Let (A,m) be an abstract complete intersection and let P a prime ideal of A. In [1] Avramov proved that AP is intersection. this article, we give elementary proof t...

We establish two criteria for certain local algebras to be complete intersections. These criteria play an important role in A. Wiles’s proof that all semi-stable elliptic curves over Q are modular. Introduction In this paper we discuss two results in commutative algebra that are used in A. Wiles’s proof that all semi-stable elliptic curves over Q are modular [11]. We first fix some notation tha...

We investigate whether surfaces that are complete intersections of quadrics and intersection in the Segre embedded product P1×Pk↪P2k+1 can belong to same Hilbert scheme. For k=2 there is a classical example; it comes from K3 projective 5-space degenerate into hypersurface on threefold. show for k≥3 only one more example. It turns out its (connected) scheme has at least two irreducible component...

let $r=k[x_1,x_2,cdots, x_n]$ be a polynomial ring over a field $k$. we prove that for any positive integers $m, n$, $text{reg}(i^mj^nk)leq mtext{reg}(i)+ntext{reg}(j)+text{reg}(k)$ if $i, j, ksubseteq r$ are three monomial complete intersections ($i$, $j$, $k$ are not necessarily proper ideals of the polynomial ring $r$), and $i, j$ are of the form $(x_{i_1}^{a_1}, x_{i_2}^{a_2}, cdots, x_{i_l...