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The Ziegler spectrum of the D-infinity plane singularity

We will describe the Cohen--Macaulay part of the Ziegler spectrum of the D-infinity plane singularity S and calculate the nilpotency index of the radical of the category of finitely generated Cohen--Macaulay S-modules.

To classify points of the space ZCM S , i.e. indecomposable pure injective Cohen-Macaulay modules, we will employ the so-called 'interval method' which is based on Ziegler's results [23].
Namely each point in this space is uniquely determined by a nontrivial cut it creates on a given interval in the lattice of positive primitive formulae. Taking intervals means to take open sets covering the category of finitely generated Cohen-Macaulay modules considered as a surface.
We will show that two basic open sets are sufficient to detect all points of this space and calculate their bases of open sets in Ziegler topology.
Due to many technicalities involved in proofs, we will be quite concise when executing them in detail. Say, the description of irreducible morphisms between finitely generated Cohen-Macaulay modules over S is known to experts in this area, but no written account seems to exist. So we rely on 'believe or check' approach when explaining this matter. Further, because the model theoretic part of this paper is similar to the A ∞ -case, which was carefully explained in [15], we will allow themselves to be quite sketchy on some instances.
The point of this paper is to propel the method rather than calculations. It proved to be successful for some classes of finite dimensional algebras and some Cohen-Macaulay singularities.
The reason for this is obvious: both finite dimensional modules, and finitely generated Cohen-Macaulay modules over complete Cohen-Macaulay rings are pure injective.

The ring
In this part we will introduce the object of our interest, the ring S, and mention its basic properties which will be used in the paper. Let F be an algebraically closed field whose characteristic is different from 2, and let S = F [[x, y]]/(x 2 y) be the factor of the ring of power series F [[x, y]] by the ideal generated by x 2 y, the so-called D ∞ plane singularity. Many properties of S can be extracted from fairly general commutative algebra.
In particular, S is a local complete 1-dimensional commutative noetherian ring with zero divisors. Namely if P is a prime ideal of S then x 2 y = 0 yields x ∈ P or y ∈ P (or both). It follows easily that P = xS, P = yS or P = m = (x, y), the maximal ideal of S. The ring theoretical properties of R and its modules can be found in [15], -we will use them freely. For instance the quotient ring Q R of R is a uniform module, and the integral closure R of R in Q R is a non-noetherian valuation ring of Krull dimension one. Further R can be identified with R + xQ R .
Because S is 1-dimensional, it suffices to invert any nonzero divisor, say x + y, to obtain its ring of quotients Q S , for instance Q S = S[(x + y) −1 ]. Namely T = (x + y) −k , k = 1, 2, . . . is a multiplicative closed set, and the localization ST −1 has only two prime ideals xST −1 and yST −1 , both consisting of zero divisors. It follows that each nonzero divisor in this ring is invertible, hence ST −1 = Q S . For basic facts in the theory of Cohen-Macaulay rings and modules see [11] or [22]. We will use the homological definition of this notion. Because S is 1-dimensional, a finitely generated Comparing this with the injective resolution 0 → S → Q S → Q S /S → 0 (see Matlis [12, Thm. 13.1]) we conclude that Q S ∼ = F ((x)) ⊕ Q R and Q S /S ∼ = I(S/m), the minimal injective cogenerator in the category of S-modules.
By S we will denote the integral closure of S in Q S . If u = x(x + y) −1 then u 2 = u 3 , hence u ∈ S, and e = u 2 is an idempotent in S. Now we can see the above decomposition of Q S more clearly.
Finally yQ S ∼ = Q R by sending y to 1.
We will consider Q S as the universal domain which contains all objects of our interest. In particular we identify R S with yS ⊂ Q S ; and Q R , considered as an S-module, with yQ S .
Recall that an overring of S is any subring of the integral closure S containing S. Since the ideal m 2 requires 3 generators, it follows (see [11,Thm. 4.18]) that S is not Bass, i.e. there is an overring of S which is not Gorenstein. We will pinpoint such an overring.
There is the least proper overring S ′ = End S (m) of S generated over S by z = x 2 (x + y) −1 . Further S ′ is a local non-Gorenstein ring which coincides with the endomorphism ring of S ′ considered as an S-module.
Proof. Because S is Gorenstein, it follows (this fact is called the rejection lemma in Kiev school) that S has the least proper overring S ′ . It can be identified with the endomorphism ring of m: For instance multiplication by z defines the endomorphism of m which sends x to x 2 and y to zero.
In fact (see [11,Exam. 14.23]) S ′ is isomorphic to the ring obtained from F [[x, y, z]] by imposing relations z 2 = zx = x 2 and yz = 0, the so-called E ∞ -singularity. Clearly S ′ is a local ring whose Jacobson radical is generated by x, y, z.
Further S ′ is no longer Gorenstein. Namely x + y is a nonzero divisor in S ′ , and S ′ /(x + y)S ′ is isomorphic to the ring obtained from F [[x, z]] by imposing relations z 2 = zx = x 2 = 0. This ring has a 2-dimensional socle, hence is not Gorenstein, therefore the same holds true for S ′ .
Since the image of each f ∈ End S (S ′ ) is uniquely determined by f (1) ∈ S ′ it follows that End S (S ′ ) coincides with S ′ , in particular S ′ S is an indecomposable module.
Next we will describe the integral closure of S. Recall that u = x(x + y) −1 ∈ S and e = u 2 = Proof. Clearly xyQ S ⊆ S, because this ideal is nilpotent, therefore S[u] + xyQ S ⊆ S.
To prove the converse inclusion note that u = x(x + y) −1 ∈ S implies x k (x + y) −k ∈ S; and similarly y k (x + y) −k ∈ S. Further clearly x(x + y) −2 / ∈ S. By multiplying this by e we conclude that x k (x + y) −l is not in S as soon as 1 ≤ k < l, and the same holds true for y k (x + y) −l .
It remains to show that each nonzero element From this we obtain the following decomposition of S S .
Proof. Since exy = 0 and eu = e, by Lemma 2.3 we conclude that e S = eS. Also eS = x 2 (x + y) −2 S is isomorphic to x 2 S as an S-module.
we obtain (1 − e) S = z ′ S + xyQ S . It is easily checked that the map z ′ → 1 extends uniquely to an isomorphism from this submodule onto R S .
It will be easier to deal with another copy of R within Q S . Namely is its easily shown that the submodule yS + xyQ S is isomorphic to R S via the map y → 1. We will identify R S with this submodule.
We extend the above definition of finitely generated Cohen-Macaulay modules to arbitrary modules. Namely an S-module M is said to be Cohen-Macaulay if Hom(F, M ) = 0, i.e. if Soc(M ) = 0.
Clearly CM-modules over S form a definable class (category), i.e. (see [5] or [9]) this class is closed with respect to direct products, direct limits and pure submodules. It is defined by the first order sentence ∀ x (vx = 0 ∧ vy = 0 → v = 0), where v denotes a variable; therefore we obtain the theory T CM whose models are CM-modules.
Recall (see [13,Sect. 3.3.4]) that there is a natural one-to-one correspondence between definable categories of modules over any ring U and first order theories of U -modules whose models are closed with respect to direct products. We will use these terms as synonymous.

Finitely generated CM-modules
In this section we recall the classification of indecomposable finitely generated CM-modules over S. It can be combined from [22,20,11], for instance we will use notation of [22, p. 78]. In particular each indecomposable finitely generated CM-module is isomorphic to either an ideal of S, or a submodule of S 2 . We will mention this realization, but prefer to define such modules using generators and relations.
2). For each k ≥ 1 the module M k is given by generators m, n and relations mx = ny k , nx = 0; for instance here is a diagram for M 2 .
It is easily seen that the map m → y k+1 and n → xy gives an isomorphism from M k onto the ideal (y k+1 , xy) of S. Further M is annihilated by x 2 , hence is a CM-module over R.
Also M k is isomorphic to the submodule of Q S generated by m = y and n = xy(x + y) −k .
3). For each k ≥ 1 the module Y k is given by generators m, n and relations mx = ny k , nxy = 0; here is a diagram for Y 2 .
For instance Y k is isomorphic to the ideal (y k , x) of S via the map m → y k and n → x, in The module X k , k ≥ 1 is generated by m, n with relations mxy = ny k and nx = 0; here is a diagram for X 1 .
For instance X 1 is isomorphic to S ′ S via the map m → 1 and n → x − z. Further X k is isomorphic to the submodule of S 2 generated by m = (y k , x) and n = (xy, 0); and can be identified with the submodule of Q S generated by m = 1 and n = xy(x + y) −k . 5). The module N k , k ≥ 1 is generated by m, n with relations mxy = ny k+1 and nxy = 0; here is the diagram for N 1 .
For instance N k is isomorphic to the submodule of S 2 generated by m = (y k , x) and n = (x, 0).
The following is the AR-quiver of the category of finitely generated CM-modules over S extrapolated from Yoshino [22, p. 78] and [20, p. 25], except of the precise form of irreducible maps.
The irreducible morphisms on this diagram resemble irreducible morphisms between string modules (see [4]) given by graph maps: first factor, and then embed. They could be guessed and checked. Here is a complete list.
1). Y k → M k is given by factoring out nx.

4)
. M k → X k is given by m → my and n → n. 5). X k → N k is provided by m → m and n → ny.

7)
. N k → Y k is provided by gluing mx with ny k . 8). N k → X k+1 is given by annihilating nx.
Here we show by the black square the element of the first module which is send to zero by this morphism. 9). B → Y 1 sends y → m, hence is the inclusion yS ⊂ m. 10). Y 1 → A is given by n → x 2 and m → 0. If we identify Y 1 with m by setting m = y and n = x, then this map is given by multiplication by z.

11
). X 1 → B is given by gluing mx and n, hence by factoring out z in S ′ = X 1 .
12). S → X 1 sends 1 → m, hence is the inclusion S ⊂ S ′ . 13). Y 1 → S sends m → y and n → x, hence is the inclusion m ⊂ S. 14). A → X 1 is given by sending x 2 → mx − n.
15). C → D is the inclusion and D → C is given by multiplication by y.
On Figure 1 we show the unbridged version of the AR-quiver, which includes the relations between irreducible morphisms. We mark the finitely generated CM-modules over R by solid circles, -they comprise roughly a quarter of the category of finitely generated CM-modules over

S.
For instance there are the following almost split sequences in the category of finitely generated CM-modules.
Symmetrically we obtain the following AR-sequences.
Note that S is projective and Ext-injective in the category of finitely generated CM-modules, hence no almost split sequence starts or ends in S. Also neither C nor D is a source or sink of an AR-sequence in this category.
Thus the AR-quiver consists of 2 components: the top one consisting of modules C and D, and the bottom one. To explain why C or D are located as shown on Figure 1, note that the inverse limit (i.e. the intersection) along the coray of morphisms going through the Y k in the northwestern direction equals xS = D. This is why D is positioned at the end of this coray.
Similarly the composite map M k → X k → M k−1 is the inclusion of ideals (y k+1 , xy) ⊂ (y k , xy), therefore C = xyS is the inverse limit of the coray of morphisms going through the M k in the northwestern direction.
Note (see [6,Thm. 21.21] for a general statement) that there is a duality of the category of finitely generated CM-modules over S given by the functor Hom(−, S). It is easily checked that this duality interchanges X k with Y k , but fixes the M k , N k and A, B, C, D, S.

Some basics in model theory
In this section we will recall few notions from the model theory of modules which are relevant to this paper. We will use Prest's book [13] as a main source for references.
The implication relation defines a partial ordering ≤ on the set of pp-formulas. We say that the pp-formulae ϕ and ψ are equivalent if both implications ϕ → ψ and ψ → ϕ hold. If we identify equivalent pp-formulas, the resulting poset is a modular lattice L called the lattice of pp-formulae (in one variable) over U . For instance the zero formula v = 0 is the least element in this lattice, and the trivial formula v = v is the largest element of L. Further the meet in this lattice is the conjunction of pp-formulae, and the join is the sum of pp-formulae: 1) ϕ implies ϕ in the theory of all S-modules.
2) ϕ is equivalent to ϕ in the theory of CM-modules over S.
We will call the formulae of the form ϕ CM the CM-formulae and use pointed modules m ∈ M to denote them. Thus the lattice L CM can be described as follows. It consists of equivalence classes of pointed finitely generated CM-modules over S, where (M, m) ≤ (N, n) if there exists a pointed morphism from the latter module to the former. For instance the formula x 2 ∈ S implies x ∈ S, but these formulae are not equivalent.
Further, the sum of CM-formulas m ∈ M and n ∈ N is given by the pointed module (m, n) ∈ M ⊕ N , and their conjunction is defined using the following pushout.

The first interval
In this section we will create the first cut along the category CM f g of finitely generated CM-modules over S, i.e. we calculate a particular interval in the lattice of CM-formulae, which later will be used as a net to catch indecomposable pure injective modules. For this we use Because (see Section 3) we know indecomposable finitely generated CM-modules, the calculation of patterns is straightforward. Here is an important example.
Proposition 5.1. The diagram on Figure 2 shows the pattern of the pointed module (S, x 2 ). Proof.
To calculate this pattern we should look at the image of x 2 with respect to morphisms f from S to indecomposable finitely generated CM-modules M . Further we will identify morphism f and g such that f (x 2 ) and g(x 2 ) are free realizations of equivalent pp-formulas. For example if f (x 2 ) = 0 then f represent the zero element in the pattern. Also the irreducible morphism S → X 1 which sends 1 → m shows that (S, x 2 ) covers (X 1 , mx 2 ) in the pattern.
Note that there is an almost split morphism f : X 1 → N 1 ⊕ B, and the image of mx 2 via this map is (mx 2 , 0); hence f is equivalent to the irreducible morphism X 1 → N given by the first component of f . It follows that mx 2 ∈ N 1 is the largest formula strictly below mx 2 ∈ X 1 . The rest is by inspection.
We have already noticed that Figure 2 represents the frame of the interval below the formula x 2 | v in the lattice of CM-formulae. We will recover the whole structure of this interval. First we will prove an auxiliary lemma.
Proof. Recall that a module is said to be uniserial if its lattice of submodules is a chain. By [14,Prop. 4.4] it suffices to prove that the formula x 2 | v, when evaluated on an indecomposable finitely generated CM-module M over S, defines a submodule which is uniserial as a module over the endomorphism ring of M . For instance, there is nothing to prove for M k , since The most difficult case is N k , -let V = End(N k ). Note that there exists an endomorphism of this module sending m → n, n → 0; and there is an endomorphism such that m → my − n, n → mx. Using this we conclude that nonzero endosubmodules of N k contained in N k x 2 form the following chain: V mx 2 ⊃ V nx 2 ⊃ V mx 3 ⊃ . . . . Now we are in a position to describe the lattice of CM-formulae in the above interval. Since every finitely generated CM-module is a direct sum of indecomposable modules, it follows that each formula in this interval is equivalent to a sum ϕ 1 + · · · + ϕ n of formulae with free realization given by modules on Figure 2. Thus to get the interval [v = 0, x 2 | v] from the pattern we need only insert additional formulas (A, x k+1 ) + (S, x k ), k ≥ 3 above each summand and strictly below the formula (Y 1 , nx k−1 ). For instance we will obtain the following diagram below (Y 1 , nx 2 ), where the top interval is simple.
It corresponds to the projection Y 1 → A ⊕ S onto the first two coordinates of the left almost It also follows that the conjunction of pp-formulas in this interval coincides with the intersection in the pattern. Thus the pattern consists of +-irreducible elements of the interval, and is a lattice by itself, but the sums in these lattices may be different.

The Ziegler spectrum
In this section we classify points of the Cohen-Macaulay part of the Ziegler spectrum of S. A module M is said to be pure injective if it is injective with respect to pure embeddings.
For instance each injective module is pure injective, so as (see [18]) any module which is linearly compact over its endomorphism ring. Since U is commutative, each linearly compact module is pure injective. Definable classes (categories) D of U -modules can be identified with closed subsets of the Ziegler spectrum whose points are indecomposable pure injective modules in D, and the topology is induced from the whole spectrum. This way we obtain the Ziegler spectrum of this category.
For instance, by considering the category of Cohen-Macaulay S-modules we will obtain the Cohen-Macaulay part of the Ziegler spectrum of S, written ZCM S . To investigate this topological space is the main objective of this paper.
First we create a sufficient supply of points in this space.
Lemma 6.1. Each indecomposable finitely generated CM-module over S is pure injective, hence is the point of ZCM S . Further the same holds true for Q R , R, and the generic modules G y = F ((y)), G x = F ((x)).
Proof. It is not difficult to check that each module in the above list is linearly compact, which implies pure injectivity. For instance for R-modules in this list, hence for B, C, M k and Q R , R, G y , this follows from [15].
We need to add one more module to this list. If we identify the module X k with the submodule of Q S generated by 1 and n k = xy(x + y) −k , k = 1, . . . , then these modules form the natural ray of inclusions, via n k → n k+1 y, going in the northeastern direction on Figure 1. The direct limit along this ray will give us the CM-module N , which is the submodule of Q S generated by 1 and the n k .
• n 0 =xy Lemma 6.2. N is indecomposable and pure injective as an S-module, hence is a point in ZCM S .
Proof. Let N ′ be the submodule of N generated by the n k , hence N ′ = xyQ S . Clearly N ′ is isomorphic to G y , hence uniserial. Further N ′ is a linearly compact module, so as the factor module N/N ′ ∼ = D = xS. It follow that N is linearly compact, hence pure injective.
It is straightforward to check that the endomorphism of N S is N itself, which is a local ring.
Thus N S is an indecomposable module, hence is a point in ZCM S .
The following property is similar to what is known for finite dimensional algebras (see [13, Cor. 5.3.37]). Lemma 6.3. Finitely generated points are dense in the CM-part of the Ziegler spectrum of S.
Proof. Clearly each CM-module over S is a direct limit of its finitely generated submodules, which are Cohen-Macaulay, hence the result follows.
To This module is called the pure injective envelope of p, written PE(p).
We say that a pp-type p is indecomposable if its pure injective envelope is an indecomposable module. There is an important syntactical criterion which recognizes the indecomposability of a type. By p − we will denote the complement of p, i.e. the collection of pp-formulae which are not in p. For an indecomposable pp-type this collections is almost a cofilter.

.4]) A pp-type
Suppose that ϕ and ψ are pp-formulae such that ϕ does not imply ψ, hence we have the nontrivial interval [ψ ∧ ϕ, ϕ] in the lattice of pp-formulae. Suppose that p is a pp-type such that ϕ ∈ p and ψ ∈ p − . Then p defines, by restriction, a cut on this interval, whose upper part (i.e. the formulae in p from this interval) is a filter. The following fact shows that (like analytic functions) the pure injective envelope of p can be recovered form any cut, whatever small it be.  For instance, the irreducibility criterion from Fact 6.4 can be used locally. Namely suppose that p is predefined by a filter in a given interval, and is indecomposable in there, i.e. satisfies the above conditions with the ϕ i and ϕ within the interval. Then p can be extended to an indecomposable pp-type, and all such extensions (whatever different) lead to isomorphic indecomposable pure injective modules.
All this relativizes to any definable class of modules, with obvious changes. From this we derive the general approach to a classification of CM-modules over S. First choose a 'nice' interval whose lattice structure is known. Then using Ziegler's criterion describe indecomposable pptypes in this interval (or rather cuts of such types). Finally find indecomposable pure injective modules which realize these pp-types locally. This will complete the classification. Now we are in a position to classify points of ZCM S . Theorem 6.6. Let M be an indecomposable pure injective Cohen-Macaulay module over S.
Then either it is finitely generated, or M is isomorphic to one of modules Q R , R, N , G y = F ((y)) and G x = F ((x)).
Proof. If M is annihilated by x 2 , then it is pure injective indecomposable CM-module over R. It follows from [15] that M is one of the finitely generated modules B, C, M k ; or one of the modules Q R , R or G y .
Otherwise there exists a nonzero m ∈ M x 2 . Thus if p = pp M (m) then x 2 | v ∈ p and v = 0 ∈ p − , hence this type (therefore the module) is visible within the interval on Figure 2.
Furthermore to apply the Ziegler criterion for indecomposability (see Fact 6.4) note that this interval is a distributive lattice. Thus a pp-type p within this interval is indecomposable iff p − is a cofilter, i.e. ϕ 1 + ϕ 2 ∈ p − for all pp-formulas ϕ i in the negative part of p.
It easily follows that any cut on the frame gives an indecomposable pp-type. The principal cut whose upper part is generated by one pp-formula is realized in the corresponding indecomposable finitely generated CM-module. For instance the cut whose positive part is generated by What remains is to identify the modules which are produced by the infinitely generated cuts p i , i ≥ 1, and by q (see Figure 2). The latter case is easy to resolve. Namely the element 1 ∈ G x = F ((x)) is divisible by any power of x, hence defines the required cut. Thus PE(q) ∼ = G x .
Now it is easily checked that the element x 2 ∈ N realizes the pp-type whose restriction to our interval is p 1 ; furthermore x k , k > 2 realizes in N the pp-type, whose restriction coincides with p k−1 . Thus we have filled in all vacant positions for points.
Recall that a pair (ϕ/ψ) of pp-formulae is said to be minimal in a theory T if ψ < ϕ, and there is no pp-formula strictly between ψ and ϕ in the lattice of pp-formulas relativized to T . From Fact 6.5 it is easy to construct a basis of open sets for each point visible on Figure 2.
For instance we immediately obtain the following.
Lemma 6.7. Each finitely generated point which occurs on Figure 2, but D, is isolated by a minimal pair in the theory T CM of CM-modules over S.
For example X 1 is isolated by the pair mx 2 ∈ X 1 over mx 2 ∈ N 1 , which corresponds to the irreducible map X 1 → N 1 .
Now we proceed with calculating the Cantor-Bendixson rank of points. The Cantor-Bendixson analysis of a closed subset T of the Ziegler spectrum runs as follows. The first derivative T ′ is obtained from T by deleting isolated points. Those points are assigned the CB-rank 0. Further, T ′′ is obtained from T ′ by deleting its isolated points, which are assigned the CB-rank 1, and so on. If T ′′′ is an empty set, then we say that the CB-rank of T equals 2.
There is a parallel analysis of the corresponding definable category which leads to the notion of the Krull-Gabriel dimension (see [9,Ch. 7] We apply this analysis to the theory T CM of CM-modules over S. We will see below that each isolated point in T CM is isolated by the minimal pair. It follows that the lattice of pp-formulae L 1 of T ′ CM is obtained from the lattice L of CM-formulae over S by collapsing intervals of finite length.
All this is applicable to any given interval, say, to the one given by Figure 2. We conclude CM is the following chain of order type 1 + ω * .
Now we are in a position to calculate the rank of D. Proof. From Figure 2 we conclude that the basis of open sets for D in the Ziegler topology is given by pairs of formulae x 3 ∈ D over nx k ∈ Y k , k = 1, 2, . . . , in particular D is not isolated.
It remains to look at the above diagram.
To include into consideration the module N we need one more definition. A point M in a closed set T is said to be neg-isolated (relative to T ), if there is a nonzero m ∈ M such that the negative part of the pp-type of m contains the largest formula. By [13,Prop. 5.3.46] this notion does not depend on the choice of m. Lemma 6.9. N is isolated in T ′ CM by the minimal pair x 2 ∈ S over x 3 ∈ D, in particular N has CB-rank 1. Further N is neg-isolated in T CM .
Proof. The basis of open sets for N is given by pairs mx k ∈ X k , k = 1, 2, . . . over x 3 ∈ D, in particular this point is neg-isolated and not isolated. It remains to look at the above diagram.
Thus the only remaining point in the above interval is G x = PE(q). Lemma 6. 10. G x has Cantor-Bendixson rank 2.
Proof. From Figure 2 we see that the basis of open sets for G x is given by pairs x k ∈ S, k = 1, 2, . . . over 0, hence q is a pp-type maximal over 0 (usually called critical). It easily follows that G x is not isolated on levels zero and one of the CB-analysis.
By applying the m-dimension analysis to the interval from the above diagram, we obtain the two-point interval. It follows that G x is the only point in the interval x 2 ∈ S over 0 in T ′′ CM .

The second interval
To complete the description of topology of ZCM S we need to calculate the bases of open sets for points from this space which are not visible on Figure 2, hence for indecomposable pure injective CM-modules over R. Here is a complete list of these points.
Furthermore (from [15]) we know that the CB-ranks of these points in the relative topology of ZCM R are 0, 1 and 2, but we have recalculate these ranks in the ambient space ZCM S . For this we construct another interval in the lattice L of CM-formulae, which captures points in question.
Proposition 7.1. The diagram on Figure 3 represents the pattern of the pointed module (C, xy).
Proof. By inspection. A very similar proof (with a similar resulting diagram) is given in [15,Prop. 4.3].
. . Note that the top pair xy ∈ C is a free realization of the formula vx = 0. It follows that the interval [v = 0, vx = 0] in L is generated by pp-formulae on the figure with respect to free sums, for instance it is distributive. This diagram allows us to calculate the ranks of remaining points.
Lemma 7.2. Each finitely generated point M k is isolated in ZCM S by a minimal pair, and the same holds true for B.
Proof. Directly from the diagram, because the M k and B are sources of almost split sequences in the category of finitely generated CM-modules over S.
For instance, the module M 1 is isolated by the pair n ∈ M 1 over (X 1 , n) + (Y 2 , ny) which corresponds to the left almost split map It also follows that none of the remaining point is isolated in T CM = ZCM S , in particular each isolated point is isolated by a minimal pair. For instance, the lattice L 1 of pp-formulae in T ′ CM coincides with the lattice obtained from the lattice L of CM-formulae by collapsing intervals of finite length. In particular this holds true for each interval.
Lemma 7.3. The interval xy ∈ C over zero in L 1 is the following chain of order type 1+Z+ω * .
Proof. By direct calculations. For instance the interval n ∈ X 1 over ny ∈ N 1 includes only one additional formula (B, xy) + (N 1 , ny), hence has length 2, and therefore becomes trivial in L 1 .
Recall that C and R have CB-rank 1 in ZCM R , therefore their ranks in ZCM S cannot be smaller. To our surprise these ranks are preserved in the ambient space. Proof. From Figure 3 we see that the basis of open sets for C is given by pairs xy ∈ C over (D, xy) + (M k , n). By evaluating it is easily seen that any such pair is open on M k+1 , therefore C is not isolated.
On the other hand in L 1 the interval (C, xy) over (D, xy) is simple, hence isolates C.

Now we proceed with R.
Lemma 7.5. R is isolated in T ′ CM by the minimal pair (M 1 , m) over (X 1 , n), in particular CB( R) = 1. Further R is neg-isolated.
Proof. From Figure 3 we see that the basis for the pointed module ( R, xy) is given by the ray heading from (M 1 , n) southeastern, hence by the pairs ny k−1 ∈ M k over n ∈ X 1 . In particular the latter formula neg-isolates the pp-type of xy in R. Since each such pair opens M k+1 , this point is not isolated. It remains to look at the above diagram.
It remains to consider the points Q R and G y whose CB-rank is at least 2. From the above analysis it follows that the lattice of pp-formulae of T ′′ CM coincides with the lattice L 2 obtained from L 1 by collapsing the finite length intervals. From the above diagram we conclude the the interval xy ∈ C over zero in L 2 is the following chain of length 2.
The simple intervals in this chain will catch the remaining points of ZCM S .

Auslander-Reiten sequences
It follows from the above analysis that almost split sequences we mentioned in Section 3 preserve their defining properties in the category of all (finitely generated or not) CM-modules.
In this section we will produce few more AR-sequences that will be used in the next section to glue a surface from the category of finitely generated CM-modules over S. Though no direct reference is possible, the ideology goes back to Auslander [1] who constructed almost split sequences in the category of all U -modules (over any ring U ) targeting a finitely presented module with a local endomorphism ring. As follows from Herzog's interpretation of this result (see [7]), the source of this AR-sequence is an indecomposable pure injective module which is neg-isolated.
This was how we guessed almost split sequences in the category of CM-modules over S with infinitely generated terms. Namely there are few indecomposable infinitely generated pure injective CM-modules which are neg-isolated, hence few candidates for sources of AR-sequences.
It was difficult to guess, but it is surprisingly easy to check the required properties.
Recall that N is identified with the submodule of Q S generated by 1 and the n k = xy(x+y) −k .
Further N ′ is the submodule of N generated by the n k , hence N ′ = xyQ S . Then the factor N/N ′ is isomorphic to D = xS via the map which sends 1 to x. We will also identify R with the submodule yS + N ′ of Q S . In particular Rx = Sxy and R/ Rx ∼ = yS/xyS, which is isomorphic With this identification in mind we will produce the first AR-sequence with infinitely generated terms.
Lemma 8.1. The following is an almost split sequence in the category of CM-modules over S.
Here ı 1 denotes the inclusion R = yS + N ′ ⊂ N , and π 1 is the above epimorphism R → R/ Rx ∼ = C which sends y ∈ R to xy ∈ C, hence is given by multiplication by x.
Further π ′ 1 is a composition of the epimorphism N → N/N ′ with the isomorphism N/N ′ ∼ = D = xS such that 1 → x, hence is given by multiplication by x; and ı ′ 1 is the inclusion C ⊂ D.
Proof. It is not difficult to check that this sequence is exact. For instance uf (y) = u(y, xy) = xy − xy = 0. Further the cokernel of f is obtained by identifying y ∈ N with −xy ∈ C, hence kicks off N ′ , and the resulting module is isomorphic to N/N ′ = D.
Since the endomorphism ring of the target D is local, it follows from [1,Prop. 4.4] that it suffices to check that f is left almost split. Suppose that K is a CM-module over S, and g : R → K is a morphism which is not a split monomorphism. We need to find h : N ⊕ C → K completing the following diagram.
y y s s s s K Note that f (xy) = (xy, 0) belongs to N .
Assume first that g(xy) = 0 and let q be the pp-type of g(xy) in K. Since g is not a split monomorphism, it increases the pp-type p of xy ∈ R, which (see Figure 3) is neg-isolated by the formula n ∈ X 1 , hence this formula is in q. Since q is a filter, by taking conjunctions we obtain that each formula in the northeastern ray starting from n ∈ X 1 belongs to p. We conclude that q includes each formula in the pp-type of xy ∈ N . It follows that there exist a morphism h : N → K (considered as a morphism N ⊕ C → K) such that g(xy) = hf (xy).
Thus by considering g ′ = g − hf : R → N ⊕ K we may assume that g(xy) = 0. Since K is Cohen-Macaulay, it follows that g(xyQ S ) = 0, therefore g factors through R/xyQ S ∼ = C, hence through π 1 .
By swapping C with D, and R with N , we obtain a 'dual' almost split sequence.
Lemma 8.2. The following is an almost split sequence in the category of CM-modules over S.
Here ı 2 : N → R and ı ′ 2 : D → C are given by multiplication by y. The epimorphisms π 1 and π ′ 1 are defines in Lemma 8.1.
Proof. We will give only the beginning of the proof. The remaining part is similar to the previous lemma.
Because uf (1) = u(y, x) = xy − xy = 0, the composite map is zero. Further, because y ∈ R is identified with −x ∈ D, we conclude that xy 2 = 0 in the cokernel. Because this element equals xy(x + y), we derive xy = 0, hence the submodule xyQ S of R is annihilated.
By projecting the above almost split sequences we produce the following irreducible maps. 2) The monomorphisms R ⊂ N and C ⊂ D.

The Ringel quilt
In this section we will add indecomposable infinitely generated pure injective CM-modules over S to the AR-quiver of the category CM f g of finitely generated CM-modules to create the Ringel quilt of this category. The term is borrowed from Ringel [19], where it is called the Auslander-Reiten quilt.
We start with the unbridged version of the AR-quiver of the category CM f g , -see Figure 1.
Now we compactify (add the boundary to) this diagram taking into account the direct limits of rays, and inverse limits of corays -see Figure 4.
Namely if we identify M k with the submodule of Q S generated by y and n k = xy(x + y) −k that the composite morphism M k → M k+1 on the diagram is the inclusion. Further the pointed module xy ∈ R is the direct limit of this ray, -we add this pointed module to its end.
Going along the parallel ray (X 1 , n) → (N 1 , ny) → . . . in the northeastern direction, we obtain pointed module (N, xy) as its direct limit. Add this module to the end of this this ray and attach the irreducible map R ⊂ N of these pointed modules. Now repeat this procedure with all parallel northeastern rays, for instance add irreducible maps N y − → R.
. . Further attach the point 1 ∈ G y to the end of the ray (C, xy) → (D, xy) → (C, xy 2 ) → . . . , because this pointed module is the corresponding direct limit. On the other hand we attach the same module to the the beginning of the coray · · · → ( R, xy) → (N, xy), because the pp-type of 1 ∈ G y is the same as the pp-type of the inverse limit of pointed modules in this coray. Now add the pointed module xy ∈ Q R to the end of the ray ( R, xy) → (N, xy) → ( R, xy 2 ) → . . . heading in southwestern direction. We will also connect R with the module C by the irreducible map R x − → C; and connect the next module N in the ray to the next module D by These four modules form the following commutative square whose edges are irreducible maps.   Note that the sides of these squares change the direction on Figure 4 -from going southeastern for the top maps to heading southwestern for bottom morphisms.
Gluing the boarder lines of this surface using just discussed irreducible morphisms we obtain the Möbius stripe. For instance (see Figure 5) starting from S we may go through X 1 → There is one misleading point in this description. The initial choice of the point D to enter (after N ) the second component of the AR-quiver was arbitrary, hence in general we will not get the closed path. Furthermore, the module G x is missing from our description. It seems that a better way to give the global picture to the category CM f g is shown on Figure 6. Thus this category is the quotient of the vertical strip by the action given by the glide reflection shifting upwards.
This way we could add the point G x to the top of this diagram, because it is the direct limit of all rays heading upwards. We will also mark on this figure the points of ZCM S included into the basic open set (x 2 | v/v = 0) corresponding to the first interval (see Section 5).

The infinite radical
In this section we will calculate the nilpotency index of the radical of the category CM f g of finitely generated CM-modules over S. Recall that the radical, rad, of this category is defined to consist of all morphism f : M → N between modules in this category such that for each Gx .
indecomposable K ∈ CM f g , any composite map K → M f − → N → K is not invertible. In follows easily that each irreducible morphism is in the radical, in fact rad is generated by irreducible morphisms.
The least ordinal λ such that rad λ = 0 (if exists) is called the nilpotency index of the radical, nil(rad). If it is defined, each morphism f in CM f g is assigned its nilpotency index, nil(f ), as the least µ such that f ∈ rad µ \ rad µ+1 . Thus the nilpotency index of the radical is the supremum of indices of morphisms between indecomposable modules.
Here is he main result of this section.
Theorem 10.1. The nilpotency index of the radical of the category of finitely generated Cohen-Macaulay S-modules equals ω · 2.
One implication in this theorem is easy. Namely the 'one revolution' map f : S x − → S is in the infinite radical, hence its power f k+1 is a nonzero map in rad ω+k , as desired.
To prove the converse let us make the following remark. Suppose that f : M → N be a morphism between indecomposable modules in CM f g which belongs to the infinite radical. If It follows that if f ∈ rad ω·2 then for each m ∈ M its image f (m) is divisible by any power of x. Looking at the Figure 2 we conclude that f (m) = 0, hence f = 0. From this the result easily follows.