References and Comments

(Not yet complete)

References for some of the lectures

Comments by O. Iyama

Some aspects of homological behaviour of finite dimensional algebras A of global dimension 2 is decided by its 3-preprojective algebra Π:=TA(Ext2A(DA,A)). Throughout let A be a finite dimensional algebra of global dimension at most 2.

Definition-Proposition: The following conditions are equivalent.
(a) Π is finite dimensional.
(b) $\tau_2:=DExt^2_A(-,A):mod A\to mod A$ is a nilpotent functor.
(c) $\tau_2^-:=Ext^2_A(DA,-):mod A\to mod A$ is a nilpotent functor.
(d) The cluster category of A is Hom-finite.
In this case we say that A is $\tau_2$-finite.

Question: How extent $\tau_2$-finiteness of A can be detected from the Cartan matrix of A?

It is known that if A is $\tau_2$-finite, then Π is Iwanaga-Gorenstein of dimension at most one. A distinguished class of $\tau_2$-finite algebras is defined as algebras A such that Π is a finite dimensional selfinjective algebra. This is equivalent to that there exists a cluster tilting A-module, and in this case we say that A is 2-representation-finite [1,2]. The following is unknown.

Question: Is the quiver of any 2-representation-finite algebra acyclic?

Let us consider some properties of Π as an A-module. It is a direct sum of $\tau_2^{-i}A$ for i≥0, so they are 2-dimensional analogue of preprojective modules for hereditary algebras. They satisfy the following vanishing property [3]:

Proposition: $\Ext^1_A(\Pi,\Pi)=0$.

We say that A is $\tau_2$-infinite if the subcatgory $add\{\tau_2^{-i}A\}_{i\ge0}$ contains infinitely many indecomposable A-modules. This class of algebras is interesting from the viewpoint of representation dimension. The following is a consequence of the above vanishing property [4].

Proposition: If A is strongly $\tau_2$-infinite, then repdim A≥ 4.

Notice that it is in general hard to show that an algebra has representation dimension at most 4. For example, the Beilinson algebra of global dimension 2 is strongly $\tau_2$-infinite. Notice that a certain distinguished class of strongly $\tau_2$-infinite algebras are recently studied by Herschend, Minamoto, Mori, Oppermann and myself, and many examples are constructed by quivers with potentials (cf. [2]).

The following general question is of interest.

Question: What is the general behaviour of the indecomposable decomposition of the A-module Π?

  1. O. Iyama, S. Oppermann, Stable categories of higher preprojective algebras, arXiv:0912.3412.
  2. M. Herschend, O. Iyama, Selfinjective quivers with potential and 2-representation-finite algebras, arXiv:1006.1917.
  3. O. Iyama, Cluster tilting for higher Auslander algebras, arXiv:0809.4897.
  4. O. Iyama, Auslander correspondence, Adv. Math. 210 (2007), no. 1, 51-82.


Comments by L.Avramov and S.Iyengar on parallel developments in commutative algebra:

By now, it seems to be an accepted meta-homological point of view that when looking for analogs of properties of artin algebras in the commutative world, one should first check out the noetherian local rings.

As you will know, for such rings finite global dimension is equivalent to regularity, so at first sight there may not be too much there.

On the other hand, there exist very precise results concerning modules of projective dimension 2.

For starters, the Hilbert-Burch Theorem describes the minimal free resolutions of cyclic modules of projective dimension 2; see Theorem 1.4.17 in Bruns and Herzog, and

MR0229634 (37 #5208) Burch, Lindsay: On ideals of finite homological dimension in local rings. Proc. Cambridge Philos. Soc. 64 1968 941--948.

This leads to Hochster's work on universal resolutions of length 2. He proves that for each sequence (b_0,b_1,b_2), admissible as a sequence of Betti numbers, there exists a "universal ring," finitely generated as an algebra over the integers, and every resolution of length 2 over any local ring is obtained by specializing some resolution defined over that universal ring. This is presented in Section 7 of:

MR0371879 (51 #8096) Hochster, Melvin: Topics in the homological theory of modules over commutative rings.Expository lectures from the CBMS Regional Conference held at the University of Nebraska, Lincoln, Neb., June 24--28, 1974. Conference Board of the Mathematical Sciences Regional Conference Series in Mathematics, No. 24. Published for the Conference Board of the Mathematical Sciences by the American Mathematical Society, Providence, R.I., 1975. vii+75

The definitive treatment was given by Tchernev, who writes down generators and relations for the universal ring and computes its main homological and arithmetical invariants. The paper is:

MR1827076 (2002c:13034) Tchernev, Alexandre B.: Universal complexes and the generic structure of free resolutions. Michigan Math. J. 49 (2001), no. 1, 65-96.

Winfiried Bruns, at Osnabrueck, is an expert on this topic. He would make for an excellent presence at the October workshop. In fact, he has contributed to this topic:

MR0709861 (85d:13021) Bruns, Winfried: Divisors on varieties of complexes. Math. Ann. 264 (1983), no. 1, 53-71.

MR0769023 (86d:13016) Bruns, Winfried: The existence of generic free resolutions and related objects. Math. Scand.55 (1984), no. 1, 33--46.

At second thought, two-dimensional regular local rings might also be of some interest to representation theorists, due to Zariski's theory of complete ideals, described in an appendix to MR0120249 (22 #11006) Zariski, Oscar; Samuel, Pierre: Commutative algebra. Vol. II. The University Series in Higher Mathematics. D. Van Nostrand Co., Inc., Princeton, N. J.-Toronto-London-New York 1960 x+414 pp.

For a modern treatment, with homological overtones, see

MR1015525 (90i:13020) Huneke, Craig: Complete ideals in two-dimensional regular local rings. Commutative algebra (Berkeley, CA, 1987), 325--338, Math. Sci. Res. Inst. Publ., 15, Springer, New York, 1989.

Yet further away, but with possible implications, is the very detailed theory of resolution of singularities in dimension 2, due to Zariski, Abhyankar, and Lipman.

Comments by David Jorgensen

On the commutative side, projective dimension (pd) is a local consideration, so let's assume that R is a commutative local (meaning also noetherian) ring. Then of course such rings of global dimension 2 are the regular local rings (RLRs) of dimension 2. I don't know what one can say in general for modules over RLRs of pd 2, but for the cyclic ones (which are really the interesting ones geometrical, as they correspond to coordinate rings of subvarieties) one has the beautiful Hilbert-Burch structure theorem for the resolutions. This theorem says that the defining ideal of the cyclic module is generated by the maximal sized minors of an (n+1)xn matrix, up to multiple by a nonzerodivisor, and this matrix is then the matrix of second syzygies in the resolution. People have been doing a lot combinatorically for resolutions of things generated by monomials, but Bruns will know a lot more about this than me.

I would say what is more interesting is modules of projective dimension 2 over non-RLRs. There are two main veins of research here that I can think is relevant.

Grothendieck's Lifting Problem:

This famous problem was an outgrowth of Serre's definition of intersection multiplicity [MR0201468 (34 #1352) Serre, Jean-Pierre Algèbre locale. Multiplicités. (French) Cours au Collège de France, 1957--1958, rédigé par Pierre Gabriel. Seconde édition, 1965. Lecture Notes in Mathematics, 11], which had the right properties for modules over unramified RLRs, but not known at the time for those over ramified RLRs. Grothendieck's Lifting Problem is this: Let Q be a local ring, x a nonzerodivisor of Q, and R=Q/(x). If M is a finitely generated R-module, then does there exist a finitely generated Q-module N such that x is a nonzerodivisor on N and N/xN=M ? Answering this would settle Serre's intersection multiplicity question for modules over ramified RLRs, as they all look like R with Q being an unramified RLR (after completion). But this lifting problem makes sense even if Q is not an unramified RLR, and whether or not x is the square of the maximal ideal of Q. Knowing an answer to the lifting problem would also have settled the famous rigidity of Tor conjecture by Peskine and Szpiro [MR0374130 (51 #10330) Peskine, C.; Szpiro, L. Dimension projective finie et cohomologie locale. Applications à la démonstration de conjectures de M. Auslander, H. Bass et A. Grothendieck. (French) Inst. Hautes Ãtudes Sci. Publ. Math. No. 42 (1973), 47--119.]. It was an effort to solve the lifting problem that led Buchsbaum and Eisenbud to their famous structure theorems for finite free resolutions [MR0340240 (49 #4995) Buchsbaum, David A.; Eisenbud, David Some structure theorems for finite free resolutions. Advances in Math. 12 (1974), 84--139.], see also, [MR0340343 (49 #5098) Buchsbaum, David A.; Eisenbud, David Lifting modules and a theorem on finite free resolutions. Ring theory (Proc. Conf., Park City, Utah, 1971), pp. 63--74.]. The point is that lifting a module from R to Q is equivalent to lifting its free resolution from R to Q. The starting point for their structure theorems, in a sense, is the Hilbert-Burch Theorem stated above (which by the way holds over any local ring, not just RLRs). I talk about some of this in my paper were I show that the impact of their structure theorem on the lifting problem is, in some sense, best possible. Here is a synopsis of what's known for the general lifting problem

  1. Any cyclic module of pd 2 lifts, by Hilbert-Burch.
  2. When x is in the square of the maximal ideal, there are easy examples of unliftable (albeit not cyclic) modules of pd 2. See my paper above.
  3. Hochster gave an example of an unliftable cyclic module exactly in the case of interest for the original Grothendieck Lifting problem [MR0412169 (54 #296) Hochster, Melvin An obstruction to lifting cyclic modules. Pacific J. Math. 61 (1975), no. 2, 457--463.] but his example was pd 6.

Related to this is the

Program of Universal Resolutions due to Hochster:

This what Hochster conjectured. [MR0371879 (51 #8096) Hochster, Melvin Topics in the homological theory of modules over commutative rings. Expository lectures from the CBMS Regional Conference held at the University of Nebraska, Lincoln, Neb., June 24--28, 1974. Conference Board of the Mathematical Sciences Regional Conference Series in Mathematics, No. 24. Published for the Conference Board of the Mathematical Sciences by the American Mathematical Society, Providence, R.I., 1975.] Given any list of Betti numbers (b_0,b_1,...,b_n), there is a unique universal resolution U over A=Z[a bunch of variables]/(some relations) such that for every finite free resolution F over R with these betti numbers, there exists a ring homomorphism A --> R such that F=U \otimes_A R. Hochster then proved this holds for n=2, which was not at all trivial. This is done in the above citation. There have been attempts to push this for n>2, but with limited success, I think. People that have worked on this are J. Weyman, and Alex Tchernev. Heitmann [MR1197425 (93m:13007) Heitmann, Raymond C. A counterexample to the rigidity conjecture for rings. Bull. Amer. Math. Soc. (N.S.) 29 (1993), no. 1, 94--97.] used such a universal resolution of pd 2 in his famous counterexample to the rigidity of Tor conjecture.

Bruns was also involved in these things back then.

Comment by A. Beligiannis

Thhere is a nice paper by Ragnar Buchweitz which may be relevant: Finite representation type and periodic Hochschild (co)homology, Contemporary Mathematics, Vol. 229 (1998), 81-109. It seems that (depending on the setting) projective, Hochschild or global dimension 2 plays a special role in Ragnar's paper.

In this connection I think that many categories of interest can be approximated or appear as full subcategories (possibly with extra structure) of abelian or module categories of global dimension 2 (the latter usually appear as categories of coherent functors; this can be interpreted as a variation of the well-known theme of Auslander and Reiten).

Comments by Paul Smith

I'm sorry to say that nothing comes to mind when I think about rings/modules of gldim/pdim =2, other than things you and/or people attending will know far better than me. But your question does make me think about the theory of complex projective algebraic surfaces (versus curves or 3-folds) and the many ways in which the theory of surfaces differs from curves. The theory of surfaces is rich in quite a different way from the ways in which the theory of curves is rich. You and others will already know this: blowing up, blowing down, the way in which that can be viewed through semi-orthogonal decomposition of D^b(S); minimal surfaces; the new perspectives obtained by re-examining the Italian classification through the lens of Mori theory. And again, the trichotomy that occurs for curves (g=0,1, >1 which also can be viewed in terms of curvature +ve, 0, -ve) extends to surfaces in terms of Kodaira dimension e.g., rational surfaces, ruled surfaces, and others. The role of the canonical divisor, etc... Maybe a reference to a survey paper about classification of surfaces--perhaps one giving the traditional (Italian) approach, the other using the ideas in Mori's MMP (minimal model program). (I think Bridgeland's thesis was about derived categories and F-M transforms for certain surfaces.) One avenue some use to introduce the MMP to grad students is to look at it in the context of toric varieties. That might be more relevant to finite dimensional algebras because K_0 is then of finite rank. And of course, one can track what the Mori program is doing in terms of how it affects the combinatorial data (the fan). I think "Toric Mori theory" is the name given to this topic.

But perhaps that is too far afield for what you want---maybe a topic for a future workshop! Nevertheless, I have often thought that the classification of surfaces and the MMP might provide new ideas for classification of finite dimensional algebras, at least when one examines what some of the ideas in classification of surfaces are saying about the derived category of surfaces. Here are a few references:


Comment by Øyvind Solberg

I am interested in Gorenstein artin algebras A with domdim A = 2 = pd D(A). These I looked at with Maurice [1], and they can be described as endomorphism rings of a module T over the endomorphism ring B of the maximal injective direct summand of A, where add T is generated by add{DTr^i(D(B))}_{i> = 0} (this additive category must be of finite type) and a module M \simeq DTr M (might be zero). The module T is a generalization of 1-cluster tilting modules (from work in progress with Osamu Iyama).

Here are two other instances that I remember (from [3]):

If I is the incidence algebra of a partially ordered set, the global dimension of I although always finite, can vary with the characteristic of K, unless gldim I <= 2 in which case it is again characteristic independent (see [2, 4]).

More generally one can show the following [3]:

Theorem 5.1. Let Q be a finite quiver and let E and F be two fields. Let r_1 ,..., r_n be linear combinations of paths in Q, where the coefficients are (+/-)1, and let I_E (I_F respectively) denote the two-sided ideal of EQ (FQ respectively) generated by the elements r_1,..., r_n. Assume that I_E and I_F are both admissible ideals, and, that gldim EQ/I_E <= 2. Then gldim FQ/I_F < infinity.

  1. Auslander, M., Solberg, Ø., Gorenstein algebras and algebras with dominant dimension at least 2, Comm. in Alg., 21(11), 3897-3934.
  2. Cibils, C., Cohomology of incidence algebras and simplicial complexes, J. Pure Appl. Algebra 56 (1989), no. 3, 221-232.
  3. Green, E. L., Solberg, Ø., Zacharia, D., Minimal projective resolutions, Trans. Amer. Math. Soc., 353 (2001), 2915-2939.
  4. Igusa, K., Zacharia, D., On the cohomology of incidence algebras of partially ordered sets, Comm. Algebra 18 (1990), no. 3, 873-887.


Comment by Martin Herschend

The only thing I can think of at the moment is my joint paper with Osamu:

Selfinjective quivers with potential and 2-representation-finite algebras  in arXiv: 1006.1917

In this paper we show that every cut in a selfinjective quiver with potential (QP) is algebraic meaning that the truncated Jacobian algebra (i.e. where the arrows in the cut have been removed) has global dimension at most two and the arrows in the cut correspond to minimal relations of the path algebra. So given the examples of selfinjective QPs in our paper one can construct a lot of algebras of global dimension at most two by cutting. In fact, they are even 2-representation finite.

In general I think that the question of when a truncated Jacobian algebra has global dimension at most two is interesting.

Comment by D.Eisenbud

Cyclic modules of projective dimension 2 over a reasonable ring are the same as suitably nondegenerate matrices of size n x n+1 for some n; "suitably nondegenerate" means that the ideal of n x n minors has depth 2 (the maximum possible.) This is the "Hilbert-Burch" theorem, exposed for example in my book on commutative algebra.

For non-cyclic modules Buchsbaum and I discovered a "structure theorem" that, as Hochster subsequently showed, provides a universal free resolution of length two, and thus a universal module of projective dimension 2 with fixed betti numbers. In principal this is a complete description's not nearly as useful as the Hilbert-Burch theorem because (unlike the space of n x n+1 matrices) the base space is not smooth. Thus a central problem would be to classify all the "suitably non-degenerate" maps from this universal space to smooth varieties. Still more interesting and more mysterious, one would like to understand all the graded homomorphisms to polynomial rings. Very little is known (though the recent developments around Boij-Soederberg theory provide some insight into what might be possible.)

Comments by K.Igusa and G.Todorov