Commutative algebra: Tight closure, Frobenius closure, and
Hilbert-Kunz multiplicity theories over Noetherian rings of prime
characteristic p;
and general primary decomposition theories over Noetherian rings.
Publications and preprints
Uniform test exponents in rings of finite F-representation type, preprint
— .pdf file.
A weak embedding theorem (and test exponents) for modules of
finite phantom projective dimension (with Melvin Hochster), work in progress
— work in progress.
Splitting of ^{e}M, strong F-regularity, and the existence of small Cohen-Macaulay modules
(with Melvin Hochster), work in progress
— first draft.
An embedding theorem for modules of finite projective dimension
(with Melvin Hochster), preprint
— .pdf file.
Second coefficients of Hilbert-Kunz functions for domains
(with Melvin Hochster), preprint
— .pdf file.
F-rational signature and drops in the Hilbert-Kunz
multiplicity (with Melvin Hochster), submitted
— .pdf file.
Global F-spltting ratio of modules
(with Alessandro De Stefani and Thomas Polstra),
submitted.
— .pdf file.
Global Frobenious Betti numbers and Frobenius Euler characteristics
(with Alessandro De Stefani and Thomas Polstra),
to appear in Mich. Math. J..
— .pdf file.
Globalizing F-invariants
(with Alessandro De Stefani and Thomas Polstra),
accepted by Adv. in Math. 350 (2019), 359–395.
—
available on arXiv (arXiv:1608.08580).
Lech's inequality, the Stückrad-Vogel conjecture, and uniform behavior of Koszul homology
(with Patricia Klein, Linquan Ma, Pham Hung Quy and Ilya Smirnov),
Adv. in Math. 347 (2019), 442–472.
—
available on arXiv (arXiv:1808.01051).
Tensoring with the Frobenius endomorphism
(with Olgur Celikbas and Arash Sadeghi),
Homology, Homotopy and Applications 20 (2018) no. 1, 251–257.
—
available on arXiv (arXiv:1706.00238).
Generalizing Serre's Splitting Theorem and Bass's Cancellation Theorem via free-basic elements
(with Alessandro De Stefani and Thomas Polstra),
Proc. Amer. Math. Soc. 146 (2018) no. 4, 1417–1430.
—
available on arXiv (arXiv:1608.08591).
On the Frobenius complexity of determinantal rings
(with Florian Enescu), J. Pure Appl. Algebra, 222 (2018) no. 2, 414–432.
— .pdf file. Also available on arXiv (arXiv:1503.03164).
Some extensions of Hilbert-Kunz multiplicity
(with Neil Epstein), Collect. Math. 68 (2017) no. 1, 69–85.
— .pdf file.
The Frobenius complexity of a local ring of prime characteristic
(with Florian Enescu), J. Algebra, 459 (2016), 133–156.
— .pdf file.
Also available on arXiv (arXiv:1401.0234).
Local cohomology with support in a parameter ideal
(with Melvin Hochster),
J. Pure Appl. Algebra 217 (2013) no. 4, 664–671.
— .pdf file.
Test exponents for modules with finite phantom projective dimension
(with Melvin Hochster),
Illinois J. Math. 56 (2012) no. 4, 1095–1107.
— .pdf file.
Criteria for flatness and injectivity
(with Neil Epstein),
Math. Z. 271 (2012) no. 3, 1193–1210.
— .pdf file.
The compatibility, independence, and linear growth properties,
book volume `Progress in Commutative Algebra: Ring Theory, Homology, and Decompositions' by de Gruyter.
— .pdf file.
Homological invariants of modules over contracting endomorphisms
(with Luchezar L. Avramov, Melvin Hochster and Srikanth B. Iyengar),
Math. Ann. 353 (2012) no. 2, 275–291.
— .pdf file.
The lower semicontinuity of the Frobenius splitting numbers
(with Florian Enescu),
Math. Proc. Cambridge Philos. Soc. 150 (2011) no. 1, 35–46.
— .pdf file.
The direct sum decomposability of ^{e}M in dimension 2,
Michigan Math. J., Special volume in honor of Melvin Hochster, 57 (2008), 745–755.
— .pdf file.
Frobenius test exponents for parameter ideals in generalized Cohen-Macaulay local rings
(with C. Huneke, M. Katzman, and R. Y. Sharp),
J. Algebra 305 (2006), 516–539.
— .pdf file.
Observations on the F-signature of local rings of characteristic p,
J. Algebra 299 (2006), 198–218.
— .pdf file.
Primary decomposition II: primary components and linear
growth, J. Pure Appl. Algebra 205 (2006) no. 1, 226–242.
— .pdf file.
Modules with finite F-representation type,
J. London Math. Soc. (2) 72 (2005), no. 1, 53–72.
— .pdf file.
Primary decomposition: compatibility, independence and linear growth,
Proc. Amer. Math. Soc. 130 (2002) no. 6, 1629–1637.
— .pdf file.
Unmixed local rings with minimal Hilbert-Kunz multiplicity are regular
(with Craig Huneke),
Proc. Amer. Math. Soc. 130 (2002) no. 3, 661–665.
— .pdf file.