Dr Moty Katzman
School of Mathematical and Physical Sciences
Reader in Pure Mathematics
m.katzman@sheffield.ac.uk
J16, Hicks Building
Full contact details
Dr Moty Katzman
School of Mathematical and Physical Sciences
J16
Hicks Building
Hounsfield Road
Sheffield
S3 7RH
School of Mathematical and Physical Sciences
J16
Hicks Building
Hounsfield Road
Sheffield
S3 7RH
- Research interests
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Dr Katzman's research is in the area of commutative algebra. Specifically, he is interested in the following.
Characteristic p methods
Certain theorems in algebra can be proved by showing that they hold in positive characteristic, and in characteristic p one has extra structure given by the Frobenius map x↦xp. There are several tools, notably tight closure, which exploit this extra structure to prove some remarkable theorems.
Local chomology modules
This modules derive their importance partly from the fact that they detect interesting properties of modules over commutative rings (e.g., depth.) Unfortunately, these objects tend to be very big are rather mysterious. It is very difficult to describe them in any detail even in seemingly easy cases. Dr. Katzman has recently been producing both examples showing that these objects are more complicated than previously conjectured but also instances where they can be understood fairly well.
Combinatorial aspects
One of the simplest family of modules imaginable are monomial ideals in polynomial rings and, perhaps surprisingly, these objects have a very rich structure, in some sense richer than the structure of graphs. Dr Katzman has recently been studying certain monomial ideals associated with graphs a discovering some surprising connections between the algebraic and combinatorial properties of these objects.
- Publications
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Show: Featured publications All publications
Featured publications
Journal articles
- Lyubeznik numbers, F-modules and modules of generalized fractions. Transactions of the American Mathematical Society, 375, 6621-6651. View this article in WRRO
- The Support of Local Cohomology Modules. International Mathematics Research Notices, 2018(23), 7137-7155.
- D-module and F-module length of local cohomology modules. Transactions of the American Mathematical Society, 370, 8551-8580. View this article in WRRO
- Global parameter test ideals. Journal of Algebra.
- An extension of a theorem of Hartshorne. Proceedings of the American Mathematical Society , 144(3), 955-962. View this article in WRRO
- Rings of Frobenius operators. Mathematical Proceedings of the Cambridge Philosophical Society, 157(1), 151-167.
All publications
Journal articles
- Lyubeznik numbers, F-modules and modules of generalized fractions. Transactions of the American Mathematical Society, 375, 6621-6651. View this article in WRRO
- A data-driven approach for predicting printability in metal additive manufacturing processes. Journal of Intelligent Manufacturing, 31(7), 1769-1781. View this article in WRRO
- The TestIdeals package for Macaulay2. Journal of Software for Algebra and Geometry, 9(2), 89-110. View this article in WRRO
- Multiplicity bounds in prime characteristic. Communications in Algebra, 47(6), 2450-2456. View this article in WRRO
- Strong F-regularity and generating morphisms of local cohomology modules. Journal of Algebra, 525, 19-41. View this article in WRRO
- FFRT properties of hypersurfaces and their F-signatures. Journal of Algebra and its Applications, 18(11). View this article in WRRO
- The Support of Local Cohomology Modules. International Mathematics Research Notices, 2018(23), 7137-7155.
- D-module and F-module length of local cohomology modules. Transactions of the American Mathematical Society, 370, 8551-8580. View this article in WRRO
- Global parameter test ideals. Journal of Algebra.
- An extension of a theorem of Hartshorne. Proceedings of the American Mathematical Society , 144(3), 955-962. View this article in WRRO
- An algorithm for producing F-pure ideals. Archiv der Mathematik, 103(5), 421-433. View this article in WRRO
- Rings of Frobenius operators. Mathematical Proceedings of the Cambridge Philosophical Society, 157(1), 151-167.
- Castelnuovo-Mumford regularity and the discreteness of F-jumping coefficients in graded rings. Transactions of the American Mathematical Society, 7(366), 3519-3533.
- Annihilators of Artinian modules compatible with a Frobenius map. Journal of Symbolic Computation, 60, 29-46.
- An algorithm for computing compatibly Frobenius split subvarieties. Journal of Symbolic Computation, 47(8), 996-1008.
- Two interesting examples of D-modules in characteristic p>0. Bulletin of the London Mathematical Society, 44(6), 1116-1122.
- An upper bound on the number of F-jumping coefficients of a principal ideal. Proceedings of the American Mathematical Society, 139(12), 4193-4197.
- Some properties and applications of $F$-finite $F$-modules.
- Frobenius maps on injective hulls and their applications to tight closure. J LOND MATH SOC, 81, 589-607.
- A NON-FINITELY GENERATED ALGEBRA OF FROBENIUS MAPS. P AM MATH SOC, 138(7), 2381-2383.
- On the discreteness and rationality of F-jumping coefficients. J ALGEBRA, 322(9), 3238-3247.
- Parameter-test-ideals of Cohen-Macaulay rings. COMPOS MATH, 144(4), 933-948.
- On ideals of minors of matrices with indeterminate entries. COMMUN ALGEBRA, 36(1), 104-111.
- Frobenius test exponents for parameter ideals in generalized Cohen-Macaulay local rings. J ALGEBRA, 305(1), 516-539.
- Characteristic-independence of Betti numbers of graph ideals. J COMB THEORY A, 113(3), 435-454.
- On the arithmetic of tight closure. J AM MATH SOC, 19(3), 659-672.
- Uniform behaviour of the Frobenius closures of ideals generated by regular sequences. J ALGEBRA, 295(1), 231-246.
- The Hilbert series of algebras of the Veronese type. COMMUN ALGEBRA, 33(4), 1141-1146.
- Counting monomials. J ALGEBR COMB, 22(3), 331-341.
- Some properties of top graded local cohomology modules. J ALGEBRA, 259(2), 599-612.
- An example of an infinite set of associated primes of a local cohomology module. J ALGEBRA, 252(1), 161-166.
- Associated primes of graded components of local cohomology modules. T AM MATH SOC, 354(11), 4261-4283.
- Bipartite graphs whose edge algebras are complete intersections. Journal of Algebra, 220 (1999) pp. 519-530.
- Finiteness of $bold{bigcup_e Ass F^e(M)}$ and its connections to
tight closure. Illinois Journal of Mathematics 40 (1996) pp. 330-337.
- Weak F-Regularity Deforms in Q-Gorenstein Rings. Journal of Algebra, 204(1), 281-285.
- The Complexity of Frobenius Powers of Ideals. Journal of Algebra, 203(1), 211-225.
- The Cohomological Dimension of a Resultant Variety in Prime Characteristicp. Journal of Algebra, 191(2), 510-517.
- Finite criteria for weak $mathrm{F}$-regularity. Illinois Journal of Mathematics, 40(3).
Conference proceedings
- The support of top graded local cohomology modules. Commutative Algebra: Geometric, Homological, Combinatorial and Computational Aspects, Vol. 244 (pp 165-174)
Preprints
- Lyubeznik numbers, $F$-modules and modules of generalized fractions, arXiv.
- The TestIdeals package for Macaulay2, arXiv.
- Strong $F$-regularity and generating morphisms of local cohomology modules, arXiv.
- FFRT Properties of hypersurfaces and their F-signature, arXiv.
- Global parameter test ideals, arXiv.
- An extension of a theorem of Hartshorne, arXiv.
- An algorithm for producing F-pure ideals, arXiv.
- Rings of Frobenius operators, arXiv.
- An algorithm for computing compatibly Frobenius split subvarieties, arXiv.
- Two interesting examples of $mathcal{D}$-modules in characteristic $p>0$, arXiv.
- An upper bound on the number of F-jumping coefficients of a principal ideal, arXiv.
- Lyubeznik numbers, F-modules and modules of generalized fractions. Transactions of the American Mathematical Society, 375, 6621-6651. View this article in WRRO
- Research group
- Grants
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Past grants, as Principal Investigator
Common threads in the theories of Local Cohomology, D-modules and Tight Closure and their interactions EPSRC Prime characteristic methods in commutative algebra EPSRC Graded components of local cohomology modules EPSRC Past grants, as Coinvestigator
Tailorable Adaptive Connected Digital Additive Manufacturing (TACDAM)
- Teaching activities
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MAS346 Groups and Symmetry MAS348 Game Theory