## Algebra & Number Theory

### Local cohomology with support in generic determinantal ideals

#### Abstract

For positive integers $m≥n≥p$, we compute the $GLm×GLn$-equivariant description of the local cohomology modules of the polynomial ring $S= Sym(ℂm⊗ℂn)$ with support in the ideal of $p×p$ minors of the generic $m×n$ matrix. Our techniques allow us to explicitly compute all the modules $ExtS∙(S∕Ix¯,S)$, for $x¯$ a partition and $Ix¯$ the ideal generated by the irreducible subrepresentation of $S$ indexed by $x¯$. In particular we determine the regularity of the ideals $Ix¯$, and we deduce that the only ones admitting a linear free resolution are the powers of the ideal of maximal minors of the generic matrix, as well as the products between such powers and the maximal ideal of $S$.

#### Article information

Source
Algebra Number Theory, Volume 8, Number 5 (2014), 1231-1257.

Dates
Revised: 25 February 2014
Accepted: 26 March 2014
First available in Project Euclid: 20 December 2017

https://projecteuclid.org/euclid.ant/1513730230

Digital Object Identifier
doi:10.2140/ant.2014.8.1231

Mathematical Reviews number (MathSciNet)
MR3263142

Zentralblatt MATH identifier
1303.13018

Subjects

#### Citation

Raicu, Claudiu; Weyman, Jerzy. Local cohomology with support in generic determinantal ideals. Algebra Number Theory 8 (2014), no. 5, 1231--1257. doi:10.2140/ant.2014.8.1231. https://projecteuclid.org/euclid.ant/1513730230

#### References

• K. Akin and J. Weyman, “Primary ideals associated to the linear strands of Lascoux's resolution and syzygies of the corresponding irreducible representations of the Lie superalgebra ${\bf gl}(m\vert n)$”, J. Algebra 310:2 (2007), 461–490.
• K. Akin, D. A. Buchsbaum, and J. Weyman, “Resolutions of determinantal ideals: the submaximal minors”, Adv. Math. 39:1 (1981), 1–30.
• W. Bruns and R. Schwänzl, “The number of equations defining a determinantal variety”, Bull. London Math. Soc. 22:5 (1990), 439–445.
• W. Bruns and U. Vetter, Determinantal rings, Lecture Notes in Math. 1327, Springer, Berlin, 1988.
• C. de Concini, D. Eisenbud, and C. Procesi, “Young diagrams and determinantal varieties”, Invent. Math. 56:2 (1980), 129–165.
• D. Eisenbud, Commutative algebra: with a view toward algebraic geometry, Graduate Texts in Mathematics 150, Springer, New York, 1995.
• D. Eisenbud, The geometry of syzygies: a second course in commutative algebra and algebraic geometry, Graduate Texts in Mathematics 229, Springer, New York, 2005.
• W. Fulton and J. Harris, Representation theory: a first course, Graduate Texts in Mathematics 129, Springer, New York, 1991.
• D. R. Grayson and M. E. Stillman, “Macaulay 2: a software system for research in algebraic geometry”, 2013, http://www.math.uiuc.edu/Macaulay2.
• R. Hartshorne, Algebraic geometry, Graduate Texts in Mathematics 52, Springer, New York, 1977.
• M. Hochster and J. A. Eagon, “Cohen–Macaulay rings, invariant theory, and the generic perfection of determinantal loci”, Amer. J. Math. 93 (1971), 1020–1058.
• G. Lyubeznik, A. Singh, and U. Walther, “Local cohomology modules supported at determinantal ideals”, preprint, 2013.
• C. Peskine and L. Szpiro, “Dimension projective finie et cohomologie locale: applications à la démonstration de conjectures de M. Auslander, H. Bass et A. Grothendieck”, Inst. Hautes Études Sci. Publ. Math. 42 (1973), 47–119.
• C. Raicu, J. Weyman, and E. E. Witt, “Local cohomology with support in ideals of maximal minors and sub-maximal Pfaffians”, Adv. Math. 250 (2014), 596–610.
• J. Weyman, Cohomology of vector bundles and syzygies, Cambridge Tracts in Mathematics 149, Cambridge University Press, 2003.
• E. E. Witt, “Local cohomology with support in ideals of maximal minors”, Adv. Math. 231:3-4 (2012), 1998–2012.