Math 711: Lecture of November 22, 2006
Our next objective is to exhibit linear maximal Cohen-Macaulay modules for rings
dened by the vanishing of the minors of a matrix of indeterminates in two special cases:
one is the case of maximal minors, and the oth
Math 711: Lecture of December 1, 2006
Recall the if R is a Noetherian ring of prime characteristic p > 0, R is called F-nite if
F : R R makes R into a module-nite algebra over
F (R) = cfw_rp : r R,
a subring of R that is also denoted Rp . When R is F -nit
Math 711: Lecture of November 29, 2006
We aim to prove the following result of Paul Monsky, following his paper [P. Monsky,
The Hilbert-Kunz function, Mathematische Annalen 263 (1983) 4349].
Theorem (Monsky). Let (R, m, K) be local where R has prime chara
Math 711: Lecture of November 20, 2006
The following result can be deduced easily from the Buchsbaum-Eisenbud acyclicity
criterion, but we give a short, self-contained argument.
Proposition. Let (R, m, K) be a local ring and N a maximal Cohen-Macaulay mod
Math 711: Lecture of December 6, 2006
We can now analyze all graded maximal Cohen-Macaulay modules for a Veronese subring of the polynomial ring in two variables, and show, as a corollary, that there is a unique
indecomposable linear maximal Cohen-Macaula
Math 711: Lecture of December 4, 2006
The next two results suggest that characteristic p techniques may be helpful in proving
the existence of linear maximal Cohen-Macaulay modules.
Let R be a ring of prime characteristic p > 0. The Frobenius closure I F
Math 711: Lecture of November 27, 2006
Before giving a second proof of the existence of linear maximal Cohen-Macaulay modules
for K[X]/I2 (X) and the extension of this result to the case of more general Segre products,
we want to note that the Segre produ
Math 711: Lecture of November 15, 2006
We want to establish that in the twisted tensor product of two Zd -graded K-algebras,
C K C , one has that if u C and v C are forms of degree 1, then
(u 1 + 1 v)d = ud 1 + 1 v d ,
a property reminiscent of the behavi
Math 711: Lecture of November 13, 2006
We are going to use certain matrix factorizations to construct linear maximal CohenMacaulay modules over hypersurfaces.
Discussion: Cohen-Macaulay modules over hypersurfaces R/f R of nite projective dimension over R.
Math 711: Lecture of November 10, 2006
Before proceeding further with our study of Lechs conjecture, we want to discuss a
result known as the associativity of multiplicities whose proof uses Lechs theorem on
computing multiplicities using ideals generated
Math 711: Lecture of November 17, 2006
Our next objective is to give a matrix factorization of a generic form over Z instead of
Z. The idea is to replace the ring Z by a ring of matrices over Z.
Discussion: block form. Let R be any associative ring wi