1. Basic properties of the tor functor For exercises 24–26 on pages 34–35 of Atiyah-Macdonald, let us review the definition and basic properties of the Tor functor. Let M and N be A-modules and consider a free resolution of M, say d3 d2 d1 d0 · · · −−−−→ F2 −−−−→ F1 −−−−→ F0 −−−−→ M −−−−→ 0. Consider the complex d3 d2 d1 · · · −−−−→ F2 −−−−→ F1 −−−−→ F0 −−−−→ 0 obtained by deleting M. A The A-modules Torn (M, N) may be defined as the homology modules of the complex obtained by tensoring with N. Thus A (ker dn ⊗ 1) Torn (M, N) = . (im dn+1 ⊗ 1) An important fact is that one gets the same modules no matter what free (or projective) resolution one takes A ∼ A of M. It is also the case that Torn (M, N) = Torn (N,M). A basic property of the Tor functor is that if 0 −−−−→ M ′ −−−−→ M −−−−→ M ′′ −−−−→ 0 is a short exact sequence of A-modules, then tensoring this sequence over A with N induces a long exact sequence A ′′ A ′ A · · · −−−−→ Tor2 (M , N) −−−−→ Tor1 (M , N) −−−−→ Tor1 (M, N) −−−−→ A ′′ ′ ′′ Tor1 (M , N) −−−−→ M ⊗A N −−−−→ M ⊗A N −−−−→ M ⊗A N −−−−→ 0. In particular, if F is a free A-module and 0 −−−−→ K −−−−→ F −−−−→ M −−−−→ 0 A ∼ A is a short exact sequence, then we have Tori+1(M, N) = Tori (K, N) for each positive integer i and A Tor1 (M, N) measures how far tensoring with N fails to preserve exactness of the given sequence. Let I and J be ideals in a ring A. One can show that: ∼ (1) (A/I) ⊗A (A/J) = A/(I + J). A ∼ (I∩J) (2) Tor1 (A/I, A/J) = IJ . 1 Let k be a field, let x,y be indeterminates over k, and let R be either the polynomial ring k[x,y] or the formal power series ring k[[x,y]]. Let m = (x,y)R. R m m m One can calculate the modules Tori (R/ , R/ ) by considering the free resolution of R/ α β 0 −−−−→ R −−−−→ R2 −−−−→ R −−−−→ R/ m −−−−→ 0 where α(1) = (y, −x), β(1, 0) = x, and β(0, 1) = y, and then tensoring with R/ m. One sees that R m m R m m Tor1 (R/ , R/ ) is nonzero while Tori (R/ , R/ ) = 0 for i > 1. Let I = (f1, . , fn)R be an m-primary ideal in R. Assume that there exists a free resolution of I α β 0 −−−−→ Rn−1 −−−−→ Rn −−−−→ I −−−−→ 0 such that for some choice of bases for Rn−1 and Rn the map α is defined by left multiplication by a matrix M ∈ Rn×n−1. One can show that (1) the ideal I is minimally generated by n elements ⇐⇒ all the entries of M are in the maximal ideal m of R, and R m ∼ R m ∼ m (2) Tor1 (I,R/ ) = Tor2 (R/I, R/ ) = (I : )/I. (3) If I is minimally generated by n elements, then (I : m)/I =∼ (R/ m)n−1. It then follows that (I : m)/I is minimally generated by n − 1 elements. The ideal (I : m)/I of the ring R/I is called the socle of R/I. An ideal I is said to be irreducible if I is not the intersection of two properly bigger ideals of the ring. For R the polynomial ring k[x,y] or the formal power series ring k[[x,y]] and m = (x,y)R, if the ideal I is m-primary, then I is irreducible ⇐⇒ (I : m)/I is one-dimensional as a vector space over R/ m. 2.