A counterexample to the small Cohen–Macaulay module conjecture

The explanation

Start with a test that uses only multiplication. The paper builds one curved surface for each prime n of 3 or more. Each surface comes with two whole numbers. The first is 5n³. The second is n³(n² − 10). The paper's test asks whether the second number is at most six times the first.

Try n = 3. The numbers are 135 and −27, and −27 is far below 6 × 135 = 810. The test passes. Try n = 5. The numbers are 625 and 1,875, and 1,875 is below 3,750. It passes again. Now try n = 7. The numbers are 1,715 and 13,377, while six times 1,715 is 10,290. The test fails, and that failure is the whole counterexample.

To see why, some words are needed. A ring is a number system where you can add and multiply. A local ring zooms in on a single point of a shape, the way power series zoom in on zero. Complete means infinite power series are allowed. A domain is a ring where two nonzero things never multiply to zero. A module over a ring is like a vector space whose scalars come from that ring.

Depth measures how well a module behaves. Pick elements of the ring one by one, so that multiplying by each one never sends a nonzero part of the module to zero. Depth is the longest such run. A small Cohen–Macaulay module is a nonzero module built from finitely many generators whose depth equals the dimension of the ring. The domain form of the conjecture asks whether every complete local domain has oneSource 1.

Melvin Hochster raised this question in his early workSource 2. His first paper on it appeared in 1973. In dimension one or two the answer is yesSource 3. If infinitely many generators are allowed, so-called big Cohen–Macaulay modules exist, by work running from Hochster in 1975 to André in 2018. In 2014, Bhargav Bhatt found rings with no finite Cohen–Macaulay extension ringSource 4. That ruled out one route, but not general modules. Hochster later wrote that he had come to expect counterexamplesSource 5.

This paper claims a counterexample in dimension threeSource 6. The ring R is built from a cone over a surface. Take all polynomial-like functions on the surface, graded by degree, and zoom in on the tip of the cone. The paper claims R has no nonzero finite module of depth three. It also gives an uncompleted version, essentially defined by finitely many polynomial equations, with the same propertySource 7.

The engine is an inequalitySource 8. If the cone ring over a surface has a small Cohen–Macaulay module, the surface must pass the test. The surfaces come from six lines through four points of the plane, followed by a branched coveringSource 9. For n = 7 the test fails, so the cone ring has no such moduleSource 10.

The one obstacle is that a module over the completed ring need not respect degreesSource 11. Earlier inequalities of this kind worked only for graded modules or special sheaves on the surface. The paper keeps the module's actual geometry near the tip instead. It compares two ways of extending that geometry across the surface sitting at the tip. Then Bogomolov's classical inequality supplies the upper bound that clashes with the testSource 12.

The work also overlaps with two recent preprints. One excludes graded modules over the same rings. Another, by Chen, proposes a similar completed counterexample by a different bound. The paper says its existence claim overlaps with that workSource 13.

How sure is this? A computer proof checker called Lean has checked only part of it. Lean confirms that a ring with the stated basic properties exists. Lean does not check that the ring lacks a small Cohen–Macaulay module. So the main claim rests on the paper's own argument. No specialist review of the paper has been published.

What it doesn't show. This does not rule out the infinite (big) Cohen–Macaulay modules, which are known to exist.