show that the Poincar´e series of all finitely generated modules over a stretched Cohen-Macaulay local ring are rational, sharing a common denominator.

INTRODUCTION

Let(R,m, k)denote a local Noetherian ringRwith maximal idealmand residue fieldk. LetK^R denote the Koszul complex on a minimal generating set ofm, and letH^Rdenote its homology. The Koszul complex K^Rcan be endowed with the structure of a differential graded (DG) algebra, and is the first step in constructing a DG algebra minimal free resolution ofkoverR(called aTate resolutionofkoverR) through the process of adjoining DG algebra variables. It is thus natural to expect that the properties of the homology algebraH^R are related to other homological properties ofR. Indeed, it is known that both the Gorenstein and complete intersection properties ofRcan be characterized in terms ofH^R.

Certain higher order homology operations on Koszul homology, introduced by Golod [10], can be used to characterize extremality in the growth of the minimal free resolution ofkoverR. If the ringRis Golod, then it has the property that for all finitely generatedR-modulesM the Poincar´e seriesP

i≥0rank_k(Tor^R_i (M, k))zⁱ are rational and share a common denominator, see Ghione and Gulliksen [9]. This property is also satisfied by other large classes of rings, and recent work of Rossi and S¸ega [24] and Kustin, S¸ega, and Vraciu [15]

provides insight into the fact that the multiplicative structure of Koszul homology plays a role in establishing such results. In this paper we further explore how the structure ofH^R can be used to derive rationality of Poincar´e series and other homological properties ofR. In particular, we give special attention in the graded case to the Koszul property.

Recall that the Koszul homology of a Golod ring has trivial multiplication, see [10]. WhenRis not Golod, we find it useful to consider conditions onH^Rthat, to some extent, generalize the condition that multiplication is trivial. We require that cycles living “deep” enough inK^R(i.e. ones that are contained inmⁱK^Rfor large enough values ofi) can be expressed, up to a boundary, in terms of certain cycles that have trivial products among themselves. More precisely, we consider the following conditions onK^R, depending on integerst, b, s:

Zt,b,s: There exists a finite setZ ⊆ Z(m^tK^R)such thatzz⁰ = 0inZ(K^R)for allz, z⁰ ∈Z and for every v ∈ m^sK^Rthere existsm ∈Nandzi ∈ Z,ui ∈ Z(m^bK^R)for eachiwith1 ≤ i≤ m, such that v−Pm

i=1ziui∈B(m^s−1K^R).

Date: May 8, 2018.

2010Mathematics Subject Classification. 13D07, 13D02, 16S37.

This material is based upon work supported by the National Science Foundation under Grant #1321794, as part of the Mathematical Research Communities 2015 program in Snowbird, Utah, and by a grant from the Simons Foundation (# 354594, Liana S¸ega). The third author was supported by the Istituto Nazionale di Alta Matematica “Francesco Severi” fellowship.

Key words and phrases:Koszul homology, Golod homomorphism, Koszul algebra.

1

P_t: There exists[l]∈H₁(K^R)such that for everyz∈Z(m^tK^R)there existsz⁰∈Z(m^t−1K^R)such that z−z⁰l∈B(m^t−1K^R).

After setting some ground work in the first two sections, in Sections 3 and 4 we prove various homologi- cal implications of the conditionsZt,b,sandPt, under specific conditions on the integerst, b, s. The main results regarding these conditions are Theorems 3.1 and 4.2. We note that the hypotheses of Theorem 3.1 also requireR to be artinian, while the hypotheses of Theorem 4.2 and its corollaries do not. The con- clusions of these theorems and their corollaries are formulated in terms of vanishing of the natural maps Tor^R_∗(m^j, k)→Tor^R_∗(mⁱ, k)induced by the inclusionsm^j ⊆mⁱfor certain values ofi,j, as well as identi- fying Golod homomorphisms, establishing generation of the Yoneda algebraExtR(k, k)in low degrees, and deducing rationality of Poincar´e series. Our arguments utilize the DG algebra structure of the minimal free resolution ofkoverR. This approach is inspired by, and generalizes, work of Levin and Avramov [18], where homological properties of local Gorenstein artinian rings are derived from the fact that the Koszul homology algebra of such a ring is a Poincar´e algebra.

For appropriate values oft, the propertyPtholds for the class of compressed Gorenstein artinian local rings discussed in [24] and also for the class of compressed level local artinian rings of odd socle degree, see [15].

In particular, our results in Section 4 can be used to recover the results of [24] and [15] regarding the fact that when the socle degree is different than three, these rings can be obtained as homomorphic images of a hypersurface, via a Golod homomorphism.

In Section 5 we show that the propertyP2is satisfied in the case of stretched artinian rings satisfyingm³6= 0 andrank_k(m/m²) 6= rankk(0 : m). The class of stretched Cohen-Macaulay local rings was considered by Sally in [25], where she proves that the Poincar´e series of the residue field over such a ring is rational. A consequence of our results on generation in the Koszul homology algebra is that the Poincar´e series of all finitely generatedR-modules over a stretched Cohen-Macaulay local ringR are rational, sharing a common denominator. Theorem 5.4 also states that the Yoneda algebraExtR(k, k)of a stretched artinian local ring (R,m, k)is generated in degree1ifrankk(m/m²)6= rankk(0 :m)and in degrees1and2ifrankk(m/m²) = rankk(0 : m).

For the remainder of the introduction, assume thatRis a standard gradedk-algebra. LetK^Rdenote the Koszul complex on a set of minimal generators of the irrelevant ideal ofRand letH^R denote the homology algebra ofK^R. Thek-algebraRis said to be aKoszul algebraif the resolution ofkoverRis linear, that is to say, the differentials in the minimal graded free resolution ofkcan be represented by matrices of linear forms (see e.g., [26] and [12]). The algebraH^Ris bigraded; when writing the bidegree(i, j)of an element, the index idenotes homological degree and the indexj denotes internal degree. Thelinear strandofH^Ris the set of elements of bidegree(i, i+ 1), and thenonlinear strandsare composed of elements of bidegree(i, i+r)with r >1. We say that the nonlinear strands ofH^R are generated by a setZ ⊆ H^R if the nonlinear strands are contained in the ideal generated byZinH^R. If the nonlinear strands are generated by a subsetZof the linear strand, it follows thatH^Ris generated by the linear strand as ak-algebra.

In Section 6 we interpret our earlier results in the graded setting, with special attention to the Koszul property.

In particular, we obtain the following statements, which provide new homological criteria for verifying that an algebra is Koszul:

(1) If the nonlinear strands ofH^Rare generated by one element of bidegree(1,2), thenRis absolutely Koszul, hence Koszul. (See [14] or Section 6 regarding absolutely Koszul algebras.)

(2) IfR_>3= 0and there exists a set of cyclesZrepresenting elements in the linear strand inH^R, with the property thatzz⁰ = 0inZ(K^R)for allz, z⁰∈Zand such that the nonlinear strand ofH^Ris generated byZ={[z]|z∈Z}, thenRis Koszul.

IfRis Koszul, then part of the Koszul homology algebraH^Ris generated by elements in the linear strand;

this point was made by Avramov, Conca, and Iyengar [4] and Boocher et al. [7]. More precisely, in [4, Theorem 4.1] it is shown that ifRis Koszul thenH^R_i,j = 0forj >2iandH^R_i,2i= (H^R_1,2)ⁱfor alli≥0and in [7, Theorem 3.1] it is proved that one has alsoH^R_i,2i−1= (H^R_1,2)ⁱ⁻²H^R_2,3for alli≥2. Section 6 and Section 7 provide some further insight into the connections between the fact thatRis Koszul and the structure ofH^R. In Proposition 6.2 we note that ifR is Koszul then the nonlinear strands ofH^R are contained in the set of matric Massey

products ofK^R. However, generation ofH^Rby the linear strand (which implies that the nonlinear strands of H^Rare contained in the set of matric Massey products) does not imply thatRis Koszul. This can be seen by means of the example in 7.4, which relies on a ring from a paper of Roos [23]. (The fact that the linear strand need not generateH^Ras ak-algebra whenRis Koszul is also noted in [7, Remark 3.2].) On the other hand, 7.2 describes a Koszul algebraRfor whichH^Rhas the same bigraded Hilbert series as the homology algebra of the ring of 7.4 and is also generated by the linear strand. It turns out that the ring in 7.2 satisfies the hypothesis of statement (2) above. This observation sheds some light on our effort to understand what distinguishes one homology algebra from the other in the two examples.

The examples in Section 7 utilize the Macaulay2 packageDGAlgebraswritten by Frank Moore, which provides an efficient way to verify rings for which statements (1) or (2) hold; using this, we apply our Theorem 6.1 to the rings studied in Roos [23]. The last section also contains a concrete example of how our results can be used towards establishing homological properties of the ring and computations of Poincar´e series, see 7.6.

Given the evidence that the properties considered in this paper show up in a large variety of situations, we hope that this study will be useful in further explorations of homological properties of local rings.

Acknowledgments:We are grateful to Jan-Erik Roos and Aldo Conca, as well as the anonymous referee, for useful comments and suggestions.

The work on this paper started during the 2015 Mathematical Research Communities program in Commu- tative Algebra, under the guidance of Liana S¸ega. The authors would also like to thank the other organizers of this program, Srikanth Iyengar, Karl Schwede, Gregory Smith, and Wenliang Zhang, for their support, and also the AMS staff that coordinated the program. The third author joined the project following conversations during the conference in honor of Craig Huneke held in Ann Arbor in July 2016, and is thankful for support to travel to this conference from the Department of Mathematics at the University of Michigan and IIT Bombay.

1. BACKGROUND

In this section we set notation and provide needed definitions. We recall the definition of a small homomor- phism and provide some preliminary results centered on this concept.

1.1. Let (R,m, k) be a local ring and M a finite (meaning finitely generated)R-module. Fix a minimal generating set ofmand letK^Rdenote the Koszul complex on this set. LetH^Rdenote the homology algebra of K^R. The complexK^Rhas a natural structure of a graded commutative algebra, and this structure is inherited byH^R. We denote byK^M the Koszul complexK^R⊗RM.

ThePoincar´e seriesP^R_M(z)ofM is defined as P^R_M(z) =X

i≥0

rankk(Tor^R_i (M, k))zⁱ.

Ifφ: (R,m, k)→(S,n, k)is a surjective homomorphism of local rings then the following coefficientwise inequality holds

P^S_M(z)4 P^R_M(z) 1−z(P^R_S(z)−1). If equality holds forM =kthen we say thatφis aGolod homomorphism.

The homomorphismφinduces maps

Extⁱ_φ(k, k) : Extⁱ_S(k, k)→Extⁱ_R(k, k).

IfExt^∗_φ(k, k)is surjective then we sayφissmall. Recall that ifφis Golod thenφis small (cf. Avramov [1, 3.5]).

WhenR is artinian, specific conditions formulated in terms of the concepts above allow for an explicit computation of the seriesP^R_k(z).

Lemma 1.2. Let (R,m, k)be an artinian local ring withm^s+1 = 0. Let ndenote the minimal number of generators ofmand letadenote the dimension ofm^s.

If the canonical projectionR→R/m^sis small and the ringR/m^sis Golod, then the Poincar´e series ofk overRis rational, satisfying the formula:

(1.2.1) P^R_k(z) = (1 +z)ⁿ

1−z(H^R/ms(z)−1) +az²(1 +z)ⁿ

whereH^R/ms(z)stands for the Hilbert series (which is in this case a polynomial of degreen) of the Koszul homology algebraH^R/ms.

Proof. Asm^s∼=k^a, we haveP^R_R/ms(z) =azP^R_k(z) + 1.

SinceR/m^s is Golod andR → R/m^s is small, R → R/m^s is a Golod homomorphism by [26, 6.7].

Therefore, we have

P^R_k(z) = P^R/m

s

k (z)

1−z(P^R_R/ms(z)−1)

= P^R/m

s

k (z) 1−az²P^R_k(z) .

By rearranging, we have that

(1.2.2) P^R_k(z) = P^R/m_k ^s(z)

1 +az²P^R/m_k ^s(z) .

Finally, sinceR/m^sis Golod, we have that

(1.2.3) P^R/m_k ^s(z) = (1 +z)ⁿ

1−z(H^R/ms(z)−1).

The conclusion follows from (1.2.2) and (1.2.3).

In order to apply the lemma, we need to verify that the canonical projectionR→R/m^sis Golod or small.

In [18], Levin and Avramov prove that this homomorphism is Golod (thus small) whenever the artinian ringR is Gorenstein. Their proof relies on the fact thatH^Ris a Poincar´e algebra whenRis Gorenstein. This leads us to believe that, more generally, the structure of the Koszul homologyH^R can be used to understand the homological properties of the canonical projectionR → R/m^s. The next lemma shows that matric Massey products play a role.

1.3. Letϕ: R→Sbe a homomorphism of local rings. Theusual productsofH^Rare understood to be the set of productsH^R_>1·H^R_>1. We denote byM H(K^R)the set ofmatric Massey productsofH^R_>1, as defined in [20].

This set of higher order homology operations is a submodule ofH^R_>1and contains the usual products; see also [2, (1.4.1)] for a more concise definition.

The induced mapH(K^ϕ) : H(K^R)→H(K^S)satisfiesH(K^ϕ)(M H(K^R))⊆M H(K^S). By [1, 4.6], ifϕ is small then the induced homomorphism

H_≥1(K^R)/M H(K^R)→H_≥1(K^S)/M H(K^S) is injective. From here we derive immediately the following statement:

Lemma. Let(R,m, k)be a local ring and leti≥0. Consider the conditions:

(1) The canonical projectionR→R/mⁱis small;

(2) H≥1(mⁱK^R)⊆M H(K^R).

Then(1)implies(2).

Example 7.4 in Section 7 shows that the implication (2)=⇒ (1) does not hold wheni = 2. Ideally, one would like to replace condition (2) with a stronger one, that is equivalent to (1). While such a condition is not yet known, we identify in Sections 3 and 4 two conditions on Koszul homology that imply (1), for certain values ofi.

2. APROPERTY OF THETATE RESOLUTION

The purpose of this section is to record in Proposition 2.8 a general property of the Tate resolution. This result will be used later, in the proof of one of the main theorems. We start with a description of the Tate resolution, and we invite the reader to consult [3] for more details, in particular for the definition of a DG algebra. We then build the ingredients of the proof of the proposition by means of a couple of lemmas.

2.1. Adjunction of variables.LetBbe a DG algebra overRand supposezis a cycle inB. We embedBinto a DG algebraB⁰=Bhyiby freely adjoining a variableysuch that∂(y) =zas follows:

If|z|is even, the variableysuch that∂(y) = zis called anexterior variableand satisfiesy² = 0. Denote bykhyithe exterior algebra overkof a freek-module on a generator of degree|z|+ 1. The differential on Bhyi=B⊗_kkhyiis given by

∂(b₀+b₁y) =∂(b₀) +∂(b₁)y+ (−1)^|b1|b₁z.

If|z|>0is odd,yis adivided powers variable. Thek-algebrakhyion a divided powers variableyis the freek-module with basis{y⁽ⁱ⁾:|y⁽ⁱ⁾|=i|y|}i≥0and multiplication table

y⁽ⁱ⁾y^(j)= i+j

i

y^(i+j), fori, j≥0.

We sety⁽¹⁾=y,y⁽⁰⁾= 1, andy⁽ⁱ⁾= 0fori <0. Forgetting the differentials,Bhyi=B⊗kkhyi. Ifz∈Bis a cycle of positive odd degree, then

∂ X

i

biy⁽ⁱ⁾

!

=X

i

∂(bi)y⁽ⁱ⁾+X

i

(−1)^|bi|bizy⁽ⁱ⁻¹⁾

is a differential onBhyithat extends that ofBand satisfies the Leibniz rule.

The notationBhy1, . . . , ynistands for the DG algebra obtained by repeated adjunction of variables as above.

2.2. The Tate resolution. Letx1, ..., xn be a minimal generating set form andK^R the Koszul complex on x1, ..., xn. Note that we can interpretK^Ras the DG algebra

K^R=RhT₁, ..., T_ni,

whereTi are degree 1 exterior variables (these exterior variables in degree 1 will be referred to asKoszul variables) with∂(T_i) =x_i. One can continue to “kill” homology by adjoining variables toK^R, following the construction in [3, 6.3.1]. The resulting DG algebraAis a minimal free resolution ofkoverR, often referred to as theTate resolutionofkoverR.

Forgetting differentials,Ais a freeR-module: see [3, Remark 6.2.1] for a description of the basis in terms of the variables adjoined. In particular, one can see thatAis also a freeK^R-module.

2.3. Notation.We writeKforK^Rwhen the ringRis understood. SinceAis a free algebra overK, we consider a homogeneousK-basis ofA. For eachj, letχ_i,jwith1≤i≤q_jdenote the elements of homological degree jin this basis. Ifz∈ Ap, then we writezin terms of this basis as

(2.3.1) z=

p

X

j=0 q_j

X

i=1

z_i,jχ_i,p−j

withzi,j∈Kjfor eachj.

As usual,Z(A)denotes the set of cycles ofAandB(A)denotes the set of boundaries.

The following lemma provides the inductive step for our key lemma, Lemma 2.7, below.

Lemma 2.4. Letz∈Zp(A)and write it as in(2.3.1). Letabe an integer with0≤a≤p. Assumezi,j∈m^tK for alljwith0≤j≤a−1and alliwith1≤i≤qj. Then

∂(z_i,a)∈m^t+1K for alliwith1≤i≤qa.

Proof. Sincez ∈Z(A), we have∂(z) = 0. On the other hand we can compute∂(z)from (2.3.1), using the Leibniz rule; this yields:

(2.4.1) 0 =

p

X

j=0 q_j

X

i=1

(−1)^jz_i,j∂(χ_i,p−j) +∂(z_i,j)χ_i,p−j.

Letibe such that1≤i≤qa. We express all terms in the right hand side of (2.4.1) in terms of theK-basis of A, and collect the terms to compute the coefficient of each basis element in the sum. We see that the coefficient ofχ_i,p−ain this sum is

∂(z_i,a) +

a−1

X

j=0 q_j

X

i⁰=1

z_i⁰_,jw_i⁰_,j withw_i⁰_,j ∈mK_a−j .

The coefficientswi⁰,j come from expressing∂(χ_i,p−j)in terms of theK-basis, forj ≤a−1. In particular, wi⁰,j ∈mKsinceAis minimal. (Note that ifj ≥athenzi,j∂(χ_i,p−j)does not have any contribution to the coefficient ofχ_i,p−a, for degree reasons.) These coefficients ofχ_i,p−amust equal0, hence

∂(zi,a) =−

a−1

X

j=0 qj

X

i⁰=1

zi⁰,jwi⁰,j ∈(m^tK)(mK)⊆m^t+1K

for alliwith1≤i≤qa.

2.5. LetRbdenote the completion ofRwith respect tom. We may writeRb =Q/I, with(Q,n, k)a regular local ring andI⊆n²; this presentation is called aminimal Cohen presentation. We set

v(R) = max{j|I⊆n^j}.

As noted in [13], this integer is independent of the choice of the minimal Cohen presentation.

Remark 2.6. Ifv(R)≥t+ 1, that is,I⊆n^t+1, then the map

(2.6.1) Hi(K^R/m^t+1K^R)→Hi(K^R/m^tK^R)

induced by the canonical homomorphismK^R/m^t+1K^R→K^R/m^tK^Ris zero for alli≥1. In particular, we have: If∂(z)∈m^t+1K^R, thenz∈B(K^R) +m^tK^Rfor allz∈K^R_>1.

Indeed, to justify this statement it suffices to assume thatRis complete, withR=Q/Ias above. We can writeK^R = K^Q⊗QR, whereK^Qis the Koszul complex on a minimal generating set ofnobtained by lifting the minimal generating set picked form. SinceI ⊆ n^t+1 by assumption, we can make the identifications K^R/m^t+1K^R = K^Q/n^t+1K^Q andK^R/m^tK^R = K^Q/n^tK^Q. The map in (2.6.1) can then be identified with the induced map

H_i(K^Q/n^t+1K^Q)→H_i(K^Q/n^tK^Q)

which is zero for alli≥1because the induced mapHi(n^t+1K^Q)→Hi(n^tK^Q)is zero for alli≥0, sinceQ is regular (for example, see [26, Theorem 3.3]).

We are now prepared to prove a key lemma; a reformulation of this will yield Proposition 2.8 below.

Lemma 2.7. Suppose(R,m, k)is a local ring,Kis the Koszul complex on a minimal generating setx1, ..., xn

ofm, andAis the Tate resolution ofk. Lett≥1be an integer such thatv(R)≥t+ 1. Ifx∈ A, then there existsy∈K₁Asuch that∂(x−y)∈m^tK₁A.

Proof. Letx∈ Ap+1. Ifp= 0, we may takey=xand the result follows trivially. Now assumep≥1. Since Ais minimal, we have∂(x) =P

xigiwithxi ∈mandgi ∈ Ap. ChoosingAi ∈K1such that∂(Ai) =xi, we have

∂ x−X

Aigi

=∂(x)−X

∂(Aigi)

=X

xigi−X

∂(Ai)gi+X

Ai∂(gi)

=X

Ai∂(gi).

Sety⁰ =PA_ig_i. Theny⁰∈K₁Aand∂(x−y⁰)∈K₁A.

Apply Lemma 2.4 withz = ∂(x−y⁰)anda= 1(noting thatzi,0 = 0for alliwith1 ≤i ≤q0, so the hypothesis is satisfied). For alliwith0≤i≤q1this yields∂(zi,1)∈m^t+1K(indeed, this holds for alltin Lemma 2.4, hence∂(zi,1) = 0), and then Remark 2.6 shows

zi,1=∂(ei,1) +fi,1

for someei,1∈K2andfi,1∈m^tK. Consequently, we have:

∂(x−y⁰) =

q1

X

i=1

(∂(ei,1) +fi,1)χ_i,p−1+V, with V ∈K2A.

Now takey1=Pq₁

i=1ei,1χi,p−1and we have:

∂(x−y⁰−y1) =

q1

X

i=1

(∂(ei,1) +fi,1)χ_i,p−1+V −

q1

X

i=1

∂(ei,1χ_i,p−1)

=

q₁

X

i=1

f_i,1χ_i,p−1+ V +

q₁

X

i=1

e_i,1∂(χ_i,p−1)

!

=

q₁

X

i=1

fi,1χ_i,p−1+V1, withV1∈K2A.

Setm= min{p, n}, and let us assume inductively, fora−1< m, that we constructedy1, y2, . . . , ya−1∈K1A such that

∂(x−y⁰−y1− · · · −y_a−1) =

a−1

X

j=1 qj

X

i=1

fi,jχ_i,p−j+V_a−1,

withf_i,j∈m^tKandV_a−1∈K_aA. Applying again Lemma 2.4 and Remark 2.6, withz=∂(x−y⁰−y₁−

· · · −y_a−1)we have that

zi,a=∂(ei,a) +fi,a

withei,a∈Ka+1andfi,a ∈m^tKa. Consequently, we can write

∂(x−y⁰−y₁− · · · −y_a−1) =

a−1

X

j=1 q_j

X

i=1

f_i,jχ_i,p−j+

q_a

X

i=1

(∂(e_i,a) +f_i,a)χ_i,p−a+V,

withV ∈Ka+1A. Now takeya =Pqa

i=1ei,aχ_i,p−aand, as above, we get:

∂(x−y⁰−y₁− · · · −y_a) =

a

X

j=1 q_j

X

i=1

f_i,jχ_i,p−j+V_a,

withVa∈Ka+1A. Note thatVa= 0whena≥m, by degree reasons (ifp < n) and sinceK>n= 0.

Sety=y⁰+y1+· · ·+ym. Then the cyclez=∂(x−y)satisfies the conclusions of our statement.

We can now prove the useful decomposition property of the Tate resolution advertised above, which was inspired by the work in [18].

Proposition 2.8. Lett≥1be an integer such thatv(R)≥t+ 1, and letAbe the Tate resolution ofkoverR.

Denote byA⁰the DG subalgebra ofAgiven by

A⁰ ={x∈ A |∂(x)∈m^tK1A}.

ThenAis generated byA⁰as aK-algebra, that is:A=A⁰+ K1A⁰+ K2A⁰+· · ·+ KnA⁰= KA⁰.

Proof. Forx∈ A, Lemma 2.7 provides an elementy∈K₁Asuch thatx−y∈ A⁰. A reformulation of Lemma 2.7 therefore gives thatA=A⁰+ K1A. Applying this fact repeatedly, and noting thatKn+1= 0, we get:

A=A⁰+ K₁A=A⁰+ K₁(A⁰+ K₁A) =A⁰+ K₁A⁰+ K₂A=· · ·

=A⁰+ K₁A⁰+ K₂A⁰+· · ·+ K_n(A⁰+ K₁A)

=A⁰+ K₁A⁰+ K₂A⁰+· · ·+ K_nA⁰

= KA⁰.

3. GENERATION BY A SPECIAL SET

We continue with the notation of the previous sections for the Koszul complex and the Tate resolution of a local ringR. In this section, we prove one of the main theorems, Theorem 3.1 below, and we point out its applications. In particular, these applications include a computation of the Poincar´e seriesP^R_k(z)and conditions under which the mapR→R/m^sis Golod.

Recall that the invariantv(R)was introduced in 2.5.

Theorem 3.1. Let(R,m, k)be a local ring and letsbe an integer such thatm^s+1= 0. Lettandbbe integers such thats−t≤b≤s−1andv(R)≥t+ 1≥2, and assume that the following condition holds:

Zt,b,s: There exists a finite setZ ⊆Z(m^tK^R)such thatzz⁰ = 0for allz, z⁰ ∈Zand for everyv ∈m^sK^R there existsm∈Nandzi∈Z,ui∈Z(m^bK^R)for eachiwith1≤i≤m, such thatv−Pm

i=1ziui∈ B(m^s−1K^R).

The mapsTor^R_i (m^s, k)→Tor^R_i (m^b, k)induced by the inclusionm^s⊆m^bare then zero for alli≥0.

We postpone the proof of the theorem in order to give some corollaries. Concrete examples for which these results can be applied will be given in Section 7. We will use below, and also in the next section, the following result of Rossi and S¸ega.

3.2. [24, Lem. 1.2] Letκ: (R,m, k)→(R,m, k)be a surjective homomorphism of local rings. If there exists a positive integerasuch that:

(a) the mapTor^R_i (R, k)→Tor^R_i (R/m^a, k)induced by the canonical quotient mapR→R/m^ais zero for all positivei, and

(b) the mapTor^R_i(m^2a, k)→ Tor^R_i (m^a, k)induced by the inclusionm^2a ⊆ m^a is zero for all non-negative integersi,

thenκis a Golod homomorphism.

Corollary 3.3. Under the hypotheses of Theorem3.1, if2b≥sthen the homomorphismR→R/m^sis Golod.

Proof. We apply 3.2 to the natural projectionκ:R →R/m^s, withR =R/m^s,m =m/m^sanda=b. We need to check that conditions (a) and (b) hold. Since2a= 2b≥s, we havem^2a= 0, so condition 3.2(b) holds trivially. Theorem 3.1 gives that the induced mapsTor^R_i (m^s, k)→Tor^R_i (m^b, k)are zero for alli >0, and this

implies condition 3.2(a), sinceR/m^s=RandR/m^b=R/m^a.

Remark 3.4. LetR=Q/Ibe a minimal Cohen presentation ofR, with(Q,n, k)a regular local ring. As first noted by L¨ofwall [19], the ringRis Golod whenever there exists an integertsuch that

(3.4.1) n^2t⊆I⊆n^t+1

Assumesandtare integers such thatm^s+1 = 0andv(R)≥t+ 1, and son^s+1 ⊆I ⊆n^t+1. Ifs <2t, then the inclusions in (3.4.1) hold, and it follows thatRis Golod. Ifs= 2tthenRis not necessarily Golod, but it follows that the quotient ringR/m^s=Q/(I+n^s)is Golod.

Corollary 3.5. Assume the hypothesis of Theorem3.1is satisfied. Ifs= 2tandb=t, then the hypotheses of Lemma1.2are satisfied, and thusP^R_k(z)satisfies the formula(1.2.1).

Proof. The homomorphismR→R/m^sis Golod by Theorem 3.1, and thus small (see 1.1). The ringR/m^sis

Golod by Remark 3.4.

Proof of Theorem3.1. Let|Z|denote the cardinality ofZ. Let{z1, ..., z_|Z|}be the cycles inZand letIdenote the set of all finite ordered lists of elements in{1, . . . ,|Z|}, including the empty set.

LetI ∈ I. If I = (i) has length 1, we set I⁻ = ∅. If I = (i1, . . . , ir) has length r ≥ 2, we set I⁻ = (i1, . . . , i_r−1). We now define for eachI∈ Ian elementyI ∈ A, whereAis the Tate resolution ofkas in 2.2, such that

(1) yI = 1∈ A0, ifI=∅;

(2) ∂(yI) =zi_ry_I⁻, ifI= (i1, . . . , ir)withr≥1.

The details of constructing these elements are as follows. IfI = ∅, we choose yI as in (1). Ifr = 1and I = (i), we can chooseyI ∈ Asuch that∂(yI) =zi, sincezi ∈Zis a cycle. Assuming thatr ≥2and the elementsyI have been defined for allI ∈ Iof lengthr−1, we can construct elementsyI satisfying (2) for I= (i1, . . . , ir)by noting thatzi_ry_I⁻is a cycle, so such ayIexists sinceAis acyclic. Indeed, we have

∂(ziry_I⁻) = (−1)^|zir|zir∂(y_I⁻) = (−1)^|zir|zirzir−1y_(I⁻₎⁻ = 0, where the last equality is due to the hypothesis on the setZ.

We identify the mapTor^R_i(m^s, k)→Tor^R_i (m^b, k)with the mapH_i(m^sA)→H_i(m^bA). In order to show this map is trivial fori≥0, we letx∈m^sA_cfor somecand must show thatx∈∂(m^bA).

SetY ={y_I ∈ A|I∈ I}. Fori, j≥0,define

A(i, j) := (m^sKiYjA⁰)∩ Ac, whereYjis the set of elements ofY of degreej.

Claim: Fori, j≥0,

(3.5.1) A(i, j)⊆∂(m^bA) + X

p+q=i+j

p>i

A(p, q) + X

p+q>i+j

A(p, q)

where only finitely many terms in this sum are not zero for degree reasons.

Proof of Claim: To prove the inclusion, it suffices to consider elements ofA(i, j)of the formvyIa⁰for some v∈m^sKi,a⁰∈ A⁰andyI ∈Yj,withI= (i1, . . . , ir)∈ IorI =∅; in the last case, we setr= 0. (Note that every element ofA(i, j)can be written as a sum of elements of this form.)

By assumption there existzι,v∈Zanduι,v∈Z(m^bK)with1≤ι≤msuch that v−

m

X

ι=1

z_ι,vu_ι,v∈∂(m^s−1K_i+1).

We need to show thus that(Pm

ι=1z_ι,vu_ι,v+w)y_IA⁰is contained in the right hand side of (3.5.1), for all w∈∂(m^s−1Ki+1),zι,v ∈Z anduι,v ∈Z(m^bK). Note that it suffices to prove this statement whenm= 1.

We assume thus thatv−zir+1u ∈∂(m^s−1Ki+1)for somezir+1 ∈Z andu∈ Z(m^bK)and we show that vy_IA⁰is contained in the right hand side of (3.5.1).

In what follows, we will simplify the notation whenIhas length 1and we will writey_i = y_(i). Define I⁺ = (i1, . . . , ir, ir+1). Recall that we defined a setI⁻for any nonemptyI∈ I. Although we do not define a set∅⁻, we agree to sety_I− = zi_r = yi_r = 0whenI = ∅. With this convention, note that the formula

∂(yI) =zi_ry_I− also holds whenI=∅, so we will not treat this case separately.

Note that

|u|=i− |z_ir+1|=i+ 1− |y_ir+1|,

|y_I⁻|=j− |zi_r| −1 =j− |yi_r| when I6=∅, and

|y_I+|=j+|zi_r+1|+ 1 =j+|yi_r+1|.

We now have

vyIA⁰ ⊆ ∂(m^s−1Ki+1) +zir+1u yIA⁰

= ∂(m^s−1Ki+1)yIA⁰+u∂(y_I+)A⁰

= ∂(m^s−1K_i+1y_IA⁰) +u∂(y_I+)A⁰+m^s−1K_i+1z_iry_I−A⁰+m^s−1K_i+1y_I∂(A⁰)

⊆ ∂(m^s−1A) +∂(uy_I+A⁰) +uy_I+∂(A⁰) +m^s−1Ki+1zi_ry_I−A⁰+m^s−1Ki+1yI∂(A⁰), where in line 2 we used the formulaz_ir+1y_I = ∂(y_I+), in line 3 we used the Leibniz rule and the formula

∂(y_I) =z_iry_I−, and in line 4 we used again the Leibniz rule and the fact thatuis a cycle.

Consider the terms from the last line in the previous display. Sinceu ∈ Z(m^bK) andb ≤ s−1, we have∂(m^s−1A) +∂(uy_I+A⁰) ⊆ ∂(m^bA). Additionally, since∂(A⁰) ⊆ m^tK1Aby the definition of A⁰, we haveuy_I+∂(A⁰)⊆m^b+tK_|u|+1y_I+Aandm^s−1Ki+1yI∂(A⁰)⊆m^s−1+tKi+2yIA. Furthermore, since zi_r =∂(yi_r), we havem^s−1Ki+1zi_ry_I−A⁰⊆m^sK_i+|yir|y_I−A⁰. Thus

vyIA⁰⊆∂(m^bA) +m^b+tK_|u|+1y_I+A+m^sK_i+|yir|y_I−A⁰+m^s−1+tKi+2yIA.

Using Proposition 2.8 and the factsb+t≥sandt≥1, we have

vyIA⁰⊆∂(m^bA) +m^sK_|u|+1y_I+A⁰+m^sK_i+|yir|y_I−A⁰+m^sKi+2yIA⁰,

with the provision that ifI=∅, the third term on the right of this inclusion is0by convention. Finally, since

|u|=i+ 1− |yi_r+1|, we conclude vy_IA⁰⊆∂(m^bA) + X

i⁰≥i+2−|y_ir+1|

A(i⁰, j+|yir+1|) +A(i+|yir|, j− |yir|) + X

i⁰≥i+1

A(i⁰+ 1, j),

with the caveat that we remove the termA(i+|yi_r|, j− |yi_r|)from the right-hand sum whenI =∅. In the display above, the second and last terms of the right-hand side are sums of the formA(p, q)withp+q > i+j and the third term is of the formA(p, q)withp+q=i+jandp > i. The Claim is thus proved.

To finish the proof of the theorem, consider an order on the setM ={(i, j)|i, j≥0}as follows: Order the elements byi+jfirst, then byias a tiebreak. In other words:

(i, j)>(i⁰, j⁰) ⇐⇒ i+j > i⁰+j⁰ or (i+j=i⁰+j⁰ and i > i⁰).

Recall thatx ∈ m^sA_c and we need to show x ∈ ∂(m^bA). Using Proposition 2.8, we know thatx ∈ Pn

i=0m^sK_iA⁰. Thus ifM₀={(i,0)|0≤i≤n}, then

x∈ X

(i,j)∈M₀

A(i, j).

Using the Claim, we see that there exists a finite setM₁and an element x1∈ X

(i,j)∈M1

A(i, j)

such thatx−x₁∈∂(m^bA), and such that the smallest element ofM₁is strictly larger (in the order described above) than the smallest element ofM₀.

Applying the Claim again, this time usingx₁, we see that there exists a finite setM₂and an elementx₂such that

x₂∈ X

(i,j)∈M₂

A(i, j)

such thatx₁−x₂ ∈ ∂(m^bA), hencex−x₂ ∈ ∂(m^bA), and such that the smallest element ofM₂ is strictly larger (in the order described above) than the smallest element ofM₁. A repeated use of the argument ensures the construction of elementsx_aand setsM_asuch that for each integerathe smallest element ofM_ais greater than the smallest element ofM_a−1 and such thatx−x_a ∈ ∂(m^bA). Whenais sufficiently large (so that p+q > cfor all(p, q) ∈M_a) one sees that, for degree reasons, we haveA(p, q) = 0for all(p, q)∈ M_a, since the elements ofA(p, q)are inAc, withcfixed. We conclude thatxa = 0forasufficiently large, hence

x∈∂(m^bA).

4. GENERATION BY ONE ELEMENT

We now turn our focus to the case where the Koszul homology algebra is generated by a single element.

Namely, here we are concerned with rings that satisfy the following condition, depending on integerstandr:

P_t,r: There exists [l] ∈ H_r(K)such that for everyz ∈ Z(m^tK) there existsz⁰ ∈ Z(m^t−1K) such that z−z⁰l∈B(m^t−1K).

Note that this condition is independent of the choice of the representativelof the class[l]∈Hr(K).

Remark 4.1. The conditionPt,ris particularly strong when the cyclelis not a minimal generator ofZr(K).

In this case, we can choosel∈mZr(K). SinceKis constructed using a minimal generating set form, we have that any cycle inZr(K)is also inmKand thus inZr(mK). Leti≥0. We have thuslz⁰ ∈mZi(m^tK)for all z⁰ ∈Zi−r(m^t−1K). The hypothesis thatPt,rholds then implies

Zi(m^tK)⊆mZi(m^tK) +Bi(m^t−1K)

for alli, and so by Nakayama’s Lemma, we haveZi(m^tK)⊆Bi(m^t−1K), hence the induced mapHi(m^tK)→ Hi(m^t−1K)is zero for alli≥0.

Whenlis part of a minimal generating set forZr(K)andris odd, we see below that a similar statement can be deduced, only that the complexKneeds to be replaced with a larger complex.

We denote byBthe following DG algebra

B= Khy|∂(y) =li.

Theorem 4.2. Let(R,m, k)be a local ring, lett≥2be an integer, andr≥1be an odd integer. IfPt,rholds, then the map

Hi(m^tB)→Hi(m^t−1B) induced by the inclusionm^t,→m^t−1is zero for alli≥0.

Proof. We will first prove two claims.

Claim 1: For a cyclez∈K, ifz=z⁰+wwithz⁰ ∈m^tK_py^(q)andw∈m^tK_p+1B, thenz⁰∈Z_p(m^tK)y^(q). Proof of Claim 1: Set|z|=c, hencep+ (r+ 1)q=c. Writez⁰ =vy^(q)withv∈m^tKp. We have

(4.2.1) 0 =∂(z) =∂(vy^(q)) +∂(w) =∂(v)y^(q)+ (−1)^pvly^(q−1)+∂(w). Sincew∈m^tKp+1Band|w|=cwe can write∂(w) =P

i,jkiy^(j)withki∈m^tKi, where the sum is taken over non-negativei, jwithi+ (r+ 1)j =c−1, andi≥p. Sincep+ (r+ 1)q=c, we must havej < qfor all suchj. Since1, y, y⁽²⁾, y⁽³⁾, . . . is a basis ofBoverK, we conclude from (4.2.1) that∂(v) = 0, and hence v∈Zp(m^tK).

Claim 2: Ifz⁰ ∈Z_p(m^tK)y^(q), then

z⁰ ∈∂(m^t−1B) +m^tKp+r+1y^(q−1).

Proof of Claim 2: Let z⁰ = vy^(q) withv ∈ Z_p(m^tK). The hypothesis of the theorem gives v−v⁰l ∈

∂(m^t−1K_p+1)for somev⁰∈Z_p−r(m^t−1K), and soz⁰−v⁰ly^(q)∈∂(m^t−1K_p+1)y^(q). An application of the Leibniz rule yields

z⁰−v⁰ly^(q)=z⁰−v⁰∂(y^(q+1)) =z⁰−(−1)^|v0|∂(v⁰y^(q+1)), and therefore

z⁰ ∈∂(m^t−1B) +∂(m^t−1Kp+1)y^(q)

⊆∂(m^t−1B) +∂(m^t−1Kp+1y^(q)) +m^t−1Kp+1∂(y^(q)), by the Leibniz rule,

⊆∂(m^t−1B) +m^tKp+1+ry^(q−1), which verifies Claim 2.

Now supposez ∈ B represents a nontrivial class in Hc(m^tB) and letpbe the largest integer such that z∈m^tKpB. Writez =Pkiy^(j), where the sum is taken over integersi, jwithi+ (r+ 1)j=candi≥p,

andk_i ∈ m^tK_i. Setz⁰ =k_py^(c−pr+1)andw =z−z⁰. Thenz =z⁰+wand the hypotheses of Claim 1 are satisfied. Putting together Claim 1 and Claim 2, we have:

(4.2.2) z∈∂(m^t−1B) +z1, for somez1∈m^tKp+1B.

Asz1is also a cycle, we may repeat the argument forz1, and so on. Inductively, we obtainz ∈∂(m^t−1B) + m^tKiBfor alli > p. SinceKi= 0fori0, we getz∈∂(m^t−1B).

We recall below a result of Levin [17, Lemma 2].

4.3. LetFbe a differential gradedR-algebra andEa differential gradedF-module which is free as a graded F-module (i. e. forgetting differentials) and such that∂(E)⊆mE. LetM,N beR-modules such thatmM ⊆ N ⊆M, and such that the canonical mapN⊗_RE →M ⊗_REis injective. If the induced homomorphism H(N⊗RF)→H(M⊗RF)is zero, then so is the induced homomorphismH(N⊗RE)→H(M ⊗RE).

Corollary 4.4. IfPt,1holds, then the following hold:

(1) The map

Tor^R_i(m^t, k)→Tor^R_i (m^t−1, k) induced by the inclusionm^t⊆m^t−1is zero for alli≥0.

(2) Ifv(R)≥t, then the algebraExtR(k, k)is generated in degrees1and2.

(3) Ift= 2, then the algebraExtR(k, k)is generated in degree1.

Proof. Ifl /∈ mZ₁(K), we can then construct a minimal Tate resolutionAofkoverRby starting with the Koszul complex K, then adjoining a variableywith∂(y) = l, and then the rest of the needed variables as described in 2.2. The description of the basis ofAin [3, Remark 6.2.1] shows that, forgetting differentials,A is free over the algebraB= Khy|∂(y) =li. Using 4.3 and Theorem 4.2, we conclude that the induced map

Hi(m^tA)→Hi(m^t−1A) is zero for alli≥0, and this yields the desired conclusion.

Ifl∈mZ₁(K), then we showed in Remark 4.1 that the induced mapH_i(m^tK)→H_i(m^t−1K)is zero for alli≥0, and the conclusion follows again by applying Levin’s result in 4.3.

In view of (1), part (2) follows then from [13] and part (3) from [22, Corollary 1].

Corollary 4.5. Let(Q,n, k)be a regular local ring, andIan ideal withI⊆n², and consider the local ring (R,m, k)defined byR=Q/I. Leth∈IrnIand letL∈K^Qsuch that∂(L) =h. Letldenote the image of LinK = K^Q⊗QR. Assume thatPt,1holds, withlas above.

IfP =Q/(h), then the induced mapTor^P_i (m^t, k)→Tor^P_i (m^t−1, k)is zero for alli≥0.

Furthermore, letabe an integer such thatI ⊆ n^a+1. If h ∈ Irn^a+2 anda+ 1 ≤ t ≤ 2a, then the canonical projectionP →Ris a Golod homomorphism.

Proof. The hypothesis implies thathis part of a minimal generating set ofI.

SetB= Khy|∂(y) =liandB⁰= K^Qhy|∂(y) =Li. Note thatB= (B⁰⊗QP)⊗PRandB⁰⊗QPis a min- imal free resolution ofkoverP. We can identify thus the induced mapTor^P_i (m^t, k)→Tor^P_i (m^t−1, k)with the induced mapHi(m^tB)→Hi(m^t−1B), and the latter is zero, by the theorem. Under the additional hypothe- ses in the last paragraph of the statement, it also follows that the induced mapTor^P_i (m^2a, k)→Tor^P_i (m^a, k) is zero for alli≥ 0, sincet ≤2aandt−1 ≥a. SinceI ⊆ n^a+1, [24, Lemma 1.4] gives that the induced mapTor^P_i (R, k)→Tor^P_i (R/m^a, k)is zero for all positivei, and then applying 3.2 to the canonical projection

κ:P →Rgives the final conclusion.

Remark 4.6. The existence of a surjective Golod homomorphism from a complete intersection ring to the local ringRis a rather remarkable property: Using a result of Levin recorded in [6, Proposition 5.18], this property allows one to conclude that the Poincar´e series of all finitely generatedR-modules are rational, sharing a common denominator.

Compressed Gorenstein artinian local rings, and, more generally, compressed level artinian local rings are defined in terms of an extremal condition involving the length, embedding dimension, and the socle of the ring.

We refer to [24] and [15] for the precise definitions. Such rings can be viewed as being “generic”, in a sense explained in more detail in [15, Theorem 3.1], for example.

Remark 4.7. Let(R,m, k)be a compressed Gorenstein local ring of socle degree s 6= 3and assumekis infinite. Sett=s+ 2−v(R)anda=v(R)−1. Whens6= 3, the inequalitiesa+ 1≤t≤2afollow from general properties of compressed Gorenstein rings, and more precisely from the fact thats = 2v(R)−1or s= 2v(R)−2, as noted in [24].

The proof of [24, Proposition 4.6] shows thatZn(m^tK)⊆lZ_n−1(m^t−1K)for some cyclel∈K1, and [24, Lemma 4.4] shows that the induced mapHi(m^tK) →Hi(m^t−1K)is zero for alli < n, wherendenotes the minimal number of generators ofm. It follows that the ringRsatisfiesPt,1. We can then constructh∈Isuch that∂(L) =h, whereLis a preimage oflinK^Q. With this data, the hypotheses of Corollary 4.5 are satisfied, hence we recover the main structural result in [24] stating thatRis a homomorphic image of a hypersurface via a Golod homomorphism.

Remark 4.8. The conditionPt,1is also satisfied for compressed level artinian local rings of socle degrees= 2t−1, withs6= 3. This can be seen from the proof of [15, Lemma 6.3] and [15, Lemma 4.4]. The conclusion that such rings are homomorphic images of a hypersurface via a Golod homomorphism is established there using the results of [24]. Corollary 4.5 recovers the same conclusion, as well.

Another class of rings for which the conditionPt,1holds is discussed in the next section.

5. STRETCHEDCOHEN-MACAULAY LOCAL RINGS

Let(R,m, k)be an artinian local ring. We set

v= rankk(m/m²); e=length(R); r= rankk(0 :m); h=e−v .

Note thatm^h+1= 0. The ringRis said to bestretchedifhis the least integerisuch thatmⁱ⁺¹= 0. Assume further thatRis not a field. ThenRis stretched if and only ifm²is principal. Sally [25] computed the series P^R_k(t)for a stretched artinian local ringRas follows:

(5.0.1) P^R_k(t) =

(1/(1−vt) ifr=v;

1/(1−vt+t²) ifr6=v.

In this section we show that finitely generated modules overRhave rational Poincar´e series as well, sharing a common denominator. The main result is Theorem 5.4; its proof involves an application of the results in Section 4.

5.1. Structure of stretched artinian local rings. Assume that(R,m, k)is a stretched artinian local ring, not a field. We further set

p=v−r and q=r−1.

Assume thath ≥3. (The caseh ≤2 has been treated in [5] and will be recalled later.) As described in [25], we choose elementst, z1, . . . , zp, w1, . . . , wq forming a minimal generating system ofmsuch that the following hold:

(1) mⁱ= (tⁱ)for alli≥2;

(2) The elementsw1, . . . , wr−1, t^hform a basis of(0 : m); in particular,twj= 0andziwj= 0for alli,j with1≤i≤pand1≤j≤q;

(3) tzi= 0for alliwith1≤i≤p;

(4) zizj=aijt^h, withaij = 0oraija unit ofR, for alli,jwith1≤i, j≤p.

(Note that there are no elementswiifr= 1and there are no elementsziifr=v.)

Ifr6= v, letaij denote the image ofaij ink=R/mand note that the matrix(aij)is invertible. Indeed, if this matrix is not invertible, then there exists an elementz =Pp

i=1bizi, withbi ∈Rsuch thatbiis a unit for at least one indexi, and such thatzzj = 0for alljfor all1≤j ≤p. This implies thatz∈(0 : m), hence z∈(w1, . . . , w_r−1, t^h), contradicting the fact thatt, z1, . . . , zp, w1, . . . , wqis a minimal generating set ofm.