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Over a Cohen-Macaulay local ring, the minimal number of generators of a maximal Cohen-Macaulay module is bounded above by its multiplicity. In 1984 Ulrich [Math. Z. 188 (1984), pp. 23–32] asked whether there always exist modules for which equality holds; such modules are known nowadays as Ulrich modules. We answer this question in the negative by constructing families of two dimensional Cohen-Macaulay local rings that have no Ulrich modules. Some of these examples are Gorenstein normal domains; others are even complete intersection domains, though not normal.
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Let be a standard graded algebra over a field. We investigate how the singularities of or affect the -vector of , which is the coefficient of the numerator of its Hilbert series. The most concrete consequence of our work asserts that if satisfies Serre’s condition and has reasonable singularities (Du Bois on the punctured spectrum or -pure), then , …, . Furthermore the multiplicity of is at least . We also prove that equality in many cases forces to be Cohen-Macaulay. The main technical tools are sharp bounds on regularity of certain modules, which can be viewed as Kodaira-type vanishing statements for Du Bois and -pure singularities. Many corollaries are deduced, for instance that nice singularities of small codimension must be Cohen-Macaulay. Our results build on and extend previous work by de Fernex-Ein, Eisenbud-Goto, Huneke-Smith, Murai-Terai and others.more » « less
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Let be a Noetherian local ring of dimension . We prove that if , then the classical Lech’s inequality can be improved uniformly for all -primary ideals, that is, there exists such that for all -primary ideals . This answers a question raised by Huneke, Ma, Quy, and Smirnov [Adv. Math. 372 (2020), pp. 107296, 33]. We also obtain partial results towards improvements of Lech’s inequality when we fix the number of generators of .more » « less
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A Noetherian local ring ( R , m ) (R,\frak {m}) is called Buchsbaum if the difference ℓ ( R / q ) − e ( q , R ) \ell (R/\mathfrak {q})-e(\mathfrak {q}, R) , where q \mathfrak {q} is an ideal generated by a system of parameters, is a constant independent of q \mathfrak {q} . In this article, we study the tight closure analog of this condition. We prove that in an unmixed excellent local ring ( R , m ) (R,\frak {m}) of prime characteristic p > 0 p>0 and dimension at least one, the difference e ( q , R ) − ℓ ( R / q ∗ ) e(\mathfrak {q}, R)-\ell (R/\mathfrak {q}^*) is independent of q \mathfrak {q} if and only if the parameter test ideal τ p a r ( R ) \tau _{\mathrm {par}}(R) contains m \frak {m} . We also provide a characterization of this condition via derived category which is analogous to Schenzel’s criterion for Buchsbaum rings.more » « less
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Abstract We establish the Minimal Model Program for arithmetic threefolds whose residue characteristics are greater than five. In doing this, we generalize the theory of global $$F$$ F -regularity to mixed characteristic and identify certain stable sections of adjoint line bundles. Finally, by passing to graded rings, we generalize a special case of Fujita’s conjecture to mixed characteristic.more » « less
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Abstract This paper extends the results of Boij, Eisenbud, Erman, Schreyer and Söderberg on the structure of Betti cones of finitely generated graded modules and finite free complexes over polynomial rings, to all finitely generated graded rings admitting linear Noether normalizations. The key new input is the existence of lim Ulrich sequences of graded modules over such rings.more » « less
